Truth: An Introduction (1)

.

Truth: An Introduction (1).

On the Structure of Reality.

-1-

.

-2-

.

ISBN 978 - 1 - 8479 - 9120 - 1

2007, Robert-Jan Milleker

-3-

.

-4-

.

To My Teachers

Also see associated Web Site:

http://www.geocities.com/robert.milleker

-5-

.

-6-

Section Index Introduction

11

Section 1

19

Introduction Unary Languages Section 2

22

Basic Propositions Section 3 Negatively Conjunctive Propositions Negatively Conjunctive Unary Relations Complementary Propositions Complementary Unary Relations

33

Section 4 Candidate World Identifying Unary Relation Propositions’ Normalisation Basic Syllogism

40

Section 5 Basic Disjunctively Forbidden Proposition Basic Disjunctively Necessary Proposition Basic Disjunctively Forbidden Unary Relation Basic Disjunctively Necessary Unary Relation Basic Conjunctively Forbidden Proposition Basic Conjunctively Necessary Proposition Basic Conjunctively Forbidden Unary Relation Basic Conjunctively Necessary Unary Relation Law of the Excluded Middle Proposition Law of the Contradictory Proposition Law of the Excluded Middle Unary Relation Law of the Contradictory Unary Relation

64

Section 6 Disjunctive Propositions Conjunctive Propositions Disjunctive Unary Relations Conjunctive Unary Relations

84

-7-

Section 7 Disjoint Propositions Joint Propositions Disjoint Unary Relations Joint Unary Relations

88

Section 8 Disjunctively Conditional Propositions Conjunctively Conditional Propositions Disjunctively Conditional Unary Relations Conjunctively Conditional Unary Relations

122

Section 9 Analytic Propositions Synthetic Propositions Analytic Unary Relations Synthetic Unary Relations Primary Propositions Secondary Propositions Primary Unary Relations Secondary Unary Relations A Priori Propositions A Posteriori Propositions A Priori Unary Relations A Posteriori Unary Relations

157

Section 10

182

Deterministic World Attributes Section 11 Dirac Propositions Dirac Unary Relations Reduced Dirac Propositions Particular Element Identifyig Unary Relations Particular World Identifyig Unary Relations Basic Composed Propositions Extended Composed Propositions Basic Composed Unary Relations Extended Composed Unary Relations Basic Composed, Primary Unary Relations Extended Composed, Primary Unary Relations

-8-

199

P.

224 Probabilistic Languages

D.

240 Unary Languages’ Dependencies

A.1.

293

Axiomatisation Unary Languages A.2.

525

Axiomatisation Unary Languages’ Dependencies A.3.

640

Axiomatisation Probabilistic Languages I.1.

670

Index Laws Unary Languages I.2.

688

Index Laws Unary Languages’ Dependencies I.3.

697

Index Laws Probabilistic Languages

-9-

.

- 10 -

Introduction

i.1.1 All knowledge will be expressed in a language. i.1.2 A language will be defined with reference to a set of propositions and a set of relations. i.1.3 The set of a language’s propositions will also be called the language’s vocabulary. i.1.4 The subset of a language’s vocabulary the members of which are those and only those members of the set that are true will be called the set of the language’s elements, or the language’s narrative. i.1.5 A language’s vocabulary will circumscribe the totality of that which is expressible in terms of the language. i.1.6 The members of a language’s vocabulary will also be called well formed formulae, or candidate states of affairs. i.1.7 The members of a language’s narrative will also be called theorems, or states of affairs, or events. i.1.8 The members of the set of a language’s relations will also be called functions, or predicates, or attributes, or Universals, or qualities. i.2.1 A suitably irreducible subset of the set of a language’s relations will be called an atomic set of relations, its members, atomic relations or atoms with respect to the subset.

- 11 -

i.2.2 For any language there may be different irreducible subsets of the set of the language’s relations, that is, for any language there may be different sets of atoms. i.2.3 All members of the set of a language’s relations that are not atoms with respect to an irreducible subset of rls will be called divisible, or composed relations with respect to the subset. i.2.4 A language’s vocabulary will be defined with reference to a set of the language’s atoms. i.3.1 A language will define a set of elements. i.3.2 A world will be a set the members of which are the world’s elements. i.3.3 There will be a subset of the set of elements defined by a language the members of which are all those and only those members of the set of elements defined by the language that are characterised by the fact that they contain elements, or that they constitute the empty world. i.3.4 A language will define a set of worlds. i.3.5 A language will constitute a world. i.3.6 There will be a member of the set of worlds defined by a language constituting a world of candidate elements. i.3.7 The member of the set of worlds defined by a language constituting a world of candidate elements will contain all elements defined by the language. All members of all worlds defined by a language will be members of the world of candidate elements defined by the language. i.3.8 The member of the set of worlds defined by a language constituting a world of candidate elements will also be called a world of conceivable elements. i.3.9 Any member of the set of worlds defined by a language will be characterised in its entirety by the membership of the members of the world of candidate elements defined by the language that are its members.

- 12 -

i.4.1 Any member of the world of candidate elements defined by a language and any member of any world defined by the language - will be characterised in its entirety by the subset of the set of the language’s attributes the members of which are those members of the set that pertain to it. i.4.2 There will not be a thing in itself. i.4.3 Attributes defining an element’s material composition will not be fundamentally different from the element’s other attributes. i.4.4 There will not be a fundamental distinction of the positions known, respectively, as ‘materialism’ and ‘idealism’. i.4.5 Any two members of the world of candidate elements defined by a language will be related by a relation of identity in the case and only in the case that they agree in all of their attributes. Any two members of the world of candidate elements defined by a language will be related by a relation of difference in the case and only in the case that there is at least one attribute with respect to which they disagree. i.4.6 The members of pairs of members of the world of candidate elements defined by a language related by a relation of identity will be called the same element, the members of pairs of members of the world of candidate elements defined by the language related by a relation of difference, different elements. i.4.7 Relations of identity and difference of members of the world of candidate elements defined by a language will be established with reference to relations of identity and difference of sets of attributes. i.4.8 Any member of the world of candidate elements defined by a language and any member of the set constituting an epiphenomenon with respect to the former will constitute the same element. i.4.9 The subset of the set of a language’s attributes the members of which pertain to a given member of the world of candidate elements defined by the language will be called the element’s form, or the element’s context. i.4.10 A member of the set of a language’s Universals will be called a contextual element.

- 13 -

i.4.11 Any member of the world of candidate elements defined by a language will be characterised in its entirety by its form, or context. i.4.12 Any member of the world of candidate elements defined by a language will obtain meaning with reference to its form. i.4.13 Establishment of the meaning of a member of the world of candidate elements defined by a language will entail distinguishing it from all other elements defined by the language. i.4.14 Establishment of attributes pertaining to a member of the world of candidate elements defined by a language will progressively narrow the element’s meaning, thereby defining it. i.5.1 A world will be defined by a world underlying the world. i.5.2 A world underlying a world will be called a language. i.5.3 A world will be defined by a language. i.5.4 A world will be defined in a hirarchy of worlds. i.5.5 A language will be defined in a hirarchy of languages. i.5.6 Plato’s cave will be a member of a hirarchy of caves, the outside of any given cave being the inside of an encompassing cave. i.5.7 The world underlying a world will be called the world’s grammar. i.5.8 The language underlying a language will be called the language’s grammar. i.5.9 The language underlying a world will also be called the world’s interpretation, or the world’s justification.

- 14 -

i.5.10 The language underlying a given world will also be called a physics with respect to the world, the language underlying the language a metaphysics, or a metaphysics of order ’one’ with respect to the world. i.5.11 The members of the set of elements of a language defining a given world will also be called laws of physics, the members of the set of elements of the language underlying the language, laws of metaphysics with respect to the world. i.5.12 The language constituting an interpretation of the language constituting a given world’s physics will be the world’s metaphysics. i.5.13 A language defining a world will be called a logic of order ‘zero’ with respect to the world, or a metaphysics of order ‘zero’ with respect to the world. i.5.14 A language constituting a world’s physics will constitute a metaphysics of order ‘zero’ with respect to the world. i.5.15 A language defining a language constituting a logic of order ‘n’ with respect to a world will be called a logic of order ‘n+1’ with respect to the world, or a metaphysics of order ’n+1’ with respect to the world. i.5.16 The sets of rules governing the scientific method and the method of deduction with respect to a world defined by a language will be subsets of the set of elements of the language defining the language, that is, they will be part of the world’s metaphysics. This will answer the set of beliefs defining the position known as logical positivism. i.6.1 A world cannot establish its own interpretation. i.6.2 A world cannot affirm or deny its own existence. Descartes’s cogito fails. i.6.3 Members of the set of a language’s Universals constituting attributes regarding elements’ existence with respect to a given world will not constitute a priori attributes unless the latter is the world of candidate elements. This will answerAnselm’s argument. i.6.4 A world will obtain meaning from without itself.

- 15 -

i.6.5 Although, according to the terminology advanced here, neither synthetic, nor a priori, those members of the vocabulary of the language underlng a language that are true with respect to the latter may be identified as Kant’s synthetic, a priori propositions with respect to the former language. i.7.1 There will not be two members of the set of worlds defined by a language such that there is a member of the second world constituting a member of the first world in the case and only in the case that it is not a member of the first world. i.7.2 In the case that a member of the set of worlds defined by a language contains all those, and only those, members of a world defined by the language that are not members of the themselves the former will not be a member of the latter. i.7.3 There will not be a member of the set of worlds defined by a language constituting a Russell set, that is, a set the members of which are all those and only those members of the world of candidate elements defined by the language that are not members of themselves. A Russell set will be inconceivable. i.8.1 Members of the set of a language’s unary relations will pertain to individual members of the set of candidate elements defined by the language. Members of the set of a language’s binary relations will jointly pertain to the members of ordered pairs of members of the set. Members of the set of a language’s n-ary relations will jointly pertain to the members of ordered n-tuples of members of the set of candidate elements defined by the language. i.8.2 The members of the set of a language’s relations will occur in pairs the members of which constitute complements with respect to each other. i.8.3 To each member of the set of elements defined by a language one, and only one, member of each complementary pair of unary attributes will pertain. i.8.4 Any member of the set of candidate elements defined by a language may be identified positively, with reference to the set of those unary attributes that pertain to it, or negatively, with reference to the set of those unary attributes that do not pertain to it: Any member of the set of candidate elements defined by a language may be defined by that which it is, or alternatively, by that which it is not.

- 16 -

i.9.1 In practise, a world’s definition will begin with an original language the interpretation of which reveals itself. A thinking mind’s original language will be its language of experience. i.9.2 A language of experience will constitute a conscious mind. i.9.3 Members of the set of elements of a language constituting a conscious mind’s language of experience will be called beliefs, the set, ‘Knowledge(0)’, or ‘Knowledge by acquaintance’ with respect to the mind. i.9.4 The subset of the vocabulary of a language constituting a conscious mind’s language of experience the members of which are those members of the set that are true with respect to that language and with respect to a second language characterised by the same vocabulary as the former will be called ‘Knowledge(1)’ or ‘Knowledge by description’ with respect to the conscious mind, and the second language. i.9.5 Knowledge(0) may be considered a special case of Knowledge(1), the two languages being the same language. i.9.6 A conscious mind cannot be mistaken about its own beliefs, that is, it cannot be mistaken about its Knowledge(0). However, any given mind will reach its own consclusions as to which of another mind’s beliefs it will consider Knowledge(1) with respect to itself. i.9.7 Members of the vocabulary of a language constituting a conscious mind’s interpretation that are true with respect to the former will not constitute beliefs with respect to the mind as they are not part of its vocabulary. To a conscious mind the mind’s interpretation will reveal itself. i.10.1 All members of all worlds defined by a language will be characterised by their attributes. As attributes confer meaning any world defined by a language will be meaningful. Chaos, that is, the meaningless world, will be inconceivable. i.10.2 Nothingness, that is, the empty world, will be conceivable. i.10.3 Any language, insofar as it defines any element characterised by the pertinence of attributes, will define at least two elements, the element, and an element that is related to the former by difference.

- 17 -

i.10.4 It may be argued that an empty language, that is, a language the set of attributes of which, and the vocabulary of which are empty will define a single unspeakable - and unspoken - entity characterised by the absence of attributes.

- 18 -

1. 1.1.1 An unary natural language w.ll be an ordered 10 - tuple L : {L, Le , Lp , LD , LI , S0 R , S0 p , S0 D , S0 I , S0 Iw } of sets, the members of which are, respectively, a s.t L the members of which are the language’s propositions, a set Lp the members of which are the language’s primary propositions, a set Le the members of which are the language’s elemental propositions, a set LD the members of which are the language’s Dirac propositions, a set LI the members of which are the language’s reduced Dirac propositions, a set S0 R the members of which are the language’s unary relations, a set S0 p the members of which are the language’s primary unary relations, a set S0 D the members of which are the language’s Dirac unary relations, a set S0 I the members of which are the language’s particular element identifying unary relations and a set S0 Iw the members of which are the language’s particular world identifying unary relations. 1.2.1 Set S0 I of a natural language’s particular element identifying unary relations will be a subset of set S0 R of the language’s unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of particular element identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( S0 I R⊂ S0 R ).

1.2.2 Members of set S0 I of a natural language’s particular element identifying unary relations w.ll also be called, the language’s characteristic unary names, or the language’s characteristic names, or the language’s unary names. 1.2.3 Set S0 I of a natural language’s particular element identifying unary relations will constitute an atomic set of unary relations with respect to set S0 p of the language’s primary unary relations.

- 19 -

1.2.4 Each member Ix of set S0 I of a natural language’s particular element identifying unary relations will pertain to one, and only one, member x of the set Scd.te of elements defined by the language. For each member x of the set Scd.te of elements defined by a natural language there will be a member Ix of set S0 I of the language’s particular element identifying unary relations that pertains to it. 1.2.5 The members of subset S0 p of set S0 R of a natural language’s unary relations constituting primary unary relations will also be called generalised element identifying unary relations, or generalised element identifying relations. 0 1.2.6 Member Q∧ T of set S R of a natural language’s unary relations constituting a basic conjunctively necessary unary relation, or a basic disjunctively forbidden unary relation, w.ll also be called, a general element identifying unary relation, or a general element identifying relation.

1.3.1 Set S0 Iw of a natural language’s particular world identifying unary relations will be a subset of set S0 R of the language’s unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of particular world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( S0 Iw R⊂ S0 R ).

1.3.2 Set S0 Iw of a natural language’s particular world identifying unary relations will be a subset of set S0 I of the language’s particular element identifying unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of particular world identifying unary relations , given as

- 20 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( S0 Iw R⊂ S0 I ).

1.3.3 The members of set S0 Iw of a natural language’s particular world identifying unary relations will also be called, the language’s characteristic unary world names, the language’s characteristic world names, or the language’s unary world names. 1.3.4 Each member Iw of set S0 Iw of a natural language’s particular world identifying unary relations will pertain to one, and only one, member w of the set of worlds defined by the language. For each member w of the set of worlds defined by a natural language there will be a member Iw of the set of the language’s particular world identifying unary relations that pertains to it.

- 21 -

2. 2.1.1 For any ordered triple {Iw , Ix , Q} of members of set S0 R of a natural language’s unary relations the first two members of which are members of set S0 I of the language’s particular element identifying unary relations there will be a member p of set L of the language’s propositions constituting a particular world universal, particular element existential basic proposition with respect to the triple, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative law of the existence of the particular world universal, particular element existential basic propositions, given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix ,∀Q, ∃p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) ∧ pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q ).

Ix will also be called the proposition’s definite noun phrase Universal, or the proposition’s noun phrase Universal, Q the proposition’s verb phrase Universal. ¯ of members of set S0 R of a natural 2.1.2 For any ordered triple {Iw , Ix , Q} language’s unary relations the first two members of which are members of set S0 I of the language’s particular element identifying unary relations there will be a member p ¯ of set L of the language’s propositions constituting a particular world universal, particular element universal basic proposition with respect to the triple, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative law of the existence of the particular world universal, particular element universal basic propositions, given as

- 22 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iw ,∀Ix ,∀ Q, ∃¯ p, ¯ ∈ S0 R ) → (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q (¯ p ∈ L) ∧ p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ (Ix x → Qx)] Iw Ix Q ).

Ix will also be called the proposition’s definite noun phrase Universal, or the ¯ the proposition’s verb phrase Universal. proposition’s noun phrase Universal, Q 2.1.3 With respect to any member p of set L of a natural language’s propositions any member Q of set S0 R of the language’s unary relations may occur as a verb phrase Universal, or as a noun phrase Universal. 2.2.1 For any ordered pair {Iw , Q} of members of set S0 R of a natural language’s unary relations the first member of which is a member of set S0 I of the language’s particular element identifying unary relations there will be a member p of set L of the language’s propositions constituting a particular world universal, general element existential basic proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative law of the existence of the particular world universal, general element existential basic propositions, given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Q,∃p, (Iw ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) ∧ pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx] Iw Q ).

- 23 -

0 2.2.2 For any ordered triple {Iw , Q0 x , Q∧ T } of members of set S R of a natural language’s unary relations the first member of which is a member of set S0 I of the language’s particular element identifying unary relations, the third member, a member of set S0 R constituting a basic conjunctively necessary unary relation, there will be a member p of set L of the language’s propositions constituting a particular world universal, generalised element existential basic proposition with respect to the triple, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative law of the existence of the particular world universal, generalised element existential basic propositions, given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Qx ,∀Q∧ T, ∃p, 0 (Iw ∈ S0 I ) → (Qx ∈ S0 I ) → (Q∧ T ∈ S R) → (p ∈ L) ∧ R[Q∧ ] Q∧ T → T

pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Qx Q∧ T ).

p will also be called an open verb phrase existential proposition (‘There is a cat.’), Qx the proposition’s noun phrase Universal. ¯ of members of set S0 R of a natural language’s 2.2.3 For any ordered pair {Iw , Q} unary relations the first member of which is a member of set S0 I of the language’s particular element identifying unary relations there will be a member p ¯ of set L of the language’s propositions constituting a particular world universal, general element universal basic proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative law of the existence of the particular world universal, general element universal basic propositions, given as

- 24 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iw ,∀ Q, ∃¯ p, ¯ ∈ S0 R ) → (Iw ∈ S0 I ) → (Q (¯ p ∈ L) ∧ p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ Qx] Iw Q ).

p ¯ will also be called an open noun phrase universal proposition (‘Everything is beautiful.’), ¯ the proposition’s verb phrase Universal. Q 2.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the particular world universal, general element existential basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Q∧ T ,∀Q, ∀p, 0 0 (Iw ∈ S0 I ) → (Q∧ T ∈ S R ) → (Q ∈ S R ) → (p ∈ L) → R[Q∧ ] Q∧ T → T

( pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iw Q ↔ pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Q∧ TQ ) ). - 25 -

2.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic propositions’ conjunctive attribute transfer , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Q∧ T ,∀Qx , ∀p, 0 (Iw ∈ S0 I ) → (Qx ∈ S0 I ) → (Q∧ T ∈ S R) → (p ∈ L) ∧ R[Q∧ ] Q∧ T → T

( pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Qx Q∧ T ↔ pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Q∧ T Qx ) ).

2.3.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the particular world universal, general element universal basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iw ,∀Q∧ T ,∀ Q, ∀¯ p, 0 0 ¯ (Iw ∈ S0 I ) → (Q∧ T ∈ S R ) → (Q ∈ S R ) → (¯ p ∈ L) → - 26 -

R[Q∧ ] Q∧ T → T

( p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ Qx]Iw Q ↔ p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ (Ix x → Qx)] Iw Q∧ TQ ) ).

2.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the particular world universal, general element existential basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ¯ ∀Iw ,∀ Q, ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → ¯ 0R → (Iw ∈ S0 I ) → QS (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → ( ∀¯ p∨ ,(¯ p∨ ∈ L) → (¯ p∨ ∈ L∨ ) ↔ ( ∃Ix ,(Ix ∈ S0 I ) ∧ p ¯∨ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ (Ix x ∧ Qx)] Iw Ix Q ) )→

- 27 -

( p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ Qx] Iw Q ↔ p ¯S[L∨ ] L∨ ) ).

2.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the particular world universal, general element universal basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀Iw ,∀Q, ∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (Iw ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → ( ∀p∧ ,(p∧ ∈ L) → (p∧ ∈ L∧ ) ↔ ( ∃Ix ,(Ix ∈ S0 I ) ∧ p∧ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q ) )→

- 28 -

( pR[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → Qx] Iw Q ↔ pS[L∧ ] L∧ ) ).

2.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the particular world universal, particular element existential basic normal propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iwcd.te ,∀Ix ,∀ Q, ∀¯ p∨ T, ¯ ∈ S0 R ) → (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q (¯ p∨ T ∈ L) → R[Iw ] Iwcd.te → cd.te

p ¯∨ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q ( ¯ [Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix → QR T∨ p ¯∨ T ) ).

2.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the particular world universal, particular element existential basic normal propositions , given as - 29 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix ,∀Q, ∀p∧ T, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p∧ T ∈ L) → R[Iw ] Iwcd.te → cd.te

p∧ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q → ( QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix → T∧ p∧ T ) ).

2.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the particular world universal, particular element existential basic propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iw ,∀Ix ,∀ Q, ∀¯ p∨ , T ¯ ∈ S0 R ) → (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q (¯ p∨ ∈ L) → T p ¯∨ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ →( (Ix x ∧ Qx)] Iw Ix Q ¯ [Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix → QR T∨ p ¯∨ T ) ).

- 30 -

2.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the particular world universal, particular element universal basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix ,∀Q, ∀p∧ T, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p∧ T ∈ L) → p∧ T R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q → ( QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix → T∧ p∧ T ) ).

2.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the complements of the particular world universal, particular element existential basic normal propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ¯ ∀ Q,∀Q, ∀¯ p,∀p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q (¯ p ∈ L) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

- 31 -

p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q → ( ¯ Q QR ¯Q→ p ¯ Rp¯ p ) ).

2.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the complements of the particular world universal, general element existential basic normal propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iwcd.te ,∀ Q,∀Q, ∀¯ p,∀p, ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Iwcd.te ∈ S0 I ) → (Q (¯ p ∈ L) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ → Qx]Iwcd.te Q pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iwcd.te Q → ( ¯ Q QR ¯Q→ p ¯ Rp¯ p ) ).

- 32 -

3. 3.1.1 For any ordered pair {p00 , p0 } of members of set L of elements of a natural language’s propositions there will be a member p ¯ of the set constituting a negatively conjunctive proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the negatively conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∃¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) ∧ p ¯ S[p: p00 !∧ p0 ] p00 p0 ).

3.1.2 For any ordered pair {Q00 , Q0 } of members of set S0 R a natural language’s ¯ of the set constituting a negatively unary relations there will be a member Q conjunctive unary relation with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the negatively conjunctive unary relations , given as

- 33 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ¯ ∃ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) ∧ (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 QS ).

3.2.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the negatively conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( ( T∧ p00 ∧ T∧ p0 )↔ T∨ p ¯ ) ).

- 34 -

3.2.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the negatively conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀¯ p0 , ∀p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (p ∈ L) → pS[p: p00 !∧ p0 ] p ¯00 p ¯0 → ( ( T∨ p ¯00 ∨ T∨ p ¯0 )→ T∧ p0 ) ).

A truth table of logical connectives will be defined in a hirarchy of truth tables of logical connectives. 3.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the negatively conjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ¯ p, ∀ Q,∀¯ (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → - 35 -

(Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → ¯ ∈ S0 R ) → (¯ p ∈ L) → (Q R[Iw ] Iwcd.te → cd.te

p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q00 → p0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q0 → ¯ p ¯a R[p : ‘∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ ‘Iw ’w → (‘Ix ’x ∧ ‘Q’x)’]Iwcd.te Ix Q ¯ → p ¯b R[p : ‘∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ ‘Iw ’w → (‘Ix ’x ∧ ‘Q’x)’]Iwcd.te Ix Q ( ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS p ¯ S[p: p00 !∧ p0 ] p00 p0 ) ).

3.4.1 For any member p of set L of a natural language’s propositions there will be a member p ¯ of the set constituting a complementary proposition with respect to p, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the propositions’ complements ,given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,∃¯ p, (p ∈ L) → (¯ p ∈ L) ∧ p ¯ Rp¯ p ).

- 36 -

3.4.2 For any member Q of set S0 R of a natural language’s unary relations there ¯ of the set constituting a complementary unary relation with will be a member Q respect to Q, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the unary relations’ complements ,given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Q,∃ Q, ¯ ∈ S0 R ) ∧ (Q ∈ S0 R ) → (Q ¯ Q Q QR ¯ ).

3.5.1 The second member of any ordered pair {¯ p, p} of members of set L of a natural language’s propositions will constitute a complementary proposition with respect to the pair’s first member in the case and only in the case that the pair’s first member constitutes a complementary proposition with respect to the pair’s second member, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the reciprocity of the propositions’ complementarity , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → ( p ¯ Rp¯ p ↔ pRp¯ p ¯ ) ).

- 37 -

¯ Q} of members of the set 3.5.2 The second member of any ordered pair {Q, of a natural language’s unary relations will constitute a complementary unary relation with respect to the pair’s first mmb in the case and only in the case that the pair’s first member constitutes a complementary unary relation with respect to the pair’s second member, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the reciprocity of the unary relations’ complementarity , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀ Q,∀Q, ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ Q QR ¯Q ↔ ¯ QRQ ¯Q ) ).

3.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the propositions’ complements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → ( p ¯ Rp¯ p ↔ p ¯ S[p: p00 !∧ p0 ] pp ) ). - 38 -

3.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the unary relations’ complements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀ Q,∀Q, ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ Q QR ¯Q ↔ ¯ [Qx: Q00 x !∧ Q0 x] Q Q QS ) ).

- 39 -

4. 4.1.1 There will be a member Iwcd.te of the set S0 Iw of a natural language’s particular world identifying unary relations constituting a candidate world identifying unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the candidate world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∃Iwcd.te ,(Iwcd.te ∈ S0 R ) ∧ (Iwcd.te ∈ S0 I ) ∧ (Iwcd.te ∈ S0 Iw ) ∧ ).

4.1.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative propositional Law of the world of candidate elements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ∧ ∀Q∧ T ,∀pT , (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → 0 ∧ (Q∧ T ∈ S R ) → (pT ∈ L) → ∧ R[Q∧ ] QT → ( T

p∧ T R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iwcd.te Ix Q∧ T ↔ p∧ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q∧ T ) ). - 40 -

4.2.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative propositional Law of the world of candidate elements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix ,∀Q, ∀p∧ T, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p∧ T ∈ L) → p∧ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q → ( QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix → T∧ p∧ T ) ).

4.2.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative propositional Law of the world of candidate elements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Iwcd.te ,∀Ix ,∀ Q, ∀¯ p∧ , F ¯ ∈ S0 R ) → (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∧ (¯ pF ∈ L) → p∧ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q

- 41 -

( ¯ [Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix → QR T∨ p ¯∧ F ) ).

4.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic propositions’ conjunctive normalisation , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Iw ,∀Ix , ∀Q∧ T ,∀Q, ∀pn ,∀pp , ∀p, (Iwcd.te ∈ S0 I ) → (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → 0 0 (Q∧ T ∈ S R ) → (Q ∈ S R ) → (pn ∈ L) → (pp ∈ L) → (p ∈ L) → R[Q∧ ] Q∧ T → T

pn R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q → pp R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q∧ T → ( pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q ↔ pS[p: p00 ∧ p0 ] pn pp ) ).

- 42 -

4.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic propositions’ disjunctive normalisation , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te , ∀Iw ,∀Ix , ¯ ∀Q∧ T ,∀ Q, ∀¯ pn ,∀pp , ∀¯ p, (Iwcd.te ∈ S0 I ) → (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → 0 0 ¯ (Q∧ T ∈ S R ) → (Q ∈ S R ) → (¯ pn ∈ L) → (pp ∈ L) → (¯ p ∈ L) → R[Q∧ ] Q∧ T → T

p ¯n R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q pp R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q∧ T → ( p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ (Ix x ∧ Qx)] Iw Ix Q ↔ p ¯ S[p: p0 → p00 ] pp p ¯n ) ).

- 43 -

4.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element existential disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → ¯ 000 ∈ S0 R ) → (¯ (Q p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 00 → (Ix x ∧ Qx)] Iw Ix Q 0 p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] Iw Ix Q p ¯000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 000 → (Ix x ∧ Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element universal disjunctive syllogism , given as

- 44 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → ¯ 000 ∈ S0 R ) → (p000 ∈ L) → (Q (¯ p ∈ L) → p ¯00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 00 → (Ix x ∧ Qx)] Iw Ix Q 0 p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → (Ix x → Qx)] Iw Ix Q p ¯000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 000 → (Ix x ∧ Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.4.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element universal disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q - 45 -

∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q 00 0 (¯ p ∈ L) → (¯ p ∈ L) → ¯ 000 ∈ S0 R ) → (p000 ∈ L) → (Q (¯ p ∈ L) → p ¯00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 00 → (Ix x ∧ Qx)] Iw Ix Q p ¯0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → (Ix x → Qx)] Iw Ix Q p ¯000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 000 → (Ix x → Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.4.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element existential disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , 000 ¯ ∀ Q ,∀¯ p000 , ∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q - 46 -

(¯ p00 ∈ L) → (¯ p0 ∈ L) → ¯ 000 ∈ S0 R ) → (¯ p000 ∈ L) → (Q (¯ p ∈ L) → p ¯00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 → (Ix x → Qx)] Iw Ix Q 0 p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] Iw Ix Q p ¯000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 000 → (Ix x ∧ Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p0 → p ¯ S[p: p00 ∨ p0 ] p ¯ p000 ) ).

4.4.5 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element existential disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → 000 0 ¯ (Q ∈ S R ) → (¯ p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 → (Ix x → Qx)] Iw Ix Q - 47 -

p ¯0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] Iw Ix Q 000 p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 000 → (Ix x → Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p0 → p ¯ S[p: p00 ∨ p0 ] p ¯ p000 ) ).

4.4.6 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element universal disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q 00 ∀¯ p ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → ¯ 000 ∈ S0 R ) → (¯ (Q p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 → (Ix x → Qx)] Iw Ix Q p ¯0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → (Ix x → Qx)] Iw Ix Q p ¯000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 000 → (Ix x ∧ Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q - 48 -

( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.4.7 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element universal disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 ,∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → 000 0 ¯ (Q ∈ S R ) → (¯ p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 → (Ix x → Qx)] Iw Ix Q p ¯0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → (Ix x → Qx)] Iw Ix Q p ¯000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 000 → (Ix x → Qx)] Iw Ix Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

- 49 -

4.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element existential conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q00 → 0 p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q0 → p000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element existential conjunctive syllogism , given as

- 50 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q00 → p0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q0 → 000 p R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q000 → 000 00 0 Q S[Qx: Q00 x ∧ Q0 x] Q Q → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.5.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element universal conjunctive syllogism , given as

- 51 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q00 → p0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q0 → 000 p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q000 → 000 00 0 Q S[Qx: Q00 x ∧ Q0 x] Q Q → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.5.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element existential, particular element universal conjunctive syllogism , given as

- 52 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q00 → 0 p R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q0 → p000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.5.5 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element existential conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , - 53 -

∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q00 → p0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q0 → p000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.5.6 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element existential conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → - 54 -

(p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q00 → 0 p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q0 → p000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.5.7 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular element universal, particular element universal conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q00 → - 55 -

p0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q0 → 000 p R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] p p000 ) ).

4.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element existential, general element existential disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw , ¯ 00 ,∀ Q ¯ 0, ∀Q 00 ∀¯ p ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q 00 0 (¯ p ∈ L) → (¯ p ∈ L) → ¯ 000 ∈ S0 R ) → (¯ (Q p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 00 → Qx]Iw Q p ¯0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → Qx]Iw Q p ¯000 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 000 → Qx]Iw Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q - 56 -

( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element existential, general element universal disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 ,∀¯ p, (Iw ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → ¯ 000 ∈ S0 R ) → (¯ (Q p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 00 → Qx]Iw Q p ¯0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → Qx]Iw Q p ¯000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 000 → Qx]Iw Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

- 57 -

4.6.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element universal, general element existential disjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ 000 ,∀¯ ∀Q p000 , ∀¯ p, (Iw ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (¯ p00 ∈ L) → (¯ p0 ∈ L) → ¯ 000 ∈ S0 R ) → (¯ (Q p000 ∈ L) → (¯ p ∈ L) → p ¯00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 → Qx]Iw Q 0 p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → Qx]Iw Q p ¯000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 000 → Qx]Iw Q ¯ 000 S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 → Q ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.6.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element universal, general element universal disjunctive syllogism , given as

- 58 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀Iw , ¯ 00 ,∀ Q ¯ 0, ∀Q 00 ∀¯ p ,∀¯ p0 , ∀¯ p000 ,∀¯ p, (L(pwr) ∈ S) → (L∧ ∈ S) → (Iw ∈ S0 I ) → ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q 00 0 (¯ p ∈ L) → (¯ p ∈ L) → (¯ p000 ∈ L) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → ( ∀¯ p000 p000 ∧ ,(¯ ∧ ∈ L) → (¯ p000 ∈ L∧ ) ↔ ( ∧ ∃I00 x ,∃I0 x , ∃¯ p00∧ ,∃¯ p0∧ , (I00 x ∈ S0 I ) ∧ (I0 x ∈ S0 I ) ∧ (¯ p00∧ ∈ L) ∧ (¯ p0∧ ∈ L) ∧ 00 p ¯∧ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 ∧ (Ix x → Qx)] Iw I00 x Q p ¯0∧ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 ∧ (Ix x → Qx)] Iw I0 x Q 00 0 00 0 p ¯000 S p ¯ p ¯ ∧ [p: p ∨ p ] ∧ ∧ ) )→ p ¯00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 00 → Qx]Iw Q p ¯0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → Qx]Iw Q 000 ∧ p ¯ S[L∧ ] L →

- 59 -

( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 ∨ p0 ] p ¯p ¯000 ) ).

4.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element existential, general element existential conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀Iw ,∀Q00 ,∀Q0 , ∀p00 ,∀p0 ,∀p000 ,∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (Iw ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (p000 ∈ L) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → ( ∀p000 ∨ ,(p000 ∨ ∈ L) → (p000 ∨ ∈ L∨ ) ↔ ( ∃I00 x ,∃I0 x , ∃p00 ∨ ,∃p0 ∨ , (I00 x ∈ S0 I ) ∧ (I0 x ∈ S0 I ) ∧ (p00 ∨ ∈ L) ∧ (p0 ∨ ∈ L) ∧ p00 ∨ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw I00 x Q00 ∧ p0 ∨ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw I0 x Q0 ∧ 000 00 0 p ∨ S[p: p00 ∧ p0 ] p ∨ p ∨ ) )→

- 60 -

p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iw Q00 → 0 p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iw Q0 → p000 S[L∨ ] L∨ → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element existential, general element universal conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx] Iw Q00 → 0 p R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → Qx] Iw Q0 → 000 p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iw Q000 → - 61 -

Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

4.7.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element universal, general element existential conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Q00 ,∀Q0 , ∀p00 ,∀p0 , (Iw ∈ S0 I ) → ∀Q000 ,∀p000 ,∀p, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → Qx]Iw Q00 → p0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iw Q0 → 000 p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → Qx]Iw Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pp000 ) ).

- 62 -

4.7.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the general element universal, general element universal conjunctive syllogism , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q000 ,∀p000 , ∀p, (Iw ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q000 ∈ S0 R ) → (p000 ∈ L) → (p ∈ L) → p00 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → Qx]Iw Q00 → 0 p R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → Qx]Iw Q0 → p000 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → Qx]Iw Q000 → Q000 S[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ( pS[p: p00 ∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] p p000 ) ).

- 63 -

5. 5.1.1 There will be a member p∧ T of the set L of a natural language’s propositions constituting a basic conjunctively necessary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic conjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ∃p∧ T ,(pT ∈ L) ∧ R[p∧ ] p∧ T T

).

p∧ T will also be called a conjunctively tautological proposition, or a tautological proposition, or a conjunctive tautology, or a tautology. 5.1.2 There will be a member p ¯∧ F of the set L of a natural language’s propositions constituting a basic conjunctively forbidden proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic conjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∃¯ p∧ p∧ F ,(¯ F ∈ L) ∧ R[p¯∧ ] p ¯∧ F F

).

p ¯∧ F will also be called a conjunctively contradictory proposition, or a contradictory proposition, or a conjunctive contradiction, or a contradiction.

- 64 -

5.1.3 There will be a member p ¯∨ T of the set L of a natural language’s propositions constituting a basic disjunctively necessary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic disjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∃¯ p∨ p∨ T ,(¯ T ∈ L) ∧ R[p¯∨ ] p ¯∨ T T

).

p ¯∨ T will also be called a disjunctively tautological proposition, or a disjunctive tautology. 5.1.4 There will be a member p∨ F of the set L of a natural language’s propositions constituting a basic disjunctively forbidden proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic disjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ∃p∨ F ,(pF ∈ L) ∧ R[p∨ ] p∨ F F

).

p∨ F will also be called a disjunctively contradictory proposition, or a disjunctive contradiction.

- 65 -

0 5.2.1 There will be a member Q∧ T of the set S R of a natural language’s unary relations constituting a basic conjunctively necessary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic conjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ 0 ∃Q∧ T ,(QT ∈ S R ) ∧ ∧ R[Q∧ ] QT T

).

Q∧ T will also be called a conjunctively tautological unary relation, or a tautological unary relation. ¯ ∧ of the set S0 R of a natural language’s unary 5.2.2 There will be a member Q F relations constituting a basic conjunctively forbidden unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic conjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( 0 ¯∧ ¯∧ ∃Q F ,(QF ∈ S R ) ∧ ∧ ¯ R[Q ¯ ∧ ] QF F

).

¯ ∧ will also be called a conjunctively contradictory unary relation, or a ctrry Q F unary relation.

- 66 -

¯ ∨ of the set S0 R of a natural language’s unary 5.2.3 There will be a member Q T relations constituting a basic disjunctively necessary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic disjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( 0 ¯∨ ¯∨ ∃Q T ,(QT ∈ S R ) ∧ ∨ ¯ R[Q ¯ ∨ ] QT T

).

¯ ∨ will also be called a disjunctively tautological unary relation. Q T 0 5.2.4 There will be a member Q∨ F of the set S R of a natural language’s unary relations constituting a basic disjunctively forbidden unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the basic disjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ 0 ∃Q∨ F ,(QF ∈ S R ) ∧ ∨ R[Q∨ ] QF F

).

Q∨ F will also be called a disjunctively contradictory unary relation.

- 67 -

5.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic conjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ∀p∧ T ,(pT ∈ L) → ( R[p∧ ] p∧ T → T

T∧ p∧ T ) ).

5.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic conjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p∧ p∧ F ,(¯ F ∈ L) → ( R[p¯∧ ] p ¯∧ F → F

T∨ p ¯∧ F ) ).

- 68 -

5.3.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic disjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p∨ p∨ T ,(¯ T ∈ L) → ( R[p¯∨ ] p ¯∨ T → T

T∨ p ¯∨ T ) ).

5.3.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic disjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ∀p∨ F ,(pF ∈ L) → ( R[p∨ ] p∨ F → F

T∧ p∨ F ) ).

- 69 -

5.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic conjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ∀Iwcd.te ,∀Ix ,∀Q∧ T ,∀pT , 0 0 ∧ (Iwcd.te ∈ S I ) → (Ix ∈ S0 I ) → (Q∧ T ∈ S R ) → (pT ∈ L) → R[Iw I → w ] cd.te cd.te

p∧ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q∧ T →( ∧ ∧ R[Q∧ ] QT → R[p∧ ] pT T

T

) ).

5.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic conjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯∧ ∀Iwcd.te ,∀Ix ,∀ Q p∧ F ,∀¯ F, 0 ¯∧ (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q p∧ F ∈ S R ) → (¯ F ∈ L) → R[Iw ] Iwcd.te → cd.te

p ¯∧ F R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯∧ (Ix x ∧ Qx)] Iwcd.te Ix Q F →( ∧ ¯∧ R[Q p ¯ ∧ ¯ ∧ ] QF → R[p ¯ ] F F

F

) ).

- 70 -

5.4.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic disjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯∨ ∀Iwcd.te ,∀Ix ,∀ Q p∨ T ,∀¯ T, 0 0 ¯∨ p∨ (Iwcd.te ∈ S I ) → (Ix ∈ S0 I ) → (Q T ∈ S R ) → (¯ T ∈ L) → R[Iw I → w ] cd.te cd.te

p ¯∨ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯∨ (Ix x ∧ Qx)] Iwcd.te Ix Q T →( ∨ ∨ ¯ R[Q ¯T ¯ ∨ ] QT → R[p ¯∨ ] p T

T

) ).

5.4.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the basic disjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ∀Iwcd.te ,∀Ix ,∀Q∨ F ,∀pF , 0 ∨ (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q∨ F ∈ S R ) → (pF ∈ L) → R[Iw ] Iwcd.te → cd.te

p∨ F R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q∨ F →( ∨ R[Q∨ ] Q∨ → R p ∨ F [p ] F F

F

) ).

- 71 -

5.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic conjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p,∀p∧ T, (¯ p ∈ L) → (p ∈ L) → (p∧ T ∈ L) → p ¯ Rp¯ p → ( R[p∧ ] p∧ T T ↔ p∧ ¯p T S[p: p00 ∨ p0 ] p ) ).

5.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic conjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p,∀¯ p∧ F, (¯ p ∈ L) → (p ∈ L) → (¯ p∧ F ∈ L) → p ¯ Rp¯ p → ( R[p¯∧ ] p ¯∧ F F ↔ p ¯∧ ¯p F S[p: p00 ∧ p0 ] p ) ).

- 72 -

5.5.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic disjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p,∀¯ p∨ T, (¯ p ∈ L) → (p ∈ L) → (¯ p∨ T ∈ L) → p ¯ Rp¯ p → ( R[p¯∨ ] p ¯∨ T T ↔ p ¯∨ ¯p T S[p: p00 ∧ p0 ] p ) ).

5.5.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic disjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p,∀p∨ F, (¯ p ∈ L) → (p ∈ L) → (p∨ F ∈ L) → p ¯ Rp¯ p → ( R[p∨ ] p∨ F F ↔ p∨ ¯p F S[p: p00 ∨ p0 ] p ) ).

- 73 -

5.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic conjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ¯ ∀ Q,∀Q,∀Q T, 0 0 ¯ (Q ∈ S R ) → (Q ∈ S0 R ) → (Q∧ T ∈ S R) → ¯ Q QR ¯Q→( R[Q∧ ] Q∧ T T ↔ ¯ Q∧ T S[Qx: Q00 x ∨ Q0 x] Q Q )

).

5.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic conjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ¯∧ ∀ Q,∀Q,∀ Q F, 0 ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯∧ (Q F ∈ S R) → ¯ QRQ ¯Q→( ¯∧ R[Q ¯ ∧ ] QF F ↔ ¯∧ ¯ Q F S[Qx: Q00 x ∧ Q0 x] Q Q ) ).

- 74 -

5.6.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic disjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ¯∨ ∀ Q,∀Q,∀ Q T, 0 0 ¯∨ ¯ (Q ∈ S R ) → (Q ∈ S0 R ) → (Q T ∈ S R) → ¯ Q QR ¯Q→( ¯∨ R[Q ¯ ∨ ] QT T ↔ ¯∨ ¯ Q T S[Qx: Q00 x ∧ Q0 x] Q Q ) ).

5.6.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the basic disjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ¯ ∀ Q,∀Q,∀Q F, 0 ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q∨ (Q F ∈ S R) → ¯ QRQ ¯Q→(

R[Q∨ ] Q∨ F F ↔ ¯ Q∨ F S[Qx: Q00 x ∨ Q0 x] Q Q ) ).

- 75 -

5.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctive dominance of the basic disjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p∨ F, (¯ p ∈ L) → (p∨ F ∈ L) → ( R[p∨ ] p∨ F → F

∨ p∨ ¯ F S[p: p00 ∨ p0 ] pF p

) ).

5.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctive neutrality of the basic disjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀¯ p∨ T, (¯ p ∈ L) → (¯ p∨ T ∈ L) → ( R[p¯∨ ] p ¯∨ T → T

p ¯ S[p: p00 ∨ p0 ] p ¯∨ ¯ Tp ) ).

- 76 -

5.7.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctive dominance of the basic conjunctively forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,∀¯ p∧ F, (p ∈ L) → (¯ p∧ F ∈ L) → ( R[p¯∧ ] p ¯∧ F → F

p ¯∧ ¯∧ F S[p: p00 ∧ p0 ] p Fp ) ).

5.7.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctive neutrality of the basic conjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,∀p∧ T, (p ∈ L) → (p∧ T ∈ L) → R[p∧ ] p∧ T → T

pS[p: p00 ∧ p0 ] p∧ Tp ).

- 77 -

5.8.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctive dominance of the basic disjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ¯ ∀ Q,∀Q F, 0 0 ¯ (Q ∈ S R ) → (Q∨ F ∈ S R) → ( R[Q∨ ] Q∨ F → F

∨ ¯ Q∨ F S[Qx: Q00 x ∨ Q0 x] QF Q

) ).

5.8.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctive neutrality of the basic disjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ Q ¯∨ ∀ Q,∀ T, 0 ¯ ∈ S0 R ) → (Q ¯∨ (Q T ∈ S R) → ( ¯∨ R[Q ¯ ∨ ] QT → T

¯ [Qx: Q00 x ∨ Q0 x] Q ¯∨ ¯ QS TQ ) ).

- 78 -

5.8.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctive dominance of the basic conjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯∧ ∀Q,∀ Q F, 0 ¯∧ (Q ∈ S0 R ) → (Q F ∈ S R) → ( ¯∧ R[Q ¯ ∧ ] QF → F

¯∧ ¯∧ Q F S[Qx: Q00 x ∧ Q0 x] QF Q ) ).

5.8.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctive neutrality of the basic conjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,∀Q∧ T, 0 (Q ∈ S0 R ) → (Q∧ T ∈ S R) → ( R[Q∧ ] Q∧ T → T

QS[Qx: Q00 x ∧ Q0 x] Q∧ TQ ) ).

- 79 -

5.9.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative disjunctive Law of the excluded middle propositions , or conjunctive Law of the contradictory propositions , or Law of the contradictory propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p, ∀¯ p∨ T, (¯ p ∈ L) → (p ∈ L) → (¯ p∨ T ∈ L) → p ¯ Rp¯ p → ( p ¯∨ ¯p → T S[p: p00 ∧ p0 ] p T∨ p ¯∨ T ) ).

5.9.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative disjunctive Law of the contradictory propositions , or conjunctive Law of the excluded middle propositions , or Law of the excluded middle propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p,∀p∧ T, (¯ p ∈ L) → (p ∈ L) → (p∧ T ∈ L) → p ¯ Rp¯ p → ( p∧ ¯p → T S[p: p00 ∨ p0 ] p T∧ p∧ T ) ).

- 80 -

5.10.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative disjunctive Law of the excluded middle unary relations , or conjunctive Law of the contradictory unary relations , or Law of the contradictory unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ¯ ∀ Q,∀Q, ¯∨ ∀Q p∨ T ,∀¯ T, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q 0 ¯∨ (Q p∨ T ∈ S R ) → (¯ T ∈ L) → I → R[Iw w cd.te ] cd.te

¯ Q QR ¯Q→ p ¯∨ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯∨ (Ix x ∧ Qx)] Iwcd.te Ix Q T → ( ¯ ¯∨ Q T S[Qx: Q00 x ∧ Q0 x] Q Q → T∨ p ¯∨ T ) ).

5.10.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative disjunctive Law of the contradictory unary relations , or conjunctive Law of the excluded middle unary relations , or Law of the excluded middle unary relations given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ¯ ∀ Q,∀Q, ∧ ∀Q∧ T ,∀pT , (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) →

- 81 -

¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ∧ 0 (Q∧ T ∈ S R ) → (pT ∈ L) → R[Iw ] Iwcd.te → cd.te

¯ Q QR ¯Q→ p∧ T R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q∧ T → ( ¯ Q∧ T S[Qx: Q00 x ∨ Q0 x] Q Q → T∧ p∧ T ) ).

5.11.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the contradictory unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ ∀ Q,∀Q, ¯∧ ∀Q p∧ F ,∀¯ F, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ ∧ ∈ S0 R ) → (¯ (Q p∧ F F ∈ L) → ¯ QRQ ¯Q→ p ¯∧ F R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯∧ → (Ix x ∧ Qx)] Iw Ix Q F ( ¯ ∧ S[Qx: Q00 x ∧ Q0 x] Q ¯Q→ Q F T∨ p ¯∧ F ) ).

- 82 -

5.11.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the excluded middle unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ ∀ Q,∀Q, ∧ ∀Q∧ T ,∀pT , (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q 0 ∧ (Q∧ T ∈ S R ) → (pT ∈ L) → ¯ QRQ ¯Q→ p∧ T R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q∧ T → ( ¯ Q∧ T S[Qx: Q00 x ∨ Q0 x] Q Q → T∧ p∧ T ) ).

- 83 -

6. 6.1.1 For any ordered pair {¯ p00 , p ¯0 } of members of set L of a natural language’s propositions there will be a member p ¯ of the set constituting a disjunctive proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀¯ p0 , ∃¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) ∧ p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 ).

6.1.2 For any ordered pair {p00 , p0 } of members of set L of a natural language’s propositions there will be a member p of the set constituting a conjunctive proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∃p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) ∧ pS[p: p00 ∧ p0 ] p00 p0 ).

- 84 -

¯ 00 , Q ¯ 0 } of members of set S0 R of a natural langua6.2.1 For any ordered pair {Q ¯ of the set constituting a disjunctive ge’s unary relations there will be a member Q unary relation with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∃ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) ∧ (Q ¯ 00 Q ¯0 ¯ [Qx: Q00 x ∨ Q0 x] Q QS ).

6.2.2 For any ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations there will be a member Q of the set constituting a conjunctive unary relation with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the conjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ∃Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ QS[Qx: Q00 x ∧ Q0 x] Q00 Q0 ).

- 85 -

6.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the disjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀p00 , ∀¯ p0 ,∀p0 ,∀¯ p, (¯ p00 ∈ L) → (p00 ∈ L) → (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯00 Rp¯ p00 → p ¯0 Rp¯ p0 → ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 ↔ p ¯ S[p: p00 !∧ p0 ] p00 p0 ) ).

6.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀¯ p,∀p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → (p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( pS[p: p00 ∧ p0 ] p00 p0 ↔ p ¯ Rp¯ p ) ). - 86 -

6.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the disjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀Q00 , ∀Q ¯ 0 ,∀Q0 ,∀ Q, ¯ ∀Q 00 0 ¯ (Q ∈ S R ) → (Q00 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 00 0 0 00 ¯ ¯ Q RQ ¯ Q → Q RQ ¯Q →( ¯ 00 Q ¯ 0 ↔ QS ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 ¯ [Qx: Q00 x ∨ Q0 x] Q QS ) ).

6.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the conjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ¯ ∀ Q,∀Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → ( QS ¯ Q QS[Qx: Q00 x ∧ Q0 x] Q00 Q0 ↔ QR ¯Q ) ).

- 87 -

7. 7.1.1 For any subset L∨ of set L of a natural language’s propositions there will be a member p ¯ of L constituting a disjoint proposition with respect to L∨ , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∃¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) ∧ L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ ).

7.1.2 For any subset L∧ of set L of a natural language’s propositions there will be a member p of L constituting a conjoint, or joint, proposition with respect to L∧ , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∃p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) ∧ L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → pS[L∧ ] L∧ ). - 88 -

∨

7.2.1 For any subset S0 R of set S0 R of a natural language’s unary relations ¯ of S0 R constituting a disjoint unary relation with there will be a member Q ∨ respect to S0 R , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjoint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ¯ ∃ Q,

(pwr)

∨

,∀S0 R , ∨

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → ¯ ∈ S0 R ) ∧ (Q S0 R

(pwr)

∨

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

(pwr)

)→

¯ [S0 ∨ ] S0 R ∨ QS R ).

∧

7.2.2 For any subset S0 R of set S0 R of a natural language’s unary relations there will be a member Q of S0 R constituting a joint, or conjoint, unary relation ∧ with respect to S0 R , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the joint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) ∧ ,∀S0 R ,∃Q, ∧ 0 (pwr) (S R ∈ S) → (S0 R ∈ S) → (Q ∈ S0 R ) ∧ (pwr) 0 0 (pwr) SR R[S(pwr) ] S R → (S0 R ∈ S0 R )

∀S0 R

QS[S0 R

∧]

∧ S0 R

).

- 89 -

∧ →

7.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ → ( ∀¯ p∨ ,(¯ p∨ ∈ L) → ∨ (¯ p ∈ L∨ ) → T∨ p ¯∨ )↔ T∨ p ¯ ).

- 90 -

7.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the resolution of the disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → pS[L∨ ] L∨ → ( ∃p∨ ,(p∨ ∈ L) ∧ (p∨ ∈ L∨ ) ∧ T∧ p∨ )→ T∧ p ).

- 91 -

7.3.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → pS[L∧ ] L∧ → ( ∀p∧ ,(p∧ ∈ L) → (p∧ ∈ L∧ ) → T∧ p∧ )↔ T∧ p ).

7.3.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the resolution of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀¯ p, (L(pwr) ∈ S) → (L∧ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → p ¯S[L∧ ] L∧ → (

- 92 -

∃¯ p∧ ,(¯ p∧ ∈ L) ∧ (¯ p∧ ∈ L∧ ) ∧ T∨ p ¯∧ )→ T∨ p ¯ ).

7.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the disjoint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , (pwr)

∨

∀S0 R ,∀S0 R , ∀Iwcd.te ,∀Ix , ¯ p, ∀ Q,∀¯ (L(pwr) ∈ S) → (L∨ ∈ S) → (pwr)

∨

(S0 R ∈ S) → (S0 R ∈ S) → (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → L(pwr) R[S(pwr) ] L → S0 R ∨

(pwr)

R[S(pwr) ] S0 R → (pwr)

(L∨ ∈ L(pwr) ) → (S0 R ∈ S0 R )→ R[Iw Iwcd.te → cd.te ] ( ∀¯ p∨ ,(¯ p∨ ∈ L) → ∨ (¯ p ∈ L∨ ) ↔ ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) ∧ ∃Q ¯ ∨ ∈ S0 R ∨ ) ∧ (Q p ¯∨ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯∨ (Ix x ∧ Qx)] Iwcd.te Ix Q ) )→ - 93 -

p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q ( ¯ [S0 ∨ ] S0 R ∨ → QS R p ¯S[L∨ ] L∨ ) ).

7.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the joint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , (pwr)

∧

∀S0 R ,∀S0 R , ∀Iwcd.te ,∀Ix , ∀Q,∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (pwr)

∧

(S0 R ∈ S) → (S0 R ∈ S) → (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → S0 R ∧

(pwr)

R[S(pwr) ] S0 R → L(pwr) R[S(pwr) ] L → ∧

(L ∈ L ) → (S0 R ∈ S0 R R[Iw ] Iwcd.te → ( (pwr)

(pwr)

)→

cd.te

∀p∧ ,(p∧ ∈ L) → (p∧ ∈ L∧ ) ↔ ( ∃Q∧ ,(Q∧ ∈ S0 R ) ∧ ∨ (Q∧ ∈ S0 R ) ∧ ∧ p R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q∧ ) )→ - 94 -

pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q → ( ∧ QS[S0 R ∧ ] S0 R → ∧ pS[L∧ ] L ) ).

7.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative third Law of the composition of the disjoint propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ∨ ∀L(pwr) ,∀L∨ b ,∀La ,∀L , ∀¯ pb ,∀¯ pa ,∀¯ p, ∨ ∨ (L(pwr) ∈ S) → (L∨ b ∈ S) → (La ∈ S) → (L ∈ S) → (¯ pb ∈ L) → (¯ pa ∈ L) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (pwr) (pwr) (L∨ ) → (L∨ )→ b ∈L a ∈L ∨ (pwr) (L ∈ L )→ p ¯b S[L∨ ] L∨ ¯a S[L∨ ] L∨ b →p a → p ¯S[L∨ ] L∨ → ( ∨ L∨ S[S: S00 ∪ S0 ] L∨ b La → p ¯ S[p: p00 ∨ p0 ] p ¯b p ¯a )

).

7.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative third Law of the composition of the joint propositions , given as - 95 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ∧ ∀L(pwr) ,∀L∧ b ,∀La ,∀L , ∀pb ,∀pa , ∀p, ∧ ∧ (L(pwr) ∈ S) → (L∧ b ∈ S) → (La ∈ S) → (L ∈ S) → (pb ∈ L) → (pa ∈ L) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (pwr) (pwr) (L∧ ) → (L∧ )→ b ∈L a ∈L ∧ (pwr) (L ∈ L )→ ∧ pb S[L∧ ] L∧ b → pa S[L∧ ] La → pS[L∧ ] L∧ → ( ∧ L∧ S[S: S00 ∪ S0 ] L∧ b La → pS[p: p00 ∧ p0 ] pb pa )

).

7.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) →

- 96 -

( L∨ R[S: s.ngle el.t s.t, x] p ¯→ p ¯S[L∨ ] L∨ ) ).

7.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → ( L∧ R[S: s.ngle el.t s.t, x] p → pS[L∧ ] L∧ ) ).

7.6.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p∨ T, - 97 -

(L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p∨ T ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → ∨ p ¯∨ T S[L∨ ] L → ( R{} L∨ → R[p¯∨ ] p ¯∨ T T

) ).

7.6.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative third Law of the composition of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀p∧ T, (L(pwr) ∈ S) → (L∧ ∈ S) → (p∧ T ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → ∧ p∧ T S[L∧ ] L → ( R{} L∧ → R[p∧ ] p∧ T T

) ).

7.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative third Law of the composition of the disjoint unary relations , given as - 98 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∨

∨

∨

,∀S0 R,b ,∀S0 R,a ,∀S0 R , ∀S0 R ¯ b ,∀ Q ¯ a, ∀Q ¯ ∀ Q, ∨

∨

∨

(pwr)

∈ S) → (S0 R,b ∈ S) → (S0 R,a ∈ S) → (S0 R ∈ S) → (S0 R 0 ¯ a ∈ S0 R ) → ¯ (Qb ∈ S R ) → (Q ¯ ∈ S0 R ) → (Q S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(S0 R,b ∈ S0 R

(pwr)

∨

) → (S0 R,a ∈ S0 R

(pwr)

)→

∨ (S0 R

(pwr) ∈ S0 R )→ ∨ 0 ¯ ¯ a S[S0 ∨ ] S0 R,a ∨ Qb S[S0 R ∨ ] S R,b → Q R 0 ∨ ¯ QS[S0 R ∨ ] S R →

→

( ∨

∨

∨

S0 R S[S: S00 ∪ S0 ] S0 R,b S0 R,a → ¯bQ ¯a ¯ [Qx: Q00 x ∨ Q0 x] Q QS ) ).

7.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative third Law of the composition of the joint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∧

∧

∧

∀S0 R ,∀S0 R,b ,∀S0 R,a ,∀S0 R , ∀Qb ,∀Qa , ∀Q, (pwr)

∧

∧

∧

(S0 R ∈ S) → (S0 R,b ∈ S) → (S0 R,a ∈ S) → (S0 R ∈ S) → 0 (Qb ∈ S R ) → (Qa ∈ S0 R ) → (Q ∈ S0 R ) →

- 99 -

S0 R

(pwr)

R[S(pwr) ] S0 R →

∧

(S0 R,b ∈ S0 R

(pwr)

∧

) → (S0 R,a ∈ S0 R

(pwr)

)→

∧ (S0 R

(pwr) ∈ S0 R )→ ∧ ∧ Qb S[S0 R ∧ ] S0 R,b → Qa S[S0 R ∧ ] S0 R,a 0 ∧ QS[S0 R ∧ ] S R →

→

( ∧

∧

∧

S0 R S[S: S00 ∪ S0 ] S0 R,b S0 R,a → QS[Qx: Q00 x ∧ Q0 x] Qb Qa ) ).

7.8.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the disjoint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ¯ ∀ Q,

(pwr)

(pwr)

∨

,∀S0 R , ∨

(S0 R ∈ S) → (S0 R ∈ S) → ¯ ∈ S0 R ) → (Q S0 R

(pwr)

∨

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

(pwr)

)→

( ∨

¯ → S0 R R[S: s.ngle el.t s.t, x] Q ¯ [S0 ∨ ] S0 R ∨ QS R

) ).

7.8.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the joint unary relations , given as - 100 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ∀Q,

(pwr)

∧

,∀S0 R , ∧

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R (Q ∈ S0 R ) → S0 R

(pwr)

∧

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

(pwr)

)→

( ∧

S0 R R[S: s.ngle el.t s.t, x] Q → QS[S0 R ∧ ] S0 R

∧

) ).

7.9.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the disjoint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀S0 R ¯ ∨, ∀Q

∨

,∀S0 R ,

T (pwr)

∨

(S0 R ∈ S) → (S0 R ∈ S) → 0 ¯∨ (Q T ∈ S R) → S0 R

(pwr)

∨

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

0 ∨ ¯∨ Q T S[S0 R ∨ ] S R

→

( ∨

R{} S0 R → ¯∨ R[Q ¯ ∨ ] QT T

) ).

- 101 -

(pwr)

)→

7.9.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the joint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀S0 R ∀Q∧ T,

∧

,∀S0 R , ∧

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ∧ 0 (QT ∈ S R ) → S0 R

(pwr)

∧

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

0 ∧ Q∧ T S[S0 R ∧ ] S R

(pwr)

)→

→

( ∧

R{} S0 R → R[Q∧ ] Q∧ T T

) ).

7.10.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the associativity of the disjoint propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ,∀L(pwr) , ∀Sb ,∀Sa , ∨ ∀L∨ b ,∀La , ∀¯ p, (pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∨ (L∨ b ∈ S) → (La ∈ S) → (¯ p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → - 102 -

(pwr)

(pwr)

) → (Sa ∈ L(pwr) (Sb ∈ L(pwr) ∨ (pwr) (pwr) (Lb ∈ L ) → (L∨ )→ a ∈L ( ∀¯ p∨ ,(¯ p∨ ∈ L) → ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ (¯ p∨ ∈ S0 b ) ) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ (¯ p∨ ∈ S0 a ) ) )→ ( ∀¯ p∨ p∨ b ,(¯ b ∈ L) → ∨ (¯ pb ∈ L∨ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ 0 p ¯∨ b S[L∨ ] S b ) )→ ( ∀¯ p∨ p∨ a ,(¯ a ∈ L) → (¯ p∨ ∈ L∨ a a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ 0 p ¯∨ a S[L∨ ] S a ) )→

- 103 -

)→

( p ¯S[L∨ ] L∨ b ↔ p ¯S[L∨ ] L∨ a ) ).

7.10.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the associativity of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ,∀L(pwr) , ∀Sb ,∀Sa , ∧ ∀L∧ b ,∀La , ∀p, (pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∧ (L∧ b ∈ S) → (La ∈ S) → (p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → (pwr)

(pwr)

(Sb ∈ L(pwr) ) → (Sa ∈ L(pwr) ∧ (pwr) (pwr) (Lb ∈ L ) → (L∧ )→ a ∈L

- 104 -

)→

( ∀p∧ ,(p∧ ∈ L) → ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ (p∧ ∈ S0 b ) ) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ (p∧ ∈ S0 a ) ) )→ ( ∀pb ∧ ,(pb ∧ ∈ L) → (pb ∧ ∈ L∧ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ pb ∧ S[L∧ ] S0 b ) )→ ( ∀pa ∧ ,(pa ∧ ∈ L) → (pa ∧ ∈ L∧ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ pa ∧ S[L∧ ] S0 a ) )→

- 105 -

( pS[L∧ ] L∧ b ↔ pS[L∧ ] L∧ a ) ).

7.11.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the associativity of the disjoint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∨ ∨ ∀S0 R,b ,∀S0 R,a , ¯ ∀ Q,

(pwr)

,

(pwr) (pwr)

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 ¯ (Q ∈ S R ) → S0 R

(pwr) (pwr)

(Sb ∈ S0 R 0

∨

(S R,b ∈

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

→ S0 R

) → (Sa ∈ S0 R

(pwr) S0 R )

∨

R[S(pwr) ] S0 R →

(pwr) (pwr)

→ (S0 R,a ∈ S0 R

- 106 -

(pwr)

(pwr)

)→

)→

( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ∀Q ( ∃S0 b ,(S0 b ∈ S) ∧ (pwr)

(S0 b ∈ S0 R (S0 b ∈ Sb ) ∧ ¯ ∨ ∈ S0 b ) (Q

)∧

) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R (S0 a ∈ Sa ) ∧ ¯ ∨ ∈ S0 a ) (Q

)∧

) )→ ( 0 ¯∨ ¯∨ ∀Q b ,(Qb ∈ S R ) → ∨ ∨ 0 ¯ (Qb ∈ S R,b ) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (pwr)

(S0 b ∈ S0 R )∧ (S0 b ∈ Sb ) ∧ 0 ¯∨ Q b S[S0 R ∨ ] S b ) )→ ( 0 ¯ ∨ ,(Q ¯∨ ∀Q a a ∈ S R) → ¯ ∨ ∈ S0 R,a ∨ ) ↔ ( (Q a

∃S0 a ,(S0 a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R )∧ (S0 a ∈ Sa ) ∧ ¯ ∨ S[S0 ∨ ] S0 a Q a R ) )→

- 107 -

( ∨

¯ [S0 ∨ ] S0 R,b QS R ↔ ¯ [S0 ∨ ] S0 R,a ∨ QS R

) ).

7.11.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the associativity of the joint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∧ ∧ ∀S0 R,b ,∀S0 R,a , ∀Q,

(pwr)

,

(pwr) (pwr)

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∧ ∧ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 (Q ∈ S R ) → S0 R

(pwr) (pwr)

(Sb ∈ S0 R ∧

(S0 R,b ∈

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

→ S0 R

) → (Sa ∈ S0 R

(pwr) S0 R )

∧

R[S(pwr) ] S0 R →

(pwr) (pwr)

→ (S0 R,a ∈ S0 R

- 108 -

(pwr)

(pwr)

)→

)→

( ∀Q∧ ,(Q∧ ∈ S0 R ) → ( ∃S0 b ,(S0 b ∈ S) ∧ (pwr)

(S0 b ∈ S0 R (S0 b ∈ Sb ) ∧ (Q∧ ∈ S0 b )

)∧

) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R (S0 a ∈ Sa ) ∧ (Q∧ ∈ S0 a )

)∧

) )→ ( ∀Qb ∧ ,(Qb ∧ ∈ S0 R ) → ∧ (Qb ∧ ∈ S0 R,b ) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (pwr)

(S0 b ∈ S0 R )∧ (S0 b ∈ Sb ) ∧ Qb ∧ S[S0 R ∧ ] S0 b ) )→ ( ∀Qa ∧ ,(Qa ∧ ∈ S0 R ) → ∧ (Qa ∧ ∈ S0 R,a ) ↔ ( 0 0 ∃S a ,(S a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R )∧ (S0 a ∈ Sa ) ∧ Qa ∧ S[S0 R ∧ ] S0 a ) )→

- 109 -

( ∧

QS[S0 R ∧ ] S0 R,b ↔ ∧ QS[S0 R ∧ ] S0 R,a ) ).

7.12.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint distributivity of the disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ,∀L(pwr) , ∀Sb ,∀Sa , ∨ ∀L∨ b ,∀La , ∀¯ p, (pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∨ (L∨ b ∈ S) → (La ∈ S) → (¯ p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → (pwr)

(pwr)

(Sb ∈ L(pwr) ) → (Sa ∈ L(pwr) ∨ (pwr) (pwr) (Lb ∈ L ) → (L∨ )→ a ∈L ( ∀¯ p∨ p∨ b ,(¯ b ∈ L) → ∨ (¯ pb ∈ L∨ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ 0 p ¯∨ b S[L∨ ] S b ) )→ - 110 -

)→

( ∀¯ p∨ p∨ a ,(¯ a ∈ L) → ∨ (¯ pa ∈ L∨ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ 0 p ¯∨ a S[L∨ ] S a ) )→ Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( p ¯S[L∨ ] L∨ b ↔ p ¯S[L∨ ] L∨ a ) ).

The second member of any ordered pair {Sb , Sa } of sets of sets will constitute a rearrangement set with respect to the pair’s first member in the case and only in the case that each member of Sb contains one, and only one, member of each of the members of Sa , and every member of the set Scd.te of candidate elements containing one, and only one, member of each of the members of Sa is a member of Sb . 7.12.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint distributivity of the disjoint propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ,∀L(pwr) , ∀Sb ,∀Sa , ∨ ∀L∧ b ,∀La , ∀¯ p, (pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sb ∈ S) → (Sa ∈ S) → - 111 -

∨ (L∧ b ∈ S) → (La ∈ S) → (¯ p ∈ L) →

L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → (pwr)

(pwr)

) → (Sa ∈ L(pwr) (Sb ∈ L(pwr) (pwr) (pwr) (L∧ ) → (L∨ )→ b ∈L a ∈L ( ∀¯ p∧ p∧ b ,(¯ b ∈ L) → ∧ (¯ p∧ b ∈ Lb ) ↔ ( 0 ∃S b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ 0 p ¯∧ b S[L∨ ] S b ) )→ ( ∀¯ p∨ p∨ a ,(¯ a ∈ L) → ∨ (¯ pa ∈ L∨ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ 0 p ¯∨ a S[L∧ ] S a ) )→ Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( p ¯S[L∧ ] L∧ b ↔ p ¯S[L∨ ] L∨ a )

)→

).

7.12.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint distributivity of the joint propositions , given as - 112 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

,∀L(pwr) , ∀L(pwr) ∀Sb ,∀Sa , ∧ ∀L∨ b ,∀La , ∀p, (pwr)

∈ S) → (L(pwr) ∈ S) → (L(pwr) (Sb ∈ S) → (Sa ∈ S) → ∧ (L∨ b ∈ S) → (La ∈ S) → (p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → (pwr)

(pwr)

(Sb ∈ L(pwr) ) → (Sa ∈ L(pwr) ∨ (pwr) (pwr) (Lb ∈ L ) → (L∧ )→ a ∈L ( ∀pb ∨ ,(pb ∨ ∈ L) → (pb ∨ ∈ L∨ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ pb ∨ S[L∧ ] S0 b ) )→ ( ∀pa ∧ ,(pa ∧ ∈ L) → (pa ∧ ∈ L∧ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ pa ∧ S[L∨ ] S0 a ) )→

- 113 -

)→

Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( pS[L∨ ] L∨ b ↔ pS[L∧ ] L∧ a ) ).

7.12.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint distributivity of the joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ,∀L(pwr) , ∀Sb ,∀Sa , ∧ ∀L∧ b ,∀La , ∀p, (pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∧ (L∧ b ∈ S) → (La ∈ S) → (p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → (pwr)

(pwr)

(Sb ∈ L(pwr) ) → (Sa ∈ L(pwr) (pwr) (pwr) (L∧ ∈ L ) → (L∧ )→( b a ∈L ∀pb ∧ ,(pb ∧ ∈ L) → (pb ∧ ∈ L∧ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ L(pwr) ) ∧ (S0 b ∈ Sb ) ∧ pb ∧ S[L∧ ] S0 b ) )→ - 114 -

)→

( ∀pa ∧ ,(pa ∧ ∈ L) → (pa ∧ ∈ L∧ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ L(pwr) ) ∧ (S0 a ∈ Sa ) ∧ pa ∧ S[L∧ ] S0 a ) )→ Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( pS[L∧ ] L∧ b ↔ pS[L∧ ] L∧ a ) ).

7.13.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint distributivity of the disjoint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∨ ∨ ∀S0 R,b ,∀S0 R,a , ¯ ∀ Q,

(pwr)

,

(pwr) (pwr)

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 ¯ (Q ∈ S R ) → S0 R

(pwr) (pwr)

(Sb ∈ S0 R

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

→ S0 R

) → (Sa ∈ S0 R - 115 -

(pwr)

R[S(pwr) ] S0 R →

(pwr) (pwr)

)→

(pwr)

) → (S0 R,a ∈ S0 R (S0 R,b ∈ S0 R ( 0 ¯∨ ¯∨ ∀Q b ,(Qb ∈ S R ) → ∨ ∨ 0 ¯ (Qb ∈ S R,b ) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧

(pwr)

)→

(pwr)

(S0 b ∈ S0 R )∧ (S0 b ∈ Sb ) ∧ 0 ¯∨ Q b S[S0 R ∨ ] S b ) )→ ( 0 ¯ ∨ ,(Q ¯∨ ∀Q a a ∈ S R) → ∨ ∨ 0 ¯ (Q ∈ S R,a ) ↔ ( a

∃S0 a ,(S0 a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R )∧ (S0 a ∈ Sa ) ∧ ¯ ∨ S[S0 ∨ ] S0 a Q a R ) )→ Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( ¯ [S0 ∨ ] S0 R,b ∨ QS R ↔ ¯ [S0 ∨ ] S0 R,a ∨ QS R ) ).

7.13.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint distributivity of the disjoint unary relations , given as

- 116 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∧ ∨ ∀S0 R,b ,∀S0 R,a , ¯ ∀ Q,

(pwr)

,

(pwr) (pwr)

(pwr)

∈ S) → ∈ S) → (S0 R (S0 R (Sb ∈ S) → (Sa ∈ S) → ∧ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → ¯ ∈ S0 R ) → (Q S0 R

(pwr) (pwr)

(Sb ∈ S0 R 0

∧

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

→ S0 R

) → (Sa ∈ S0 R

(pwr) S0 R )

0

∨

(pwr)

(S0 b ∈ S0 R )∧ (S0 b ∈ Sb ) ∧ 0 ¯∧ Q b S[S0 R ∨ ] S b ) )→ ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ∀Q a a ¯ ∨ ∈ S0 R,a ∨ ) ↔ ( (Q a ∃S0 a ,(S0 a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R )∧ (S0 a ∈ Sa ) ∧ ¯ ∨ S[S0 ∧ ] S0 a Q a R ) )→

- 117 -

R[S(pwr) ] S0 R →

(pwr) (pwr)

(S R,b ∈ → (S R,a ∈ S0 R ( 0 ¯∧ ¯∧ ∀Q b ,(Qb ∈ S R ) → ∧ 0 ¯∧ ) ↔( (Q ∈ S R,b b ∃S0 b ,(S0 b ∈ S) ∧

(pwr)

(pwr)

)→

)→

Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( ¯ [S0 ∧ ] S0 R,b ∧ QS R ↔ ¯ [S0 ∨ ] S0 R,a ∨ QS R ) ).

7.13.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint distributivity of the joint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∨ ∧ ∀S0 R,b ,∀S0 R,a , ∀Q,

(pwr)

,

(pwr) (pwr)

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → (Sb ∈ S) → (Sa ∈ S) → ∨ ∧ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 (Q ∈ S R ) → S0 R

(pwr) (pwr)

(Sb ∈ S0 R ∨

(S0 R,b ∈

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

→ S0 R

) → (Sa ∈ S0 R

(pwr) S0 R )

∧

R[S(pwr) ] S0 R →

(pwr) (pwr)

→ (S0 R,a ∈ S0 R

- 118 -

(pwr)

(pwr)

)→

)→

( ∀Qb ∨ ,(Qb ∨ ∈ S0 R ) → ∨ (Qb ∨ ∈ S0 R,b ) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (pwr)

)∧ (S0 b ∈ S0 R (S0 b ∈ Sb ) ∧ Qb ∨ S[S0 R ∧ ] S0 b ) )→ ( ∀Qa ∧ ,(Qa ∧ ∈ S0 R ) → ∧ (Qa ∧ ∈ S0 R,a ) ↔ ( 0 0 ∃S a ,(S a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R )∧ (S0 a ∈ Sa ) ∧ Qa ∧ S[S0 R ∨ ] S0 a ) )→ Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( ∨ QS[S0 R ∨ ] S0 R,b ↔ ∧ QS[S0 R ∧ ] S0 R,a ) ).

7.13.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint distributivity of the joint unary relations , given as

- 119 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∧ ∧ ∀S0 R,b ,∀S0 R,a , ∀Q,

(pwr)

,

(pwr) (pwr)

(pwr)

∈ S) → ∈ S) → (S0 R (S0 R (Sb ∈ S) → (Sa ∈ S) → ∧ ∧ (S0 R,b ∈ S) → (S0 R,a ∈ S) → (Q ∈ S0 R ) → S0 R

(pwr) (pwr)

(Sb ∈ S0 R 0

∧

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

→ S0 R

) → (Sa ∈ S0 R

(pwr) S0 R )

0

∧

(pwr)

) )→ ( ∀Qa ∧ ,(Qa ∧ ∈ S0 R ) → ∧ (Qa ∧ ∈ S0 R,a ) ↔ ( 0 0 ∃S a ,(S a ∈ S) ∧ (pwr)

(S0 a ∈ S0 R )∧ (S0 a ∈ Sa ) ∧ Qa ∧ S[S0 R ∧ ] S0 a ) )→

- 120 -

R[S(pwr) ] S0 R →

(pwr) (pwr)

(S R,b ∈ → (S R,a ∈ S0 R ( ∀Qb ∧ ,(Qb ∧ ∈ S0 R ) → ∧ (Qb ∧ ∈ S0 R,b ) ↔ ( 0 0 ∃S b ,(S b ∈ S) ∧ (S0 b ∈ S0 R )∧ (S0 b ∈ Sb ) ∧ Qb ∧ S[S0 R ∧ ] S0 b

(pwr)

(pwr)

)→

)→

Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( ∧ QS[S0 R ∧ ] S0 R,b ↔ ∧ QS[S0 R ∧ ] S0 R,a ) ).

- 121 -

8. 8.1.1 For any ordered pair {¯ p00 , p ¯0 } of members of set L of a natural language’s propositions the second member of which is a basic composed proposition there will be a member p ¯ of the set constituting a disjunctively conditional proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀¯ p0 , ∃¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) ∧ R[p: b.c c.p.sed prp.s.n] p ¯0 → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 ).

p ¯00 will also be called the disjunctively conditional proposition’s major term, p ¯0 , the disjunctively conditional proposition’s minor term. 8.1.2 For any ordered pair {p00 , p0 } of members of set L of a natural language’s propositions the second member of which is a basic composed proposition there will be a member p of the set constituting a conjunctively conditional proposition with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∃p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) ∧

- 122 -

R[p: b.c c.p.sed prp.s.n] p0 → pS[p: p00 |∧ p0 ] p00 p0 ).

p00 will also be called the conjunctively conditional proposition’s major term, p0 , the conjunctively conditional proposition’s minor term. ¯ 00 , Q ¯ 0 } of members of set S0 R of a natural lan8.2.1 For any ordered pair {Q guage’s unary relations the second member of which is a basic composed unary ¯ of the set constituting a disjunctively condirelation there will be a member Q tional unary relation with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∃ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) ∧ (Q ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q 00 ¯ 0 ¯ ¯ QS[Qx: Q00 x |∨ Q0 x] Q Q ).

¯ 00 will also be called the disjunctively conditional unary relation’s major comQ ¯ 0 , the disjunctively conditional unary relation’s minor component. ponent, Q 8.2.2 For any ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations the second member of which is a basic composed unary relation there will be a member Q of the set constituting a disjunctively conditional unary relation with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the conjunctively conditional unary relations , given as - 123 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ∃Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q0 → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 ).

Q00 will also be called the conjunctively conditional unary relation’s major component, Q0 , the conjunctively conditional unary relation’s minor component. 8.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 , ∀¯ p0 ,∀p0 , ∀p, (p00 ∈ L) → (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( pS[p: p00 |∧ p0 ] p00 p0 ↔ pS[p: p00 |∨ p0 ] p00 p ¯0 ) ).

- 124 -

8.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 , ¯ 0 ,∀Q0 , ∀Q ∀Q, (Q00 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 ↔ ¯0 QS[Qx: Q00 x | Q0 x] Q00 Q ∨

) ).

8.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀¯ p00 ,∀¯ p0 , ¯ p, ∀ Q,∀¯ (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q 00 0 (¯ p ∈ L) → (¯ p ∈ L) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → - 125 -

p ¯00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ 00 → (Ix x ∧ Qx)] Iwcd.te Ix Q 0 p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] Iw Ix Q cd.te

p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q ( ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨ 00 0 p ¯ S[p: p00 |∨ p0 ] p ¯ p ¯ ) ).

8.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the resolution of the conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ∀Q00 ,∀Q0 , ∀p00 ,∀p0 , ∀Q,∀p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (p00 ∈ L) → (p0 ∈ L) → (Q ∈ S0 R ) → (p ∈ L) → p00 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q00 → p0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q0 → pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iwcd.te Ix Q →

- 126 -

( QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 → pS[p: p00 |∧ p0 ] p00 p0 ) ).

8.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀¯ p0 , 000 ∀¯ p ,∀¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p000 ∈ L) → (¯ p ∈ L) → p ¯000 S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( p ¯ S[p: p00 ∨ p0 ] p ¯000 p ¯0 → 00 0 p ¯ S[p: p00 ∨ p0 ] p ¯ p ¯ ) ).

8.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀p000 ,∀p, - 127 -

(p00 ∈ L) → (p0 ∈ L) → (p000 ∈ L) → (p ∈ L) → p000 S[p: p00 |∧ p0 ] p00 p0 → ( pS[p: p00 ∧ p0 ] p000 p0 → pS[p: p00 ∧ p0 ] p00 p0 ) ).

8.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀ Q ¯ 0, ∀Q 000 ¯ ¯ ∀ Q ,∀ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ 000 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 000 ¯ ¯ 00 Q ¯0 → Q S[Qx: Q00 x | Q0 x] Q ∨

( ¯ 000 Q ¯0 → ¯ [Qx: Q00 x ∨ Q0 x] Q QS 00 ¯ 0 ¯ ¯ QS[Qx: Q00 x ∨ Q0 x] Q Q ) ).

8.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively conditional unary relations , given as

- 128 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ∀Q000 ,∀Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q000 ∈ S0 R ) → (Q ∈ S0 R ) → Q000 S[Qx: Q00 x |∧ Q0 x] Q00 Q0 → ( QS[Qx: Q00 x ∧ Q0 x] Q000 Q0 → QS[Qx: Q00 x ∧ Q0 x] Q00 Q0 ) ).

8.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the double conditional disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0b ,∀¯ p0a , 00 ∀¯ p ,∀¯ p0 , ∀¯ p000 ,∀¯ p, (¯ p0b ∈ L) → (¯ p0a ∈ L) → 00 (¯ p ∈ L) → (¯ p0 ∈ L) → (¯ p000 ∈ L) → (¯ p ∈ L) → p ¯0 S[p: p00 ∨ p0 ] p ¯0b p ¯0a → 000 p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0a → ( p ¯ S[p: p00 |∨ p0 ] p ¯000 p ¯0b → 00 0 p ¯ S[p: p00 |∨ p0 ] p ¯ p ¯ ) ). - 129 -

8.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the double conditional conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 b ,∀p0 a , ∀p00 ,∀p0 , ∀p000 ,∀p, (p0 b ∈ L) → (p0 a ∈ L) → (p00 ∈ L) → (p0 ∈ L) → (p000 ∈ L) → (p ∈ L) → p0 S[p: p00 ∧ p0 ] p0 b p0 a → p000 S[p: p00 |∧ p0 ] p00 p0 a → ( pS[p: p00 |∧ p0 ] p000 p0 b → pS[p: p00 |∧ p0 ] p00 p0 ) ).

8.8.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the double conditional disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0b ,∀ Q ¯ 0a , ∀Q 00 ¯ ¯ 0, ∀ Q ,∀ Q ¯ 000 ,∀ Q, ¯ ∀Q ¯ 0b ∈ S0 R ) → (Q ¯ 0a ∈ S0 R ) → (Q ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q - 130 -

¯ ∈ S0 R ) → ¯ 000 ∈ S0 R ) → (Q (Q ¯ 0 S[Qx: Q00 x ∨ Q0 x] Q ¯ 0b Q ¯ 0a → Q 000 00 ¯ 0 ¯ ¯ Q S[Qx: Q00 x | Q0 x] Q Qa → ∨

( ¯ [Qx: Q00 x | Q0 x] Q ¯ 000 Q ¯ 0b → QS ∨ 00 ¯ 0 ¯ ¯ QS[Qx: Q00 x |∨ Q0 x] Q Q ) ).

8.8.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the double conditional conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 b ,∀Q0 a , ∀Q00 ,∀Q0 , ∀Q000 ,∀Q, (Q0 b ∈ S0 R ) → (Q0 a ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q000 ∈ S0 R ) → (Q ∈ S0 R ) → Q0 S[Qx: Q00 x ∧ Q0 x] Q0 b Q0 a → Q000 S[Qx: Q00 x |∧ Q0 x] Q00 Q0 a → ( QS[Qx: Q00 x |∧ Q0 x] Q000 Q0 b → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 ) ).

8.9.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive disjunctivly conditional propositions , given as - 131 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 b ,∀p00 a , ∀¯ p00 ,∀¯ p0 , ∀pb ,∀pa , ∀¯ p, (p00 b ∈ L) → (p00 a ∈ L) → (¯ p00 ∈ L) → (¯ p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → (¯ p ∈ L) → pb S[p: p00 |∨ p0 ] p00 b p ¯0 → pa S[p: p00 |∨ p0 ] p00 a p ¯0 → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( p ¯00 S[p: p00 !∧ p0 ] p00 b p00 a → p ¯ S[p: p00 !∧ p0 ] pb pa ) ).

8.9.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive conjunctivly conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 b ,∀p00 a , ∀¯ p00 ,∀p0 , ∀¯ pb ,∀¯ pa , ∀¯ p, (p00 b ∈ L) → (p00 a ∈ L) → (¯ p00 ∈ L) → (p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → (¯ p ∈ L) → - 132 -

pb S[p: p00 |∧ p0 ] p00 b p0 → pa S[p: p00 |∧ p0 ] p00 a p0 → p ¯ S[p: p00 |∧ p0 ] p ¯00 p0 → ( p ¯00 S[p: p00 !∧ p0 ] p00 b p00 a → pS[p: p00 !∧ p0 ] pb pa ) ).

8.10.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive disjunctivly conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 b ,∀Q00 a , ¯ 00 ,∀ Q ¯ 0, ∀Q ∀Qb ,∀Qa , ¯ ∀ Q, (Q00 b ∈ S0 R ) → (Q00 a ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯0 → Qb S[Qx: Q00 x |∨ Q0 x] Q00 b Q 00 ¯ 0 Qa S[Qx: Q00 x |∨ Q0 x] Q a Q → ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨ ( ¯ 00 S[Qx: Q00 x !∧ Q0 x] Q00 b Q00 a → Q ¯ [Qx: Q00 x !∧ Q0 x] Qb Qa QS ) ).

- 133 -

8.10.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive conjunctivly conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 b ,∀Q00 a , ¯ 00 ,∀Q0 , ∀Q ∀Qb ,∀Qa , ¯ ∀ Q, (Q00 b ∈ S0 R ) → (Q00 a ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ¯ ∈ S0 R ) → (Q Qb S[Qx: Q00 x |∧ Q0 x] Q00 b Q0 → Qa S[Qx: Q00 x |∧ Q0 x] Q00 a Q0 → ¯ 00 Q0 → QS[Qx: Q00 x |∧ Q0 x] Q ( ¯ 00 S[Qx: Q00 x !∧ Q0 x] Q00 b Q00 a → Q ¯ [Qx: Q00 x !∧ Q0 x] Qb Qa QS ) ).

8.11.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctivly conditional propositions’ complements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀p00 , ∀¯ p0 , ∀¯ p,∀p, (¯ p00 ∈ L) → (p00 ∈ L) → (¯ p0 ∈ L) → - 134 -

(¯ p ∈ L) → (p ∈ L) → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → 00 0 pS[p: p00 |∨ p0 ] p p ¯ → ( p ¯00 Rp¯ p00 → p ¯ Rp¯ p ) ).

8.11.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctivly conditional propositions’ complements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀p00 , ∀p0 , ∀¯ p,∀p, (¯ p00 ∈ L) → (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → (p ∈ L) → p ¯ S[p: p00 |∧ p0 ] p ¯00 p0 → pS[p: p00 |∧ p0 ] p00 p0 → ( p ¯00 Rp¯ p00 → p ¯ Rp¯ p ) ).

8.12.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctivly conditional unary relations’ complements , given as - 135 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀Q00 , ∀Q ¯ ∀ Q0 , ¯ ∀ Q,∀Q, ¯ 00 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨ 00 ¯ 0 QS[Qx: Q00 x | Q0 x] Q Q → ∨

( 00 ¯ 00 RQ Q ¯Q → ¯ QRQ ¯Q

) ).

8.12.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctivly conditional unary relations’ complements , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀Q00 , ∀Q ∀Q0 , ¯ ∀ Q,∀Q, ¯ 00 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q0 → QS ∧ QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 →

- 136 -

( 00 ¯ 00 RQ Q ¯Q → ¯ QRQ ¯Q

) ).

8.13.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ∀L(pwr) ,∀L∨ b ,∀La , ∀¯ p00 ,∀¯ p0 , ∀¯ p, ∨ (L(pwr) ∈ S) → (L∨ b ∈ S) → (La ∈ S) → (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) → (pwr) (pwr) L(pwr) R[S(pwr) ] L → (L∨ ) → (L∨ )→( b ∈L a ∈L

∀¯ p∨ ,(¯ p∨ ∈ L) → (¯ p∨ ∈ L∨ b) ↔ ( ∃¯ p00∨ ,(¯ p00∨ ∈ L) ∧ (¯ p00∨ ∈ L∨ a)∧ p ¯∨ S[p: p00 |∨ p0 ] p ¯00∨ p ¯0 ) )→ R[p: b.c c.p.sed prp.s.n] p ¯0 → 00 0 p ¯ S[p: p00 |∨ p0 ] p ¯ p ¯ → ( p ¯00 S[L∨ ] L∨ a → p ¯S[L∨ ] L∨ b ) ). - 137 -

8.13.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ∀L(pwr) ,∀L∧ b ,∀La , ∀¯ p00 ,∀¯ p0 , ∀¯ p, ∧ (L(pwr) ∈ S) → (L∧ b ∈ S) → (La ∈ S) → 00 0 (¯ p ∈ L) → (¯ p ∈ L) → (¯ p ∈ L) → (pwr) (pwr) L(pwr) R[S(pwr) ] L → (L∧ ) → (L∧ )→ b ∈L a ∈L

( ∀¯ p∧ ,(¯ p∧ ∈ L) → ∧ (¯ p ∈ L∧ b) ↔ ( ∃¯ p00∧ ,(¯ p00∧ ∈ L) ∧ (¯ p00∧ ∈ L∧ a)∧ p ¯∧ S[p: p00 |∨ p0 ] p ¯00∧ p ¯0 ) )→ R[p: b.c c.p.sed prp.s.n] p ¯0 → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( p ¯00 S[L∧ ] L∧ a → p ¯S[L∧ ] L∧ b ) ).

8.13.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint conjunctively conditional propositions , given as - 138 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨ ∀L(pwr) ,∀L∨ b ,∀La , ∀p00 ,∀p0 , ∀p, ∨ (L(pwr) ∈ S) → (L∨ b ∈ S) → (La ∈ S) → (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) → (pwr) (pwr) L(pwr) R[S(pwr) ] L → (L∨ ) → (L∨ )→ b ∈L a ∈L

( ∀p∨ ,(p∨ ∈ L) → (p∨ ∈ L∨ b) ↔ ( ∃p00 ∨ ,(p00 ∨ ∈ L) ∧ (p00 ∨ ∈ L∨ a)∧ p∨ S[p: p00 |∧ p0 ] p00 ∨ p0 ) )→ R[p: b.c c.p.sed prp.s.n] p0 → pS[p: p00 |∧ p0 ] p00 p0 → ( p00 S[L∨ ] L∨ a → pS[L∨ ] L∨ b ) ).

8.13.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint conjunctively conditional propositions , given as

- 139 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧ ∀L(pwr) ,∀L∧ b ,∀La , ∀p00 ,∀p0 , ∀p, ∧ (L(pwr) ∈ S) → (L∧ b ∈ S) → (La ∈ S) → (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) → (pwr) (pwr) L(pwr) R[S(pwr) ] L → (L∧ ) → (L∧ )→ b ∈L a ∈L

( ∀p∧ ,(p∧ ∈ L) → (p∧ ∈ L∧ b) ↔ ( ∃p00 ∧ ,(p00 ∧ ∈ L) ∧ (p00 ∧ ∈ L∧ a)∧ p∧ S[p: p00 |∧ p0 ] p00 ∧ p0 ) )→ R[p: b.c c.p.sed prp.s.n] p0 → pS[p: p00 |∧ p0 ] p00 p0 → ( p00 S[L∧ ] L∧ a → pS[L∧ ] L∧ b ) ).

8.14.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint disjunctively conditional unary relations , given as

- 140 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∨

(pwr)

∨

,∀S0 R,b ,∀S0 R,a , ∀S0 R ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∀ Q, ∨

∨

(pwr)

∈ S) → (S0 R,b ∈ S) → (S0 R,a ∈ S) → (S0 R 00 0 ¯ 0 ∈ S0 R ) → ¯ (Q ∈ S R ) → (Q ¯ ∈ S0 R ) → (Q S0 R

(pwr)

∨

R[S(pwr) ] S0 R → (S0 R,b ∈ S0 R

(pwr)

∨

) → (S0 R,a ∈ S0 R

(pwr)

)→

( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ∀Q ∨ ∨ ¯ (Q ∈ S0 R,b ) ↔ ( ¯ 00 ,(Q ¯ 00 ∈ S0 R ) ∧ ∃Q ∨ ∨ ¯ 00 ∈ S0 R,a ∨ ) ∧ (Q ∨ ¯ ∨ S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 Q ∨

∨

) )→ ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ¯ 00 Q ¯0 → ¯ [Qx: Q00 x | Q0 x] Q QS ∨

( ¯ 00 S[S0 ∨ ] S0 R,a ∨ → Q R ¯ [S0 ∨ ] S0 R,b ∨ QS R

) ).

8.14.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint disjunctively conditional unary relations , given as

- 141 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∧

(pwr)

∧

,∀S0 R,b ,∀S0 R,a , ∀S0 R ∀Q00 ,∀Q0 , ∀Q, ∧

∧

(pwr)

∈ S) → (S0 R,b ∈ S) → (S0 R,a ∈ S) → (S0 R 00 0 (Q ∈ S R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → S0 R

(pwr)

∧

R[S(pwr) ] S0 R → (S0 R,b ∈ S0 R

(pwr)

∧

) → (S0 R,a ∈ S0 R

(pwr)

)→

( ¯ ∧ ,(Q ¯ ∧ ∈ S0 R ) → ∀Q ∧ ∧ ¯ (Q ∈ S0 R,b ) ↔ ( ¯ 00 ,(Q ¯ 00 ∈ S0 R ) ∧ ∃Q ∧ ∧ ¯ 00 ∈ S0 R,a ∧ ) ∧ (Q ∧ ¯ ∧ S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 Q ∨

∧

) )→ ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ¯ 00 Q ¯0 → ¯ [Qx: Q00 x | Q0 x] Q QS ∨

( ¯ 00 S[S0 ∧ ] S0 R,a ∧ → Q R ¯ [S0 ∧ ] S0 R,b ∧ QS R

) ).

8.14.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjoint conjunctively conditional unary relations , given as

- 142 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∨

∨

,∀S0 R,b ,∀S0 R,a , ∀S0 R ∀Q00 ,∀Q0 , ∀Q, ∨

∨

(pwr)

∈ S) → (S0 R,b ∈ S) → (S0 R,a ∈ S) → (S0 R 00 0 (Q ∈ S R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → S0 R

(pwr)

∨

R[S(pwr) ] S0 R → (S0 R,b ∈ S0 R

(pwr)

∨

) → (S0 R,a ∈ S0 R

(pwr)

)→

( ∀Q∨ ,(Q∨ ∈ S0 R ) → ∨ (Q∨ ∈ S0 R,b ) ↔ ( ∃Q00 ∨ ,(Q00 ∨ ∈ S0 R ) ∧ ∨ (Q00 ∨ ∈ S0 R,a ) ∧ ∨ Q S[Qx: Q00 x |∧ Q0 x] Q00 ∨ Q0 ) )→ R[Q: b.c c.p.sed u.ry r.l.n] Q0 → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 → ( ∨ Q00 S[S0 R ∨ ] S0 R,a → QS[S0 R ∨ ] S0 R,b

∨

) ).

8.14.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint conjunctively conditional unary relations , given as

- 143 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∧

∧

,∀S0 R,b ,∀S0 R,a , ∀S0 R ∀Q00 ,∀Q0 , ∀Q, ∧

∧

(pwr)

∈ S) → (S0 R,b ∈ S) → (S0 R,a ∈ S) → (S0 R 00 0 (Q ∈ S R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → S0 R

(pwr)

∧

R[S(pwr) ] S0 R → (S0 R,b ∈ S0 R

(pwr)

∧

) → (S0 R,a ∈ S0 R

(pwr)

)→

( ∀Q∧ ,(Q∧ ∈ S0 R ) → ∧ (Q∧ ∈ S0 R,b ) ↔ ( ∃Q00 ∧ ,(Q00 ∧ ∈ S0 R ) ∧ ∧ (Q00 ∧ ∈ S0 R,a ) ∧ ∧ Q S[Qx: Q00 x |∧ Q0 x] Q00 ∧ Q0 ) )→ R[Q: b.c c.p.sed u.ry r.l.n] Q0 → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 → ( ∧ Q00 S[S0 R ∧ ] S0 R,a → QS[S0 R ∧ ] S0 R,b

∧

) ).

8.15.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively conditioned disjunctively conditional propositions , given as

- 144 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 , ∀¯ p0b ,∀¯ p0a , ∀¯ p0 , ∀¯ p000 ,∀¯ p000 a , ∀¯ p, (¯ p00 ∈ L) → (¯ p0b ∈ L) → (¯ p0a ∈ L) → (¯ p0 ∈ L) → (¯ p000 ∈ L) → (¯ p000 a ∈ L) → (¯ p ∈ L) → p ¯0 S[p: p00 ∨ p0 ] p ¯0b p ¯0a → 000 p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → 00 0 00 0 p ¯000 S p ¯ p ¯ [p: p |∨ p ] a a → ( R[p: b.c c.p.sed prp.s.n] p ¯0 ∧ 0 ¯ ∨ p R [p ] ¯ F

)→ ( p ¯ S[p: p00 ∨ p0 ] p ¯0b p ¯000 a → p ¯ S[p: p00 ∨ p0 ] p ¯0b p ¯000 ) ).

8.15.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively conditioned conjunctively conditional propositions , given as

- 145 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 , ∀p0 b ,∀p0 a , ∀p0 , ∀p000 ,∀p000 a , ∀p, (p00 ∈ L) → (p0 b ∈ L) → (p0 a ∈ L) → (p0 ∈ L) → (p000 ∈ L) → (p000 a ∈ L) → (p ∈ L) → p0 S[p: p00 ∧ p0 ] p0 b p0 a → p000 S[p: p00 |∧ p0 ] p00 p0 → p000 a S[p: p00 |∧ p0 ] p00 p0 a → ( R[p: b.c c.p.sed prp.s.n] p0 ∧ ¯ ∧ p0 R [p¯ ] F

)→ ( pS[p: p00 ∧ p0 ] p0 b p000 a → pS[p: p00 ∧ p0 ] p0 b p000 ) ).

8.16.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively conditioned disjunctively conditional unary relations , given as

- 146 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 , ∀Q ¯ 0b ,∀ Q ¯ 0a , ∀Q ¯ 0, ∀Q ¯ 000 ,∀ Q ¯ 000 ∀Q a , ¯ ∀ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0b ∈ S0 R ) → (Q ¯ 0a ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q 0 ¯ 000 ∈ S0 R ) → (Q ¯ 000 (Q a ∈ S R) → ¯ ∈ S0 R ) → (Q ¯ 0 S[Qx: Q00 x ∨ Q0 x] Q ¯ 0b Q ¯ 0a → Q 00 ¯ 0 000 ¯ ¯ Q S[Qx: Q00 x |∨ Q0 x] Q Q → ¯ 000 ¯ 00 ¯ 0 Q a S[Qx: Q00 x | Q0 x] Q Qa → ∨

( ¯0 R[Q: b.c c.p.sed u.ry r.l.n] Q ∧ ¯ ∨ Q ¯0 R [Q ] F

)→ ( ¯ 0b Q ¯ 000 ¯ [Qx: Q00 x ∨ Q0 x] Q QS a → ¯ [Qx: Q00 x ∨ Q0 x] Q ¯ 0b Q ¯ 000 QS ) ).

8.16.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively conditioned conjunctively conditional unary relations , given as

- 147 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 , ∀Q0 b ,∀Q0 a , ∀Q0 , ∀Q000 ,∀Q000 a , ∀Q, (Q00 ∈ S0 R ) → (Q0 b ∈ S0 R ) → (Q0 a ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q000 ∈ S0 R ) → (Q000 a ∈ S0 R ) → (Q ∈ S0 R ) → Q0 S[Qx: Q00 x ∧ Q0 x] Q0 b Q0 a → Q000 S[Qx: Q00 x |∧ Q0 x] Q00 Q0 → Q000 a S[Qx: Q00 x |∧ Q0 x] Q00 Q0 a → ( R[Q: b.c c.p.sed u.ry r.l.n] Q0 ∧ ¯ ¯ ∧ Q0 R [Q ] F

)→ ( QS[Qx: Q00 x ∧ Q0 x] Q0 b Q000 a → QS[Qx: Q00 x ∧ Q0 x] Q0 b Q000 ) ).

8.17.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional disjunctively conditional propositions , given as

- 148 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p∨ T, (¯ p0 ∈ L) → (¯ p∨ T ∈ L) → 0 ¯ R[p∨ ] p ¯ → F

( p ¯∨ ¯0 p ¯0 → T S[p: p00 |∨ p0 ] p R[p¯∨ ] p ¯∨ T T

) ).

8.17.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p0 , ∀p∨ F, (¯ p0 ∈ L) → (p0 ∈ L) → (p∨ F ∈ L) → p ¯0 Rp¯ p0 → ¯ ∨ p0 → R [p¯ ] T

( 0 0 p∨ ¯ → F S[p: p00 |∨ p0 ] p p R[p∨ ] p∨ F F

) ).

- 149 -

8.17.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p0 , ∀p∧ T, (¯ p0 ∈ L) → (p0 ∈ L) → (p∧ T ∈ L) → p ¯0 Rp¯ p0 → 0 ¯ ∧ p R [p ] ¯ → T

( p ¯∧ ¯0 p0 → F S[p: p00 |∧ p0 ] p ∧ R[p¯∧ ] p ¯F F

) ).

8.17.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p∧ T, (p0 ∈ L) → (p∧ T ∈ L) → ¯ ∧ p0 → R [p¯ ] F

( 0 0 p∧ T S[p: p00 |∧ p0 ] p p → R[p∧ ] p∧ T T

) ). - 150 -

8.18.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional disjunctively conditional unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀ Q ¯∨ ∀Q T, 0 0 ¯∨ ¯ (Q ∈ S0 R ) → (Q T ∈ S R) → ¯ ∨ Q ¯0 → R [Q ] F

( ¯∨ ¯0 ¯0 Q T S[Qx: Q00 x |∨ Q0 x] Q Q → ∨ ¯ R[Q ¯ ∨ ] QT T

) ).

8.18.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional disjunctively conditional unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀Q0 ,∀ Q ¯∨ ∀Q T, 0 ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯∨ (Q T ∈ S R) → 0 0 ¯ Q RQ ¯Q → ¯ ¯ ∨ Q0 → ( R [QT ] 0 ¯0 Q∨ F S[Qx: Q00 x |∨ Q0 x] Q Q → R[Q∨ ] Q∨ F F

) ).

- 151 -

8.18.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀Q0 , ∀Q ∀Q∧ T, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q 0 (Q∧ T ∈ S R) → ¯ ∧ Q ¯0 → R [Q ] T

( ¯0 0 ¯∧ Q F S[Qx: Q00 x |∧ Q0 x] Q Q → ¯∧ R[Q ¯ ∧ ] QF F

) ).

8.18.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the self conditional conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,∀Q∧ T, 0 (Q0 ∈ S0 R ) → (Q∧ T ∈ S R) → ¯ ¯ ∧ Q0 → R [Q ] F

( 0 0 Q∧ T S[Qx: Q00 x |∧ Q0 x] Q Q → R[Q∧ ] Q∧ T T

) ).

- 152 -

8.19.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic disjunctively forbidden disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p∨ p0 , F ,∀¯ ∨ (pF ∈ L) → (¯ p0 ∈ L) → R[p: b.c c.p.sed prp.s.n] p ¯0 → ( R[p∨ ] p∨ F → F

∨ 0 p∨ ¯ F S[p: p00 |∨ p0 ] pF p

) ).

8.19.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic disjunctively necessary disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p∨ p0 , T ,∀¯ ∨ (¯ pT ∈ L) → (¯ p0 ∈ L) → R[p: b.c c.p.sed prp.s.n] p ¯0 → ( R[p¯∨ ] p ¯∨ T → T

p ¯∨ ¯∨ ¯0 T S[p: p00 |∨ p0 ] p Tp ) ).

- 153 -

8.19.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic conjunctively forbidden conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( 0 ∀¯ p∧ F ,∀p , ∧ (¯ pF ∈ L) → (p0 ∈ L) → R[p: b.c c.p.sed prp.s.n] p0 → ( R[p¯∧ ] p ¯∧ F → F

0 p ¯∧ ¯∧ F S[p: p00 |∧ p0 ] p Fp

) ).

8.19.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic conjunctively necessary conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( 0 ∀p∧ T ,∀p , ∧ (pT ∈ L) → (p0 ∈ L) → R[p: b.c c.p.sed prp.s.n] p0 → ( R[p∧ ] p∧ T → T

∧ 0 p∧ T S[p: p00 |∧ p0 ] pT p

) ).

- 154 -

8.20.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic disjunctively forbidden disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯0 ∀Q∨ F ,∀ Q , ∨ ¯ 0 ∈ S0 R ) → (QF ∈ S0 R ) → (Q ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ( R[Q∨ ] Q∨ F → F

∨ ¯0 Q∨ F S[Qx: Q00 x |∨ Q0 x] QF Q

) ).

8.20.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic disjunctively necessary disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯∨ ¯0 ∀Q T ,∀ Q , 0 0 ¯∨ ¯0 (Q T ∈ S R ) → (Q ∈ S R ) → ¯ R[Q: b.c c.p.sed u.ry r.l.n] Q0 → ( ¯∨ R[Q ¯ ∨ ] QT → T

¯∨ ¯∨ ¯0 Q T S[Qx: Q00 x |∨ Q0 x] QT Q ) ).

- 155 -

8.20.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic conjunctively forbidden conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( 0 ¯∧ ∀Q F ,∀Q , ∧ 0 ¯ (QF ∈ S R ) → (Q0 ∈ S0 R ) → R[Q: b.c c.p.sed u.ry r.l.n] Q0 → ( ¯∧ R[Q ¯ ∧ ] QF → F

¯∧ 0 ¯∧ Q F S[Qx: Q00 x |∧ Q0 x] QF Q ) ).

8.20.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic conjunctively necessary conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( 0 ∀Q∧ T ,∀Q , 0 0 0 (Q∧ T ∈ S R ) → (Q ∈ S R ) → R[Q: b.c c.p.sed u.ry r.l.n] Q0 → ( R[Q∧ ] Q∧ T → T

∧ 0 Q∧ T S[Qx: Q00 x |∧ Q0 x] QT Q

) ).

- 156 -

9. 9.1.1 Any member p of set L of a natural language’s propositions will constitute an analytic proposition in the case and only in the case that it constitutes a basic conjunctively necessary proposition or a basic conjunctively forbidden proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the analytic propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → R[p: a.l.c prp.s.n] p ↔ ( R[p∧ ] p T

∨ R[p¯∧ ] p F

) ).

9.2.1 Any member p of set L of a natural language’s propositions will constitute a synthetic proposition in the case and only in the case that it does not constitute an analytic proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the synthetic propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → ( R[p: s.th.c prp.s.n] p ↔ ¯ [p: a.l.c prp.s.n] p R ) ).

- 157 -

9.3.1 Any member Q of set S0 R of a natural language’s unary relations will constitute an analytic unary relation in the case and only in the case that it constitutes a basic conjunctively necessary unary relation or a basic conjunctively forbidden unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the analytic unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → R[Q: a.l.c u.ry r.l.n] Q ↔ ( R[Q∧ ] Q T

∨ R[Q ¯ ∧] Q F

) ).

9.4.1 Any member Q of the set S0 R of a natural language’s unary relations will constitute a synthetic unary relation in the case and only in the case that it does not constitute an analytic unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the synthetic unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( R[Q: s.th.c u.ry r.l.n] Q ↔ ¯ [Q: a.l.c u.ry r.l.n] Q R ) ).

- 158 -

9.5.1 Any member p of set L of a natural language’s propositions will constitute a primary proposition or a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the propositions’ primary or secondary composition , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → R[p: a.l.c prp.s.n] p ↔ ( R[p: pr.ry prp.s.n] p ∨ R[p: s.cd.ry prp.s.n] p ) ).

9.5.2 Any member p of set L of a natural language’s propositions, in the case that it constitutes a primary proposition and a secondary proposition, will constitute an analytic proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary and secondary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → ( R[p: pr.ry prp.s.n] p ∧ R[p: s.cd.ry prp.s.n] p )→ R[p: a.l.c prp.s.n] p ).

- 159 -

9.6.1 Any member Q of set S0 R of a natural language’s unary relations will constitute a primary unary relation or a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the unary relations’ primary or secondary composition , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( R[Q: pr.ry u.ry r.l.n] Q ∨ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

9.6.2 Any member Q of set S0 R of a natural language’s unary relations, in the case that it constitutes a primary unary relation and a secondary unary relation, will constitute an analytic unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary and secondary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( R[Q: pr.ry u.ry r.l.n] Q ∧ R[Q: s.cd.ry u.ry r.l.n] Q )→ R[Q: a.l.c u.ry r.l.n] Q ).

- 160 -

9.7.1 Any member p of set L of a natural language’s propositions will constitute an a priori proposition in the case and only in the case that it constitutes a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the a priori propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → ( R[p: a pr. prp.s.n] p ↔ R[p: pr.ry prp.s.n] p ) ).

9.7.2 Any member p of set L of a natural language’s propositions will constitute an a posteriori proposition in the case and only in the case that it constitutes a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the a posteriori propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → ( R[p: a p.st prp.s.n] p ↔ R[p: s.cd.ry prp.s.n] p ) ).

- 161 -

9.8.1 Any member Q of set S0 R of a natural language’s unary relations will constitute an a priori unary relation in the case and only in the case that it constitutes a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the a priori unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( R[Q: a pr. u.ry r.l.n] Q ↔ R[Q: pr.ry u.ry r.l.n] Q ) ).

9.8.2 Any member Q of set S0 R of a natural language’s unary relations will constitute an a posteriori unary relation in the case and only in the case that it constitutes a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the a posteriori unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( R[Q: a p.st u.ry r.l.n] Q ↔ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

- 162 -

9.9.1 In the case that the members of ordered pair {p00 , p0 } of members of set L of a natural language’s propositions constitute primary propositions, member p ¯ of the set constituting a negatively conjunctive proposition with respect to the pair will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary negatively conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( ( R[p: pr.ry prp.s.n] p00 ∧ R[p: pr.ry prp.s.n] p0 )→ R[p: pr.ry prp.s.n] p ¯ ) ).

9.9.2 In the case that either of the members of ordered pair {p00 , p0 } of members of set L of a natural language’s propositions does not constitute a primary proposition, member p ¯ of the set constituting a negatively conjunctive proposition with respect to the pair will constitute a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive secondary propositions , given as

- 163 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( ( ¯ [p: pr.ry prp.s.n] p00 R ∨ ¯ [p: pr.ry prp.s.n] p0 R )→ R[p: s.cd.ry prp.s.n] p ¯ ) ).

9.10.1 In the case that the members of ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations constitute primary unary relations, ¯ of the set constituting a negatively conjunctive unary relation with member Q respect to the pair will constitute a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive primary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS

- 164 -

( ( R[Q: pr.ry u.ry r.l.n] Q00 ∧ R[Q: pr.ry u.ry r.l.n] Q0 )→ ¯ R[Q: pr.ry u.ry r.l.n] Q ) ).

9.10.2 In the case that either of the members of ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations does not constitute a ¯ of the set constituting a negatively conjuncprimary unary relation, member Q tive unary relation with respect to the pair will constitute a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the negatively conjunctive secondary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS ( ( ¯ [Q: pr.ry u.ry r.l.n] Q00 R ∨ ¯ [Q: pr.ry u.ry r.l.n] Q0 R )→ ¯ R[Q: s.cd.ry u.ry r.l.n] Q ) ). - 165 -

9.11.1 In the case that all members of subset L∨ of set L of a natural language’s propositions constitute primary propositions, member p ¯ of the set constituting a disjoint proposition with respect to L∨ will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ → ( ( ∀¯ p∨ , (¯ p∨ ∈ L) → (¯ p∨ ∈ L∨ ) → R[p: pr.ry prp.s.n] p ¯∨ )→ R[p: pr.ry prp.s.n] p ¯ ) ).

9.11.2 In the case that all members of subset L∧ of set L of a natural language’s propositions constitute primary propositions, member p of the set constituting a joint proposition with respect to L∧ will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary joint propositions , given as - 166 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → pS[L∧ ] L∧ → ( ( ∀p∧ , (p∧ ∈ L) → (p∧ ∈ L∧ ) → R[p: pr.ry prp.s.n] p∧ )→ R[p: pr.ry prp.s.n] p ) ).

9.11.3 In the case that there is a member of subset L∨ of set L of a natural language’s propositions not constituting a primary proposition, member p ¯ of the set constituting a disjoint proposition with respect to L∨ will constitute a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary disjoint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → - 167 -

L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ → ( ( ∃¯ p∨ , (¯ p∨ ∈ L) ∧ (¯ p∨ ∈ L∨ ) ∧ ¯ [p: pr.ry prp.s.n] p R ¯∨ )→ R[p: s.cd.ry prp.s.n] p ¯ ) ).

9.11.4 In the case that there is a member of subset L∧ of set L of a natural language’s propositions not constituting a primary proposition, member p of the set constituting a joint proposition with respect to L∧ will constitute a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary joint propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → pS[L∧ ] L∧ →

- 168 -

( ( ∃p∧ , (p∧ ∈ L) ∧ (p∧ ∈ L∧ ) ∧ ¯ [p: pr.ry prp.s.n] p∧ R )→ R[p: s.cd.ry prp.s.n] p ) ).

∨

9.12.1 In the case that all members of subset S0 R of set S0 R of a natural ¯ of the language’s unary relations constitute primary unary relations, member Q ∨ set constituting a disjoint unary relation with respect to S0 R will constitute a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary disjoint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) ∨ ¯ ,∀S0 R ,∀ Q, ∨ 0 (pwr) ¯ ∈ S0 R ) (S R ∈ S) → (S0 R ∈ S) → (Q (pwr) 0 (pwr) 0 0 ∨ SR R[S(pwr) ] S R → (S R ∈ S0 R )

∀S0 R

¯ [S0 ∨ ] S0 R ∨ → ( QS R ( ¯ ∨, ∀Q ¯ ∨ ∈ S0 R ) → (Q ¯ ∨ ∈ S0 R ∨ ) → (Q ¯∨ R[Q: pr.ry u.ry r.l.n] Q )→ ¯ R[Q: pr.ry u.ry r.l.n] Q ) ). - 169 -

→ →

∧

9.12.2 In the case that all members of subset S0 R of set S0 R of a natural language’s unary relations constitute primary unary relations, member Q of the ∧ set constituting a joint unary relation with respect to S0 R will constitute a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary joint unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ∀Q,

(pwr)

∧

,∀S0 R , ∧

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → (Q ∈ S0 R ) → S0 R

(pwr)

∧ (S0 R

R[S(pwr) ] S0 R → (pwr)

∈ S0 R )→ ∧ QS[S0 R ∧ ] S0 R → ( ( ∀Q∧ , (Q∧ ∈ S0 R ) → ∧ (Q∧ ∈ S0 R ) → R[Q: pr.ry u.ry r.l.n] Q∧ )→ R[Q: pr.ry u.ry r.l.n] Q ) ).

∨

9.12.3 In the case that there is a member of subset S0 R of set S0 R of a natural language’s unary relations not constituting a primary unary relation, member ¯ of the set constituting a disjoint unary relation with respect to S0 R ∨ will Q constitute a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary disjoint unary relations , given as - 170 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ¯ ∀ Q,

(pwr)

∨

,∀S0 R , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ ∈ S0 R ) → (Q S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(pwr)

)→ (S0 R ∈ S0 R ¯ [S0 ∨ ] S0 R ∨ → QS R ( ( ¯ ∨, ∃Q ¯ (Q∨ ∈ S0 R ) ∧ ¯ ∨ ∈ S0 R ∨ ) ∧ (Q ¯ [Q: pr.ry u.ry r.l.n] Q ¯∨ R )→ ¯ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

∧

9.12.4 In the case that there is a member of subset S0 R of set S0 R of a natural language’s unary relations not constituting a primary unary relation, ∧ member Q of the set constituting a joint unary relation with respect to S0 R will constitute a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary joint unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ∀Q, (S0 R

(pwr)

(pwr)

∧

,∀S0 R , ∧

∈ S) → (S0 R ∈ S) → - 171 -

(Q ∈ S0 R ) → S0 R

(pwr)

∧ (S0 R

R[S(pwr) ] S0 R → (pwr)

∈ S0 R )→ ∧ QS[S0 R ∧ ] S0 R → ( ( ∃Q∧ , (Q∧ ∈ S0 R ) ∧ ∧ (Q∧ ∈ S0 R ) ∧ ¯ R[Q: pr.ry u.ry r.l.n] Q∧ )→ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

9.13.1 In the case that the members of an ordered pair {¯ p00 , p ¯0 } of members of set L of a natural language’s propositions constitute primary propositions, member p ¯ of the set constituting a disjunctively conditional proposition with respect to the pair will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀¯ p0 , ∀¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 →

- 172 -

( ( R[p: pr.ry prp.s.n] p ¯00 ∧ R[p: pr.ry prp.s.n] p ¯0 )→ R[p: pr.ry prp.s.n] p ¯ ) ).

9.13.2 In the case that either of the members of an ordered pair {¯ p00 , p ¯0 } of members of set L of a natural language’s propositions does not constitute a primary proposition, member p ¯ of the set constituting a disjunctively conditional proposition with respect to the pair will constitute a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary disjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p00 ,∀¯ p0 , ∀¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( ( ¯ [p: pr.ry prp.s.n] p R ¯00 ∨ ¯ [p: pr.ry prp.s.n] p R ¯0 )→ R[p: s.cd.ry prp.s.n] p ¯ ) ). - 173 -

9.14.1 In the case that the members of an ordered pair {p00 , p0 } of members of set L of a natural language’s propositions constitute primary propositions, member p of the set constituting a conjunctively conditional proposition with respect to the pair will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary conjunctively conditional propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) → pS[p: p00 |∧ p0 ] p00 p0 → ( ( R[p: pr.ry prp.s.n] p00 ∧ R[p: pr.ry prp.s.n] p0 )→ R[p: pr.ry prp.s.n] p ) ).

9.14.2 In the case that the members of an ordered pair {p00 , p0 } of members of set L of a natural language’s propositions constitute primary propositions, member p of the set constituting a conjunctively conditional proposition with respect to the pair will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary conjunctively conditional propositions , given as

- 174 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) → pS[p: p00 |∨ p0 ] p00 p0 → ( ( ¯ [p: pr.ry prp.s.n] p00 R ∨ ¯ [p: pr.ry prp.s.n] p0 R )→ R[p: s.cd.ry prp.s.n] p ) ).

¯ 00 , Q ¯ 0 } of members 9.15.1 In the case that the members of an ordered pair {Q of set S0 R of a natural language’s unary relations constitute primary unary ¯ of the set constituting a disjunctively conditional unary relations, member Q relation with respect to the pair will constitute a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∀ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨

- 175 -

( ( ¯ 00 R[Q: pr.ry u.ry r.l.n] Q ∧ ¯0 R[Q: pr.ry u.ry r.l.n] Q )→ ¯ R[Q: pr.ry u.ry r.l.n] Q ) ).

¯ 00 , Q ¯ 0 } of 9.15.2 In the case that either of the members of an ordered pair {Q members of set S0 R of a natural language’s unary relations does not constitute ¯ of the set constituting a disjunctively a primary unary relation, member Q conditional unary relation with respect to the pair will constitute a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary disjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∀ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨

( ( ¯ [Q: pr.ry u.ry r.l.n] Q ¯ 00 R ∨ ¯ [Q: pr.ry u.ry r.l.n] Q ¯0 R )→ ¯ R[Q: s.cd.ry u.ry r.l.n] Q ) ). - 176 -

9.15.3 In the case that the members of an ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations constitute primary unary relations, member Q of the set constituting a conjunctively conditional unary relation with respect to the pair will constitute a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary conjunctively conditional unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ∀Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 → ( ( R[Q: pr.ry u.ry r.l.n] Q00 ∧ R[Q: pr.ry u.ry r.l.n] Q0 )→ R[Q: pr.ry u.ry r.l.n] Q ) ).

9.15.4 In the case that either of the members of an ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations does not constitute a primary unary relation, member Q of the set constituting a conjunctively conditional unary relation with respect to the pair will constitute a secondary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary conjunctively conditional unary relations , given as

- 177 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ∀Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 → ( ( ¯ [Q: pr.ry u.ry r.l.n] Q00 R ∨ ¯ [Q: pr.ry u.ry r.l.n] Q0 R )→ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

¯ 9.16.1 In the case that the third member of an ordered triple {Iwcd.te , Ix , Q} of elements the first two members of which are members of set S0 I of a natural language’s particular element identifying unary relations, the third member, a member of set S0 R of the language’s unary relations constitutes a primary unary relation member p ¯ of set L of the language’s propositions constituting a particular world universal, particular element existential normal b.c proposition will constitute a primary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ¯ p, ∀ Q,∀¯ (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → R[Iw ] Iwcd.te → cd.te

- 178 -

p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q ( ¯ → R[Q: pr.ry u.ry r.l.n] Q R[p: pr.ry prp.s.n] p ¯ ) ).

¯ 9.16.2 In the case that the third member of an ordered triple {Iwcd.te , Ix , Q} of elements the first two members of which are members of set S0 I of a natural language’s particular element identifying unary relations, the third member, a member of set S0 R of the language’s unary relations constitutes a secondary unary relation member p ¯ of set L of the language’s propositions constituting a particular world universal, particular element existential normal b.c proposition will constitute a secondary proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary basic propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te ,∀Ix , ¯ p, ∀ Q,∀¯ (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → R[Iw ] Iwcd.te → cd.te

p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q ( ¯ → R[Q: s.cd.ry u.ry r.l.n] Q R[p: s.cd.ry prp.s.n] p ¯ ) ).

- 179 -

9.17.1 Any member pe of set L of a natural language’s propositions constituting an elemental proposition with respect to an ordered pair {Iwcd.te , Ix } of members of set S0 I of the language’s particular element identifying unary relations will constitute a primary proposition in the case and only in the case that it constitutes a normal proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the normal composition of the primary elemental propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q∧ T ,∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → 0 (Q∧ T ∈ S R ) → (p ∈ L) → R[Q∧ ] Q∧ T → T

pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q∧ T → ( R[p: pr.ry prp.s.n] p ↔ R[Iw ] Iw cd.te

) ).

9.18.1 All members of set L of a natural language’s propositions will constitute a posteriori propositions, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the secondary basic propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,(¯ p ∈ L) → R[p: s.cd.ry prp.s.n] p ¯ ).

- 180 -

9.19.1 Any member p ¯ of set L of a natural language’s propositions will constitute an a priori proposition in the case and only in the case that it constitutes an analytic proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the analyticity of the primary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p, (¯ p ∈ L) → ( R[p: pr.ry prp.s.n] p ¯ ↔ R[p: a.l.c prp.s.n] p ¯ ) ).

- 181 -

10 . 10.1.1 Fao pair {I0 x , Q0 } of members of set S0 R of a nl’s unary relations the first member of which is a member of set S0 I of the language’s particular ele¯ of S0 R constituting ment identifying unary relations there will be a member Q a disjunctively deterministic particular element existential world attribute with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjunctively deterministic generalised element existential world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x ,∀Q0 , ¯ ∃ Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) ∧ (Q ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I QR

x,

0 0 Q0 ] I x Q

).

¯ 0 } of members of set S0 R of a nl’s unary relations the 10.1.2 Fao pair {I0 x , Q first member of which is a member of set S0 I of the language’s particular element identifying unary relations there will be a member Q of S0 R constituting a disjunctively deterministic particular element universal world attribute with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the disjunctively deterministic generalised element universal world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0, ∀I0 x ,∀ Q ∃Q,

- 182 -

¯ 0 ∈ S0 R ) → (I0 x ∈ S0 I ) → (Q (Q ∈ S0 R ) ∧ ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q ).

10.1.3 Fao pair {I0 x , Q0 } of members of set S0 R of a nl’s unary relations the first member of which is a member of set S0 I of the language’s particular element identifying unary relations there will be a member Q of S0 R constituting a conjunctively deterministic particular element existential world attribute with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the conjunctively deterministic generalised element existential world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x ,∀Q0 , ∃Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 ).

¯ 0 } of members of set S0 R of a nl’s unary relations the 10.1.4 Fao pair {I0 x , Q first member of which is a member of set S0 I of the language’s particular ele¯ of S0 R constituting ment identifying unary relations there will be a member Q a conjunctively deterministic particular element universal world attribute with respect to the pair, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the conjunctively deterministic generalised element universal world attributes , given as

- 183 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0, ∀I0 x ,∀ Q ¯ ∃ Q, ¯ 0 ∈ S0 R ) → (I0 x ∈ S0 I ) → (Q (Q ∈ S0 R ) ∧ ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I QR

x,

0 ¯0 Q0 ] I x Q

).

10.2.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the conjunctively deterministic generalised element existential world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ¯ 0 ,∀Q0 , ∀Q ∀Q, (I0 x ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 ↔ ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q ) ).

- 184 -

10.2.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the composition of the conjunctively deterministic generalised element universal world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, (I0 x ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ( ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I QR

x,

0 ¯0 Q0 ] I x Q

↔ ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I QR

x,

0 0 Q0 ] I x Q

) ).

There will be a member of the set of elements of the language underlying a deterministic natural language constituting a derivative deterministic Law of the composition of the disjunctively deterministic generalised element universal world attributes , given as

LR(nt.l, dt.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix , ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q,∀Q, (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → - 185 -

¯ Q QR ¯Q→ ( ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q ↔ ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q0 QR x ) ).

There will be a member of the set of elements of the language underlying a deterministic natural language constituting a derivative deterministic Law of the composition of the conjunctively deterministic generalised element universal world attributes , given as

LR(nt.l, dt.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q,∀Q, (I0 x ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ¯ Q QR ¯Q→ ( ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] I0 x Q ¯0 QR x

↔ QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 ) ).

10.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the distributivity of the deterministic world attributes , given as - 186 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

, ∀S0 R ∧ ∨ ∀S0 R,b ,∀S0 R,a , 0 0 ∀Q x ,∀Q , ¯ ∀ Q, (pwr)

∈ S) → (S0 R ∨ ∧ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 0 0 (Q x ∈ S R ) → (Q ∈ S0 R ) → ¯ ∈ S0 R ) → (Q S0 R 0

(pwr)

R[S(pwr) ] S0 R →

∧

(S R,b ∈ S0 R ∨

(pwr)

)→

(pwr) S0 R )

(S0 R,a ∈ → ( ¯ ∧ ,(Q ¯ ∧ ∈ S0 R ) → ∀Q b b ¯ ∧ ∈ S0 R,b ∧ ) ↔ ( (Q b ∃Qa ∨ ,(Qa ∨ ∈ S0 R ) ∧ ∨ (Qa ∨ ∈ S0 R,a ) ∧ ¯ ∧ R[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Qa ∨ Q0 Q b x ) )→ ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Q0 x Q0 → QR x ( ∨ Q0 x S[S0 R ∨ ] S0 R,a → ¯ [S0 ∧ ] S0 R,b ∧ QS R

) ).

10.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the distributivity of the deterministic world attributes , given as

- 187 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

, ∀S0 R ∨ ∨ ∀S0 R,b ,∀S0 R,a , 0 0 ¯ ∀Q x ,∀ Q , ∀Q, (pwr)

∈ S) → (S0 R ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 0 0 ¯ (Q x ∈ S R ) → (Q ∈ S0 R ) → (Q ∈ S0 R ) → S0 R 0

(pwr) ∨

R[S(pwr) ] S0 R →

(S R,b ∈ S0 R ∨

(pwr)

)→

(pwr) S0 R )

(S0 R,a ∈ → ( ∀Qb ∨ ,(Qb ∨ ∈ S0 R ) → ∨ (Qb ∨ ∈ S0 R,b ) ↔ ( ∨ ∨ ∃Qa ,(Qa ∈ S0 R ) ∧ ∨ (Qa ∨ ∈ S0 R,a ) ∧ ¯0 Qb ∨ R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Qa ∨ Q ) )→ ¯0 → QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Q0 x Q ( ∨ Q0 x S[S0 R ∨ ] S0 R,a → QS[S0 R ∨ ] S0 R,b

∨

) ).

10.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the deterministic world attributes’ conjunctive attribute transfer , given as

- 188 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 x ,∀Q0 , 000 ∀Q∧ T ,∀Q , ¯ ∀ Q, (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → 0 000 0 (Q∧ T ∈ S R ) → (Q ∈ S R ) → 0 ¯ (Q ∈ S R ) → R[Q∧ ] Q∧ T → T

Q000 S[Qx: Q00 x ∧ Q0 x] Q0 Q0 x → ( ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I QR

x,

↔ ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I QR

x,

0 0 Q0 ] Q x Q ∧ 000 Q0 ] QT Q

) ).

10.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the deterministic wold attributes’ disjunctive attribute transfer , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0, ∀Q0 x ,∀ Q 000 ¯ ∀Q∧ ,∀ Q , T ∀Q, ¯ 0 ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q 0 000 ¯ (Q∧ ∈ S ) → ( Q ∈ S0 R ) → R T (Q ∈ S0 R ) → R[Q∧ ] Q∧ T → T

¯ 000 S[Qx: Q0 x → Q00 x] Q0 x Q ¯0 → Q

- 189 -

( ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Q0 x Q ↔ ¯ 000 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Q∧ TQ ) ).

10.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the deterministic world attributes’ conjunctive separation , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ¯∧ ¯0 ∀Q F ,∀ Q , ∀Qp ,∀Q, (I0 x ∈ S0 I ) → 0 0 ¯∧ ¯0 (Q F ∈ S R ) → (Q ∈ S R ) → (Qp ∈ S0 R ) → (Q ∈ S0 R ) → ¯∧ R[Q ¯ ∧ ] QF → F

¯∧ Qp R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q F → ( ¯0 → QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] I0 x Q x

QS[Qx: Q00 x ∧ Q0 x] Q Qp ) ).

10.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the deterministic world attributes’ disjunctive separation , given as

- 190 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , 0 ¯∧ ∀Q F ,∀Q , ¯ 000 , ∀Qp ,∀ Q ¯ ∀ Q, (I0 x ∈ S0 I ) → 0 0 0 ¯∧ (Q F ∈ S R ) → (Q ∈ S R ) → ¯ 000 ∈ S0 R ) → (Qp ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯∧ R[Q ¯ ∧ ] QF → F

¯∧ Qp R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q F → 000 0 ¯ Q R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I x Q0 → ( ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q0 → QR x ¯ [Qx: Q0 x → Q00 x] Qp Q ¯ 000 QS ) ).

10.6.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the deterministic world attributes’ conjunctive separation , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , 0 ∀Q∧ T ,∀Q , ∀Qp ,∀Q, (I0 x ∈ S0 I ) → 0 0 0 (Q∧ T ∈ S R ) → (Q ∈ S R ) → (Qp ∈ S0 R ) → (Q ∈ S0 R ) → R[Q∧ ] Q∧ T → T

- 191 -

Qp R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q∧ T → ( QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → QS[Qx: Q00 x ∧ Q0 x] Q Qp ) ).

10.6.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the deterministic world attributes’ disjunctive separation , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ¯0 ∀Q∧ T ,∀ Q , ¯ 000 , ∀Qp ,∀ Q ¯ ∀ Q, (I0 x ∈ S0 I ) → 0 0 ¯0 (Q∧ T ∈ S R ) → (Q ∈ S R ) → 0 000 ¯ (Qp ∈ S R ) → (Q ∈ S0 R ) → ¯ ∈ S0 R ) → (Q R[Q∧ ] Q∧ T → T

Qp R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q∧ T → ¯ 000 R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q ¯0 → Q x ( ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] I0 x Q ¯0 → QR x 000 ¯ ¯ QS[Qx: Q0 x → Q00 x] Qp Q ) ).

- 192 -

10.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the resolution of the deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 w ,∀I0 x , ¯ 0 ,∀Q, ∀Q ∀¯ p, (I0 w ∈ S0 I ) → (I0 x ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q (¯ p ∈ L) → p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] I0 w I0 x Q 0 ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I x Q → ( T∨ p ¯ ↔ QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 w ) ).

10.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the resolution of the deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 w ,∀I0 x , ¯ 0 ,∀Q, ∀Q ∀¯ p, (I0 w ∈ S0 I ) → (I0 x ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q (¯ p ∈ L) → - 193 -

p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → (Ix x → Qx)] I0 w I0 x Q 0 ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I x Q → ( T∨ p ¯ ↔ QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 w ) ).

10.7.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the resolution of the deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 w ,∀I0 x , ∀Q0 ,∀Q, ∀p, (I0 w ∈ S0 I ) → (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) → pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] I0 w I0 x Q0 → QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → ( T∧ p ↔ QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 w ) ).

- 194 -

10.7.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the resolution of the deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 w ,∀I0 x , ∀Q0 ,∀Q, ∀p, (I0 w ∈ S0 I ) → (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) → pR[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] I0 w I0 x Q0 → QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → ( T∧ p ↔ QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 w ) ).

10.8.1 Any member Q of set S0 R of a natural language’s unary relations constituting a conjunctively deterministic particular element existential world attribute with respect to an ordered pair {I0 x , Q0 } of members of S0 R the first member of which is a member of set S0 I of the language’s particular element identifying unary relations will constitute a basic composed unary relation in the case that Q0 constitutes a basic composed unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the basic composed deterministic world attributes , given as

- 195 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ∀Q0 ,∀Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → ( R[Q: b.c c.p.sed u.ry r.l.n] Q0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ) ).

¯ of set S0 R of a natural language’s unary relations consti10.8.2 Any member Q tuting a conjunctively deterministic particular element universal world attribute ¯ 0 } of members of S0 R the first member with respect to an ordered pair {I0 x , Q of which is a member of set S0 I of the language’s particular element identifying unary relations will constitute a basic composed unary relation in the case that ¯ 0 constitutes a basic composed unary relation, that is, there will be a member Q of the set of elements of the language underlying a natural language constituting a derivative first Law of the basic composed deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀ Q, ¯ ∀I0 x ,∀ Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (I0 x ∈ S0 I ) → (Q ¯ ¯0 → ( QR[Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ¯ R[Q: b.c c.p.sed u.ry r.l.n] Q ) ).

- 196 -

10.9.1 Any member Q of set S0 R of a natural language’s unary relations constituting a conjunctively deterministic particular element existential world attribute with respect to an ordered pair {I0 x , Q0 } of members of S0 R will constitute a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀I0 x , ∀Q0 ,∀Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) →

( QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → R[Q: pr.ry u.ry r.l.n] Q ) ).

¯ of set S0 R of a natural language’s unary relations consti10.9.2 Any member Q tuting a conjunctively deterministic particular element universal world attribute ¯ 0 } of members of S0 R will constitute a with respect to an ordered pair {I0 x , Q primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary deterministic world attributes , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary deterministic world attributes , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q

- 197 -

(I0 x ∈ S0 I ) → ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ( ¯0 → ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] I0 x Q QR x ¯ R[Q: pr.ry u.ry r.l.n] Q ) ).

- 198 -

11 . 11.1.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the Dirac propositions’ conjunctive content completeness , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pD ,(pD ∈ L) → ( (pD ∈ LD ) → R[p: c.j.v.ly c.t.t c.pl.te ] pD ) ).

11.1.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the Dirac propositions’ conjunctive content consistency , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pD ,(pD ∈ L) → ( (pD ∈ LD ) → R[p: c.j.v.ly c.t.t c.ss.tnt ] pD ) ).

11.1.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the Dirac propositions’ joint basic composition , given as - 199 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pD ,(pD ∈ L) → ( (pD ∈ LD ) → R[p: j.nt b.c prp.s.n] pD ) ).

11.1.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the particular Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,(p0 ∈ L) → ( ∀pD ,(pD ∈ L) → (pD ∈ LD ) → p0 R[p0 : c.j.v.ly c.t.t frb.dd.n, p] pD )→ R[p¯∧ ] p0 F

).

11.2.1 Set LD of a natural language’s Dirac propositions will be a subset of set L of the language’s propositions, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of Dirac propositions , given as

- 200 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∃L(pwr) ,(L(pwr) ∈ S) ∧ L(pwr) R[S(pwr) ] L ∧ (LD ∈ L(pwr) ) ).

11.2.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the set of Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → ( (p ∈ LD ) ↔ R[p: Dirac prp.s.n] p ) ).

11.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the Dirac unary relations’ conjunctive content completeness , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀QD ,(QD ∈ S0 R ) →

- 201 -

( (QD ∈ S0 D ) → R[Q: c.j.v.ly c.t.t c.pl.te ] QD ) ).

11.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the Dirac unary relations’ conjunctive content consistency , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀QD ,(QD ∈ S0 R ) → ( (QD ∈ S0 D ) → R[Q: c.j.v.ly c.t.t c.ss.tnt ] QD ) ).

11.3.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the particular Dirac unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,(Q0 ∈ S0 R ) → ( ∀QD ,(QD ∈ S0 R ) → (QD ∈ S0 D ) → Q0 R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] QD )→ - 202 -

0 R[Q ¯ ∧] Q F

).

11.4.1 Set S0 D of a natural language’s Dirac unary relations will be a subset of set S0 R of the language’s unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of Dirac unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr) ,(S0 R ∈ S) 0 (pwr) SR R[S(pwr) ] S0 R →

∀S0 R

(S0 D ∈ S0 R

(pwr)

→

)

).

11.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the set of Dirac unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( (Q ∈ S0 D ) ↔ R[Q: Dirac u.ry r.l.n] Q ) ).

- 203 -

11.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the elemental conjunctive content completeness of the reduced Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pI ,(pI ∈ L) → ( (pI ∈ LI ) → pI R[p: s.t c.j.v.ly c.t.t c.pl.te , La ] Le ) ).

11.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the elemental conjunctive content consistency of the reduced Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pI ,(pI ∈ L) → ( (pI ∈ LI ) → pI R[p: s.t c.j.v.ly c.t.t c.ss.tnt , La ] Le ) ).

11.5.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint elemental composition of the reduced Dirac propositions , given as

- 204 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pI ,(pI ∈ L) → ( (pI ∈ LI ) → R[p: j.nt el.l pr.p.s.n] pI Le ) ).

11.5.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the particular reduced Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,(p0 ∈ L) → R[p: j.nt el.l pr.p.s.n] p0 → ( ∀pI ,(pI ∈ L) → (pI ∈ LI ) → p0 R[p0 : c.j.v.ly c.t.t frb.dd.n, p] pI )→ R[p¯∧ ] p0 F

).

11.5.5 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the elemental propositions’ basic composition , given as

- 205 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pI ,(pI ∈ L) → ( (pI ∈ LI ) → R[p: b.c c.p.sed prp.s.n] pI ) ).

11.6.1 Set LI of a natural language’s reduced Dirac propositions will be a subset of set L of the language’s propositions, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of reduced Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,(L(pwr) ∈ S) → L(pwr) R[S(pwr) ] L → (LI ∈ L(pwr) ) ).

11.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the set of reduced Dirac propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) →

- 206 -

( (p ∈ LI ) ↔ R[p: r.d.c.ed Dirac prp.s.n] p ) ).

11.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary content completeness of the particular element identifying unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 R ) → ( (Ix ∈ S0 I ) → Ix R[Q: s.t c.j.v.ly c.t.t c.pl.te , S0 R,a ] S0 p ) ).

11.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary content consistency of the particular element identifying unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 R ) → ( (Ix ∈ S0 I ) → Ix R[Q: s.t c.j.v.ly c.t.t c.ss.tnt , S0 R,a ] S0 p ) ).

- 207 -

11.7.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctive content non recursion of the particular element identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 R ) → ( (Ix ∈ S0 I ) → ¯ [Q0 : c.j.v.ly c.t.t r.crs.ve] Ix R ) ).

11.7.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the particular element identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,(Q0 ∈ S0 R ) → R[Q: pr.ry u.ry r.l.n] Q0 → ( ∀Ix ,(Ix ∈ S0 R ) → (Ix ∈ S0 I ) → Q0 R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix )→ 0 R[Q ¯ ∧] Q F

).

11.7.5 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composition of the particular element identifying unary relations , given as - 208 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 R ) → ( (Ix ∈ S0 I ) → R[Q: b.c c.p.sed u.ry r.l.n] Ix ) ).

11.7.6 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the primary composition of the particular element identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 R ) → ( (Ix ∈ S0 I ) → R[Q: pr.ry u.ry r.l.n] Ix ) ).

11.8.1 Set S0 I of a natural language’s particular element identifying unary relations will be a subset of set S0 R of the language’s unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of particular element identifying unary relations , given as

- 209 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr) ,(S0 R ∈ S) 0 (pwr) SR R[S(pwr) ] S0 R →

∀S0 R

(S0 I ∈ S0 R

(pwr)

→

)

).

11.8.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the set of particular element identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 R ) → ( (Ix ∈ S0 I ) ↔ R[Ix ] Ix ) ).

11.9.1 Any member of set S0 I of a natural language’s particular element identifying unary relations not constituting a conjunctively deterministcally empty element identifying unary relation will constitute a particular world identifying unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the particular non empty world identifying unary relations , given as

- 210 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,∀Ix , (Q ∈ S0 R ) → (Ix ∈ S0 I ) → QR[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q → ( QR[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix → R[Iw ] Ix ) ).

11.9.2 There will be a member of set S0 I of a natural language’s particular element identifying unary relations constituting a conjunctively deterministcally empty world identifying unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the particular empty world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∃Q,∃Ix , (Q ∈ S0 R ) ∧ (Ix ∈ S0 I ) ∧ QR[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q ∧ ( QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix ∧ R[Iw ] Ix ) ).

- 211 -

11.9.3 There will not be two distinct members of set S0 I of a natural language’s particular element identifying unary relations constituting conjunctively deterministcally empty world identifying unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the uniqueness of the particular empty world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q, ∀I00 x ,∀I0 x , (Q ∈ S0 R ) → (I00 x ∈ S0 I ) → (I0 x ∈ S0 I ) → QR[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q → ( QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I00 x ∧ QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 x ∧ R[Iw ] I00 x ∧ R[Iw ] I0 x )→ I00 x Rid I0 x ).

11.9.4 There will be a member Iwcd.te of set S0 I of a natural language’s particular element identifying unary relations constituting a candidate world identifying unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the candidate world identifying unary relations , given as

- 212 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∃Iwcd.te , (Iwcd.te ∈ S0 I ) ∧ R[Iw ] Iwcd.te ∧ R[Iw ] Iwcd.te cd.te

).

∨

11.9.5 For any subset S0 I of set S0 I of a natural language’s particular element identifying unary relations there will be a member Iw of set S0 I identifying an element the members of which are those and only those members of set Scd.te ∨ of elements by the language identified by members of S0 I , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the existence of the deterministic worlds , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iwcd.te , ∀Q∧ T, ∀Q,∃Iw , (Iwcd.te ∈ S0 I ) → 0 (Q∧ T ∈ S R) → 0 (Q ∈ S R ) → (Iw ∈ S0 I ) ∧ R[Iw ] Iwcd.te → cd.te

R[Q∧ ] Q∧ T → T

R[Q: pr.ry u.ry r.l.n] Q → ( ∀Ix , ∀Q00 ,∀Q0 , (Ix ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → Q0 R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Ix Q∧ T → - 213 -

Q0 R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Ix Q → ( Q00 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Iwcd.te ↔ Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Iw ) ) ).

11.9.6 Any member of set S0 I of a natural language’s particular element identifying unary relations constituting a particular world identifying unary relation and not constituting an empty element identifying unary relation will constitute a joint basic world attribute, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the joint basic composition of the particular non empty world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,∀Ix , (Q ∈ S0 R ) → (Ix ∈ S0 I ) → QR[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q → ( ( QR[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix ∧ R[Iw ] Ix )→ ¯ [Q: joint basic world attribute ] Ix R ) ).

- 214 -

11.10.1 Set S0 Iw of a natural language’s particular world identifying unary relations will be a subset of set S0 I of the language’s particular element identifying unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative first Law of the composition of the set of particular world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr) ∈ S) ,(S0 I (pwr) S0 I R[S(pwr) ] S0 I →

∀S0 I

(S0 Iw ∈ S0 I

(pwr)

→

)

).

11.10.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative second Law of the composition of the set of particular world identifying unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Ix ,(Ix ∈ S0 I ) → ( (Ix ∈ S0 Iw ) ↔ R[Iw ] Ix ) ).

- 215 -

11.11.1 Any member p of set L of a natural language’s propositions will constitute a basic composed proposition in the case and only in the case that that there is a subset L∨ of set LD of the language’s Dirac propositions such that p constitutes a disjoint proposition with respect to L∨ , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀LD ∀p,

,

(pwr)

(LD ∈ S) → (p ∈ L) → (pwr)

R[S(pwr) ] LD → R[p: b.c c.p.sed prp.s.n] p ↔ ( ∃L∨ ,(L∨ ∈ S) ∧ LD

(pwr)

(L∨ ∈ LD pS[L∨ ] L∨

)∧

) ).

11.11.2 Any member p of set L of a natural language’s propositions will constitute an extended composed proposition in the case and only in the case that that there is an ordered pair {p00 , p0 } of members of set L of the language’s propositions the members of which are basic composed propositions such that p constitutes a conjunctively conditional proposition with respect to {p00 , p0 }, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the extended composed propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → R[p: ext.d.d c.p.sed prp.s.n] p ↔

- 216 -

( ∃p00 ,∃p0 , (p00 ∈ L) ∧ (p0 ∈ L) ∧ R[p: b.c c.p.sed prp.s.n] p00 ∧ R[p: b.c c.p.sed prp.s.n] p0 ∧ pS[p: p00 |∧ p0 ] p00 p0 ) ).

11.12.1 In the case that the members of ordered pair {p00 , p0 } of members set L of a natural language’s propositions constitute basic composed propositions member p ¯ of the set constituting a negatively conjunctive proposition with respect to the pair will constitute a basic composed proposition, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed negatively conjunctive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( R[p: b.c c.p.sed prp.s.n] p00 ∧ R[p: b.c c.p.sed prp.s.n] p0 )→ R[p: b.c c.p.sed prp.s.n] p ¯ ).

- 217 -

11.12.2 All members of set L of a natural language’s propositions will constitute extended composed propositions, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the propositions’ extended composition , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → R[p: ext.d.d c.p.sed, pr.ry prp.s.n] p ).

11.13.1 Any member Q of set S0 R of a natural language’s unary relations will constitute a basic composed unary relation in the case and only in the case that ∨ that there is a subset S0 R of set S0 D of the language’s Dirac unary relations ∨ such that Q constitutes a disjoint unary relation with respect to S0 R , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀S0 D ∀Q,

,

(pwr)

(S0 D ∈ S) → (Q ∈ S0 R ) → (pwr)

S0 D

R[S(pwr) ] S0 D →

R[Q: b.c c.p.sed u.ry r.l.n] Q ↔ ( ∨

∨

∃S0 R ,(S0 R ∈ S) ∧ ∨ (S0 R

(pwr) ∈ S0 D ) 0 ∨ QS[S0 R ∨ ] S R

∧

) ).

- 218 -

11.13.2 Any member Q of set S0 R of a natural language’s unary relations will constitute an extended composed unary relation in the case and only in the case that that there is an ordered pair {Q00 , Q0 } of members of set S0 R of the language’s unary relations the members of which are basic composed unary relations such that Q constitutes a conjunctively conditional unary relation with respect to {Q00 , Q0 }, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the extended composed unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → R[Q: ext.d.d c.p.sed u.ry r.l.n] Q ↔ ( ∃Q00 ,∃Q0 , (Q00 ∈ S0 R ) ∧ (Q0 ∈ S0 R ) ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q00 ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q0 ∧ QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 ) ).

11.14.1 In the case that the members of ordered pair {Q00 , Q0 } of members set S0 R of a natural language’s unary relations constitute basic composed unary ¯ of the set constituting a negatively conjunctive unary rerelations member Q lation with respect to the pair will constitute a basic composed unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed negatively conjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS - 219 -

( R[Q: b.c c.p.sed u.ry r.l.n] Q00 ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q0 )→ ¯ R[Q: b.c c.p.sed u.ry r.l.n] Q ).

11.14.2 All members of set S0 R of a natural language’s unary relations will constitute extended composed unary relations, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the unary relations’ extended composition , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → R[Q: ext.d.d c.p.sed u.ry r.l.n] Q ).

11.15.1 Any member Q of set S0 R of a natural language’s unary relations will constitute a basic composed, primary unary relation in the case and only in the ∨ case that there is a subset S0 R of set S0 I of the language’s particular element identifying unary relations such that Q constitutes a disjoint unary relation with ∨ respect to S0 R , that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed, primary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 I ∀Q, (S0 I

(pwr)

(pwr)

, ∈ S) → - 220 -

(Q ∈ S0 R ) → S0 I

(pwr)

R[S(pwr) ] S0 I →

R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q ↔ ( ∨

∨

∃S0 R ,(S0 R ∈ S) ∧ ∨ (S0 R

(pwr) ∈ S0 I ) ∨ QS[S0 R ∨ ] S0 I

∧

) ).

11.15.2 Any member Q of set S0 R of a natural language’s unary relations will constitute an extended composed, primary unary relation in the case and only in the case that there is an ordered pair {Q00 , Q0 } of members of set S0 R of the language’s unary relations the members of which are basic composed, primary unary relations such that Q constitutes a conjunctively conditional unary relation with respect to {Q00 , Q0 }, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the extended composed, primary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → R[Q: ext.d.d c.p.sed, pr.ry u.ry r.l.n] Q ↔ ( ∃Q00 ,∃Q0 , (Q00 ∈ S0 R ) ∧ (Q0 ∈ S0 R ) ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q00 ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q0 ∧ QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 ) ).

- 221 -

11.16.1 In the case that the members of an ordered pair {Q00 , Q0 } of members of set S0 R of a natural language’s unary relations constitute basic composed, pri¯ of the set constituting a negatively conjunctive mary unary relations, member Q unary relation with respect to the pair will constitute a basic composed, primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed, primary negatively conjunctive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q0 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS ( ( R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q00 ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q0 )→ ¯ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q ) ).

11.17.1 Any member Q of set S0 R of a natural language’s unary relations will constitute a basic composed, primary unary relation in the case and only in the case that Q constitutes a basic composed unary relation and a primary unary relation, that is, there will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the basic composed and primary unary relations given as

- 222 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q,(Q ∈ S0 R ) → ( R[Q: b.c c.p.sed u.ry r.l.n] Q ∧ R[Q: pr.ry u.ry r.l.n] Q )↔ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q ).

- 223 -

P. p.1.1 A probabilistic unary natural language will be an ordered 12 - tuple Lp : {L, Le , Lp , LD , LI , S0 R , S0 p , S0 D , S0 I , S0 Iw , SR , Sq } of sets, the members of which are, respectively, a set L the members of which are the language’s propositions, a set Lp the members of which are the language’s primary propositions, a set Le the members of which are the language’s elemental propositions, a set LD the members of which are the language’s Dirac propositions, a set LI the members of which are the language’s reduced Dirac propositions, a set S0 R the members of which are the language’s unary relations, a set S0 p the members of which are the language’s primary unary relations, a set S0 D the members of which are the language’s Dirac unary relations, a set S0 I the members of which are the language’s particular element identifying unary relations, a set S0 Iw the members of which are the language’s particular world identifying unary relations, a set SR constituting the language’s real number reference set and a set Sq the members of which are the language’s probabilities. p.2.1 Ordered subset L : {L, Le , Lp , LD , LI , S0 R , S0 p , S0 D , S0 I , S0 Iw } of set Lp constituting a probabilistic unary natural language will constitute an unary natural language, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative foundational Law of the probabilistic unary natural languages , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw

(1)

).

p.3.1 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the real number reference sets , given as

- 224 -

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( RR SR

(2)

).

p.4.1 Set Sq of a probabilistic natural language’s probabilities will be a subset of set SR constituting the language’s real number reference set, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the probability sets , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → (

(3)

∀SR (pwr) ,(SR (pwr) ∈ S) → SR (pwr) R[S(pwr) ] SR → (Sq ∈ SR (pwr) ) ).

p.5.1 There will be a member qT of set Sq of a probabilistic natural language’s probabilities constituting an affirmative probability, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the existence of the affirmative probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∃qT ,(qT ∈ Sq ) ∧ R[q: aff.t.ve prb.ty] qT ).

- 225 -

(4)

p.5.2 There will be a member qF of set Sq of a probabilistic natural language’s probabilities constituting an dismissive probability, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the existence of the dismissive probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∃qF ,(qF ∈ Sq ) ∧ R[q: d.sm.ss.ve prb.ty] qF

(5)

).

p.5.3 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the complementarity of the dismissive and the affirmative probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀qT ,∀qF , (qF ∈ Sq ) → (qF ∈ Sq ) → R[q: d.sm.ss.ve prb.ty] qF → R[q: aff.t.ve prb.ty] qT → qF Rq¯ qT

(6)

).

p.5.4 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the composition of the affirmative probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀qT ,(qT ∈ Sq ) →

- 226 -

(7)

( R[q: aff.t.ve prb.ty] qT ↔ R[‘1’]qT ) ).

p.5.5 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the composition of the dismissive probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀qF ,(qF ∈ Sq ) → ( R[q: d.sm.ss.ve prb.ty] qF ↔ R[‘0’]qF )

(8)

).

p.6.1 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the probabilities’ upper bound , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀q,∀qT , (q ∈ Sq ) → (qT ∈ Sq ) →

- 227 -

(9)

( R[q: aff.t.ve prb.ty] qT → qR≤ qT ) ).

p.6.2 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the probabilities’ lower bound , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀q,∀qF , (q ∈ Sq ) → (qF ∈ Sq ) → ( R[q: d.sm.ss.ve prb.ty] qF → qF R≤ q )

(10)

).

p.7.1 For any member q of set Sq of a probabilistic natural language’s probabilities there will be a member q ¯ of the set constituting a complementary probability with respect to q, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the existence of the complementary probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀q,∃¯ q, (q ∈ Sq ) → (¯ q ∈ Sq ) ∧ q ¯Rq¯ q ).

- 228 -

(11)

p.7.2 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the reciprocity of the complementarity of probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ q,∀q, (¯ q ∈ Sq ) → (q ∈ Sq ) → ( q ¯Rq¯ q ↔ qRq¯ q ¯ )

(12)

).

p.7.3 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the composition of the complementary probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀qT , ∀¯ q,∀q, (qT ∈ Sq ) → (¯ q ∈ Sq ) → (q ∈ Sq ) → R[q: aff.t.ve prb.ty] qT → ( q ¯Rq¯ q ↔ qT S[q:q00 +q0 ] q ¯q ) ).

- 229 -

(13)

p.8.1 For any member p ¯ of a probabilistic natural language’s propositions there will be a member q of the set of the language’s probabilities constituting a disjunctive probability, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the existence of the disjunctive probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p,∃q, (¯ p ∈ L) → (q ∈ Sq ) ∧ p ¯ T∨ q

(14)

).

p.8.2 For any member p of a probabilistic natural language’s propositions there will be a member q of the set of the language’s probabilities constituting a conjunctive probability, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the existence of the disjunctive probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀p,∃q, (p ∈ L) → (q ∈ Sq ) ∧ p T∧ q

(15)

).

p.8.3 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the conjunctive probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀p, ∀¯ q,∀q, (p ∈ L) → - 230 -

(16)

(¯ q ∈ Sq ) → (q ∈ Sq ) → q ¯Rq¯ q → ( p T∧ q ↔ p T∨ q ¯ ) ).

p.9.1 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the disjunctively certain propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p,∀qT , (¯ p ∈ L) → (qT ∈ Sq ) → R[q: aff.t.ve prb.ty] qT → ( T∨ p ¯ ↔ p ¯ T∨ qT )

(17)

).

p.9.2 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the conjunctively certain propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀p,∀qT , (p ∈ L) → (qT ∈ Sq ) → - 231 -

(18)

R[q: aff.t.ve prb.ty] qT → ( T∧ p ↔ p T∧ qT ) ).

p.10.1 Any member q of set Sq of a probabilistic natural language’s probabilities will constitute a deterministic probability in the case and only in the case that it constitutes an affirmative probability, or a dismissive probability, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the deterministic probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀q,(q ∈ Sq ) → R[q: d.t.c prb.ty] ↔( R[q: aff.t.ve prb.ty] q ∨ R[q: d.sm.ss.ve prb.ty] q )

(19)

).

p.11.1 Any member q of set Sq of a probabilistic natural language’s probabilities will constitute an indeterministic probability in the case and only in the case that it does not constitute an affirmative probability, or a dismissive probability, that is, there will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the indeterministic probabilities , given as

- 232 -

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀q,(q ∈ Sq ) → R[q: i.d.t.c prb.ty] ↔( ¯ [q: aff.t.ve prb.ty] q R ∧ ¯ [q: d.sm.ss.ve prb.ty] q R )

(20)

).

p.12.1 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the complementary probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p,∀p, ∀¯ q,∀q, ∀¯ p∨ T ,∀qT , (¯ p ∈ L) → (p ∈ L) → (¯ q ∈ Sq ) → (q ∈ Sq ) → (¯ p∨ T ∈ L) → (qT ∈ Sq ) → p ¯ Rp¯ p → R[q: aff.t.ve prb.ty] qT → p ¯ T∨ q → p T∨ q ¯→ p ¯∨ T T∨ qT → ( p ¯∨ ¯p → T S[p: p00 ∧ p0 ] p qT S[q:q00 +q0 ] q ¯q ) ).

- 233 -

(21)

p.12.2 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the complementary probabilities , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p,∀p, ∀¯ q,∀q, ∀p∧ T ,∀qT , (¯ p ∈ L) → (p ∈ L) → (¯ q ∈ Sq ) → (q ∈ Sq ) → (p∧ T ∈ L) → (qT ∈ Sq ) → p ¯ Rp¯ p → R[q: aff.t.ve prb.ty] qT → p ¯ T∧ q ¯→ p T∧ q → p∧ T T∧ qT → ( p∧ ¯p → T S[p: p00 ∨ p0 ] p qT S[q:q00 +q0 ] q ¯q )

(22)

).

p.13.1 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the of the disjunctive and the conjunctive propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p00 ,∀¯ p0 , ∀¯ pb ,∀¯ pa , ∀q00 ,∀q0 , ∀qb ,∀qa , ∀q, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → - 234 -

(23)

(q00 ∈ Sq ) → (q0 ∈ Sq ) → (qb ∈ Sq ) → (qa ∈ Sq ) → (q ∈ SR ) → p ¯b S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯a S[p: p00 ∧ p0 ] p ¯00 p ¯0 → p ¯00 T∨ q00 → p ¯0 T∨ q0 → p ¯b T∨ qb → p ¯a T∨ qa → ( qS[q:q00 +q0 ] qb qa ↔ qS[q:q00 +q0 ] q00 q0 ) ).

p.13.2 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative Law of the of the disjunctive and the conjunctive propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀p00 ,∀p0 , ∀pb ,∀pa , ∀q00 ,∀q0 , ∀qb ,∀qa , ∀q, (p00 ∈ L) → (p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → (q00 ∈ Sq ) → (q0 ∈ Sq ) → (qb ∈ Sq ) → (qa ∈ Sq ) → (q ∈ SR ) → pb S[p: p00 ∨ p0 ] p00 p0 → pa S[p: p00 ∧ p0 ] p00 p0 → p00 T∧ q00 → - 235 -

(24)

p0 T∧ q0 → pb T∧ qb → pa T∧ qa → ( qS[q:q00 +q0 ] qb qa ↔ qS[q:q00 +q0 ] q00 q0 ) ).

p.14.1 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative first Law of the resolution of the disjunctively conditional propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p00 ,∀¯ p0 , 000 ∀¯ p ,∀¯ p, ∀q000 ,∀q0 , ∀q, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p000 ∈ L) → (¯ p ∈ L) → (q000 ∈ Sq ) → (q0 ∈ Sq ) → (q ∈ Sq ) → p ¯000 S[p: p00 |∨ p0 ] p ¯00 p ¯0 → 00 0 p ¯ S[p: p00 ∨ p0 ] p ¯ p ¯ → p ¯000 T∨ q000 → p ¯0 T∨ q0 → ( p ¯ T∨ q qS[q:q00 ∗q0 ] q000 q0 ) ).

- 236 -

(25)

p.14.2 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative first Law of the resolution of the conjunctively conditional propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀p00 ,∀p0 , ∀p000 ,∀p, ∀q000 ,∀q0 , ∀q, (p00 ∈ L) → (p0 ∈ L) → (p000 ∈ L) → (p ∈ L) → (q000 ∈ Sq ) → (q0 ∈ Sq ) → (q ∈ Sq ) → p000 S[p: p00 |∧ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] p00 p0 → p000 T∧ q000 → p0 T∧ q0 → ( p T∧ q qS[q:q00 ∗q0 ] q000 q0 )

(26)

).

p.14.3 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative second Law of the resolution of the disjunctively conditional propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p00 ,∀p00 , ∀¯ p0 , ∀p000 ,∀¯ p, ∀q00 ,∀q0 , ∀¯ q000 ,∀q(iv) , ∀q, - 237 -

(27)

(¯ p00 ∈ L) → (p00 ∈ L) → (¯ p0 ∈ L) → (p000 ∈ L) → (¯ p ∈ L) → (q00 ∈ Sq ) → (q0 ∈ Sq ) → (¯ q000 ∈ Sq ) → (q(iv) ∈ Sq ) → (q ∈ Sq ) → p ¯0 Rp¯ p0 → p000 S[p: p00 |∨ p0 ] p00 p ¯0 → 00 0 p ¯ S[p: p00 ∧ p0 ] p ¯ p ¯ → p ¯00 T∨ q00 → p ¯0 T∨ q0 → p000 T∨ q ¯000 → q(iv) S[q:q00 ∗q0 ] q ¯000 q0 → ( p ¯ T∨ q qS[q:q00 +q0 ] q00 q(iv) ) ).

p.14.3 There will be a member of the set of elements of the language underlying a probabilistic natural language constituting a derivative second Law of the resolution of the conjunctively conditional propositions , given as

LR(nt.l,prb.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → ( ∀¯ p00 ,∀p00 , ∀p0 , ∀¯ p000 ,∀p, ∀q00 ,∀q0 , ∀¯ q000 ,∀q(iv) , ∀q, (¯ p00 ∈ L) → (p00 ∈ L) → (p0 ∈ L) → (¯ p000 ∈ L) → (p ∈ L) → (q00 ∈ Sq ) → (q0 ∈ Sq ) → - 238 -

(28)

(¯ q000 ∈ Sq ) → (q(iv) ∈ Sq ) → (q ∈ Sq ) → p ¯0 Rp¯ p0 → p ¯000 S[p: p00 |∧ p0 ] p ¯00 p0 → pS[p: p00 ∨ p0 ] p00 p0 → p00 T∧ q00 → p0 T∧ q0 → p ¯000 T∧ q ¯000 → (iv) q S[q:q00 ∗q0 ] q ¯000 q0 → ( p T∧ q qS[q:q00 +q0 ] q00 q(iv) ) ).

- 239 -

D. d.1.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ↔ p ¯ S[p: p00 ∨ p0 ] p ¯p ¯0 ) ).

d.1.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content absent propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀¯ p, (p0 ∈ L) → (¯ p ∈ L) → ( p0 R[p0 : d.sj.v.ly c.t.t abs.nt, p] p ¯ ↔ ¯ [p0 : d.sj.v.ly c.t.t pr.s.nt, p] p p0 R ¯ ) ).

- 240 -

d.1.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ↔ pS[p: p00 ∧ p0 ] pp0 ) ).

d.1.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content absent propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p, (¯ p0 ∈ L) → (p ∈ L) → ( p ¯0 R[p0 : c.j.v.ly c.t.t abs.nt, p] p ↔ ¯ [p0 : c.j.v.ly c.t.t pr.s.nt, p] p p ¯0 R ) ).

- 241 -

d.2.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content present unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀ Q, ¯ ∀Q ¯ ∈ S0 R ) → ¯ (Q0 ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ Q ↔ ¯Q ¯0 ¯ [Qx: Q00 x ∨ Q0 x] Q QS ) ).

d.2.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content absent unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ ∀Q0 ,∀ Q, ¯ ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ( ¯ Q0 R[Q0 : d.sj.v.ly c.t.t abs.nt, Q] Q ↔ ¯ [Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ Q0 R ) ).

- 242 -

d.2.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content present unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,∀Q, (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( Q0 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ↔ QS[Qx: Q00 x ∧ Q0 x] Q Q0 ) ).

d.2.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content absent unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀Q, ∀Q ¯ 0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : c.j.v.ly c.t.t abs.nt, Q] Q Q ↔ ¯ [Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ¯ 0R Q ) ).

- 243 -

d.3.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q ∀¯ p0 ,∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] Iw Ix Q p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯ → (Ix x ∧ Qx)] Iw Ix Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ → Q p ¯0 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ) ).

d.3.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q 0 ∀¯ p ,∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q (¯ p0 ∈ L) → (¯ p ∈ L) → - 244 -

p ¯0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 → (Ix x ∧ Qx)] Iw Ix Q p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ → (Ix x → Qx)] Iw Ix Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ → Q p ¯0 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ) ).

d.3.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q ∀¯ p0 ,∀¯ p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯0 → (Ix x → Qx)] Iw Ix Q p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ → (Ix x → Qx)] Iw Ix Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ → Q p ¯0 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ) ).

- 245 -

d.4.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, ∀p0 ,∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (p0 ∈ L) → (p ∈ L) → p0 R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q0 → pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q → ( Q0 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q → p0 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ) ).

d.4.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, ∀p0 ,∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (p0 ∈ L) → (p ∈ L) → - 246 -

p0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q0 → pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Iw Ix Q → ( Q0 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q → p0 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ) ).

d.4.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content present propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, ∀p0 ,∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (p0 ∈ L) → (p ∈ L) → p0 R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q0 → pR[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → (Ix x → Qx)] Iw Ix Q → ( Q0 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q → p0 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ) ).

- 247 -

d.5.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, ∀p0 ,∀p, ∀¯ p000 , (¯ p0 ∈ L) → (¯ p ∈ L) → (p0 ∈ L) → (p ∈ L) → (¯ p000 ∈ L) → p ¯0 Rp¯ p0 → p ¯ Rp¯ p → p ¯000 S[p: p0 !→ p00 ] pp0 → ( p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ↔ p ¯000 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ) ).

d.5.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content forbidden propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p0 , ∀¯ p, (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯0 Rp¯ p0 →

- 248 -

( p0 R[p0 : d.sj.v.ly c.t.t frb.dd.n, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ) ).

d.5.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, ∀p000 , (p0 ∈ L) → (p ∈ L) → (p000 ∈ L) → p000 S[p: p0 → p00 ] pp0 → ( p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ↔ p000 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ) ).

d.5.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content forbidden propositions , given as

- 249 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p0 , ∀p, (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( p ¯0 R[p0 : c.j.v.ly c.t.t frb.dd.n, p] p ↔ p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ) ).

d.6.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀ Q, ¯ ∀Q ∀Q0 ,∀Q, ¯ 000 , ∀Q ¯ ¯ ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ¯ 000 ∈ S0 R ) → (Q 0 ¯ Q0 RQ ¯Q → ¯ Q QR ¯Q→ ¯ 000 S[Qx: Q0 x !→ Q00 x] Q Q0 → Q

- 250 -

( ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q ¯ Q ↔ ¯ 000 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ Q ) ).

d.6.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ( ¯ Q0 R[Q0 : d.sj.v.ly c.t.t frb.dd.n, Q] Q ↔ ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q ¯ Q ) ).

d.6.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content necessary unary relations , given as

- 251 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,∀Q, ∀Q000 , (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q000 ∈ S0 R ) → Q000 S[Qx: Q0 x → Q00 x] QQ0 → ( Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Q ↔ Q000 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ) ).

d.6.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀Q0 , ∀Q ∀Q, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( ¯ 0 R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Q Q ↔ Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Q ) ).

- 252 -

d.7.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ) ).

d.7.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ↔ p0 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ¯ ) ).

- 253 -

d.8.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀ Q, ¯ ∀Q ¯ ∈ S0 R ) → ¯ (Q0 ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q ¯ Q ↔ ¯ ¯ 0 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q Q ) ).

d.8.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,∀Q, (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Q ↔ Q0 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ) ).

- 254 -

d.9.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ) ).

d.9.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, p] p ↔ p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ) ).

- 255 -

d.10.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively content recursive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ¯ 0 ,∀ Q, ¯ ∀Q ¯ ∈ S0 R ) → ¯ (Q0 ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, Q] Q ¯ Q ↔ ¯ ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q Q ) ).

d.10.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively content recursive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Q0 ,∀Q, (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( Q0 R[Q0 : c.j.v.ly c.t.t r.crs.ve, Q] Q ↔ Q0 Q ) ).

- 256 -

d.11.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set disjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∨ , ∀¯ p0 , (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p0 ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, La ] L∨ ↔ ( p ¯0 S[L∨ ] L∨ → ( ∃¯ p∨ ,(¯ p∨ ∈ L) ∧ (¯ p∨ ∈ L∨ ) ∧ p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, p] p ¯∨ ) ) ).

d.11.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set conjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀L∧ , ∀p0 , (L(pwr) ∈ S) → (L∧ ∈ S) → (p0 ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) →

- 257 -

p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, La ] L∧ ↔ ( p0 S[L∧ ] L∧ → ( ∃p∧ ,(p∧ ∈ L) ∧ (p∧ ∈ L∧ ) ∧ p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, p] p∧ ) ) ).

d.12.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set disjunctively content recursive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ¯ 0, ∀Q

(pwr)

(pwr)

∨

,∀S0 R , ∨

(S0 R ∈ S) → (S0 R ∈ S) → ¯ 0 ∈ S0 R ) → (Q S0 R

(pwr)

∨

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

(pwr)

¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S0 ] S0 R ∨ ↔ ( Q R,a ¯ 0 S[S0 ∨ ] S0 R ∨ → ( Q R ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) ∧ ∃Q ¯ ∨ ∈ S0 R ∨ ) ∧ (Q ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, Q] Q ¯∨ Q ) ) ).

- 258 -

)→

d.13.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set conjunctively content recursive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R ∀Q0 ,

(pwr)

∧

,∀S0 R , ∧

(pwr)

(S0 R ∈ S) → (S0 R ∈ S) → (Q0 ∈ S0 R ) → S0 R

(pwr)

∧

R[S(pwr) ] S0 R → (S0 R ∈ S0 R

∧ Q R[Q0 : c.j.v.ly c.t.t r.crs.ve, S0 R,a ] S0 R 0 0 ∧ Q S[S0 R ∨ ] S R → ( ∃Q∧ ,(Q∧ ∈ S0 R ) ∧ ∧ (Q∧ ∈ S0 R ) ∧ 0 Q R[Q0 : c.j.v.ly c.t.t r.crs.ve, Q] Q∧ 0

(pwr)

)→

↔(

) ) ).

d.14.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set set disjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ∀Sa ,∀¯ p0 ,

,∀L(pwr) ,

(pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sa ∈ S) → (¯ p0 ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L →

(Sa ∈ L(pwr)

(pwr)

)→ - 259 -

p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, Sa ] Sa ↔ ( ∀L∨ ,(L∨ ∈ S) → (L∨ ∈ Sa ) → p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, La ] L∨ ) ).

d.14.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set set conjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr)

∀L(pwr) ∀Sa ,∀p0 ,

,∀L(pwr) ,

(pwr)

(L(pwr) ∈ S) → (L(pwr) ∈ S) → (Sa ∈ S) → (p0 ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) → L(pwr) R[S(pwr) ] L → (pwr)

(Sa ∈ L(pwr) )→ p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, Sa ] Sa ↔ ( ∀L∧ ,(L∧ ∈ S) → (L∧ ∈ Sa ) → p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, La ] L∧ ) ).

d.15.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set set disjunctively content recursive unary relations , given as

- 260 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ¯ 0, ∀Sa ,∀ Q

,∀S0 R

(pwr)

,

(pwr) (pwr)

(S0 R ∈ S) → (S0 R ¯ 0 ∈ L) → (Sa ∈ S) → (Q S0 R

(pwr) (pwr)

R[S(pwr) ] S0 R

(pwr)

(pwr)

∈ S) →

→ S0 R

(pwr)

R[S(pwr) ] S0 R →

(pwr) (pwr)

(Sa ∈ S0 R )→ ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S ] Sa ↔ ( Q a ∨

∨

∀S0 R ,(S0 R ∈ S) → ∨ (S0 R ∈ Sa ) → 0 ¯ R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S0 ] S0 R ∨ Q R,a ) ).

d.15.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the set set conjunctively content recursive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( (pwr) (pwr)

∀S0 R ∀Sa ,∀Q0 ,

,∀S0 R

(pwr)

,

(pwr) (pwr)

(S0 R ∈ S) → (S0 R (Sa ∈ S) → (Q0 ∈ L) → S0 R

(pwr) (pwr)

(Sa ∈ S0 R

R[S(pwr) ] S0 R

(pwr) (pwr)

(pwr)

(pwr)

)→

- 261 -

∈ S) →

→ S0 R

(pwr)

R[S(pwr) ] S0 R →

Q0 R[Q0 : c.j.v.ly c.t.t r.crs.ve, Sa ] Sa ↔ ( ∧

∧

∀S0 R ,(S0 R ∈ S) → ∧ (S0 R ∈ Sa ) → ∧ 0 Q R[Q0 : c.j.v.ly c.t.t r.crs.ve, S0 R,a ] S0 R ) ).

d.16.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀¯ p0 , (L(pwr) ∈ S) → (¯ p0 ∈ L) → L(pwr) R[S(pwr) ] L → ( R[p0 : d.sj.v.ly c.t.t r.crs.ve] p ¯0 ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, Sa ] L(pwr) ) ).

d.16.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content recursive propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀L(pwr) ,∀p0 , (L(pwr) ∈ S) → (p0 ∈ L) → - 262 -

L(pwr) R[S(pwr) ] L → ( R[p0 : c.j.v.ly c.t.t r.crs.ve] p0 ↔ p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, Sa ] L(pwr) ) ).

d.17.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively content recursive unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R

(pwr)

¯ 0, ,∀ Q

(pwr) ¯0 (S0 R ∈ S) → (Q 0 (pwr) 0 SR R[S(pwr) ] S R

∈ S0 R ) → →

( ¯0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve] Q ↔ ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S ] S0 R (pwr) Q a ) ).

d.17.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively content recursive unary relations , given as

- 263 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀S0 R

(pwr)

,∀Q0 ,

(pwr) ∈ S) → (Q0 (S0 R 0 (pwr) R[S(pwr) ] S0 R SR

∈ S0 R ) → →

( R[Q0 : c.j.v.ly c.t.t r.crs.ve] Q0 ↔ (pwr) Q0 R[Q0 : c.j.v.ly c.t.t r.crs.ve, Sa ] S0 R ) ).

d.18.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,(¯ p ∈ L) → ( R[p: d.sj.v.ly nc.ss.ry] p ¯ ↔ T∨ p ¯ ) ).

d.18.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively forbidden propositions , given as

- 264 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → p ¯ Rp¯ p → ( R[p: d.sj.v.ly frb.dd.n] p ↔ R[p: d.sj.v.ly nc.ss.ry] p ¯ ) ).

d.18.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively necessary propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p,(p ∈ L) → ( R[p: c.j.v.ly nc.ss.ry] p ↔ T∧ p ) ).

d.18.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively forbidden propositions , given as

- 265 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → p ¯ Rp¯ p → ( R[p: c.j.v.ly frb.dd.n] p ¯ ↔ R[p: c.j.v.ly nc.ss.ry] p ) ).

d.19.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ p, ∀ Q,∀¯ (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯ → (Ix x ∧ Qx)] Iw Ix Q ( ¯ [Q: d.sj.v.ly nc.ss.ry] Iw Ix QR ↔ R[p: d.sj.v.ly nc.ss.ry] p ¯ ) ).

- 266 -

d.19.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ ∀ Q,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ Q QR ¯Q→ ( QR[Q: d.sj.v.ly frb.dd.n] Iw Ix ↔ ¯ [Q: d.sj.v.ly nc.ss.ry] Iw Ix QR ) ).

d.19.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively necessary unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix ,∀Q,∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → pR[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ (Ix x → Qx)] Iw Ix Q → ( QR[Q: c.j.v.ly nc.ss.ry] Iw Ix ↔ R[p: c.j.v.ly nc.ss.ry] p ) ). - 267 -

d.19.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively forbidden unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ ∀ Q,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ QRQ ¯Q→ ( ¯ [Q: c.j.v.ly frb.dd.n] Iw Ix QR ↔ QR[Q: c.j.v.ly nc.ss.ry] Iw Ix ) ).

d.20.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively necessary third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, ∀p0 ,∀p, ∀¯ p000 , (¯ p0 ∈ L) → (¯ p ∈ L) → (p0 ∈ L) → (p ∈ L) → (¯ p000 ∈ L) → p ¯0 Rp¯ p0 → - 268 -

p ¯ Rp¯ p → p ¯000 S[p: p0 !→ p00 ] pp0 → ( p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ↔ R[p: d.sj.v.ly nc.ss.ry] p ¯000 ) ).

d.20.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively forbidden third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p0 , ∀¯ p, (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯0 Rp¯ p0 → ( p0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ) ).

d.20.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively necessary third propositions , given as

- 269 -

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, ∀p000 , (p0 ∈ L) → (p ∈ L) → (p000 ∈ L) → p000 S[p: p0 → p00 ] pp0 → ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ↔ R[p: c.j.v.ly nc.ss.ry] p000 ) ).

d.20.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively forbidden third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀p0 , ∀p, (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( p ¯0 R[p0 : c.j.v.ly frb.dd.n, p] p ↔ p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ) ).

- 270 -

d.21.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively necessary third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q ∀Q0 ,∀Q, ¯ 000 , ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 0 0 (Q ∈ S R ) → (Q ∈ S0 R ) → ¯ 000 ∈ S0 R ) → (Q 0 ¯ Q0 RQ ¯Q → ¯ Q QR Q → ¯ ¯ 000 S[Qx: Q0 x !→ Q00 x] Q Q0 → Q ( ¯ Iw Ix ¯ 0 R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q Q ↔ ¯ 000 R[Q: Q

d.sj.v.ly nc.ss.ry] Iw Ix

) ).

d.21.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively forbidden third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → - 271 -

¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 000 ¯ Q000 RQ ¯Q → 0 ¯ 0 RQ Q Q → ¯ ( Q0 R[Q0 : ↔ ¯ 0 R[Q0 : Q

¯

d.sj.v.ly frb.dd.n, Q] Q Iw Ix

¯

d.sj.v.ly nc.ss.ry, Q] Q Iw Ix

) ).

d.21.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively necessary third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, ∀Q000 , (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q000 ∈ S0 R ) → Q000 S[Qx: Q0 x → Q00 x] Q Q0 → ( Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix ↔ Q000 R[Q: c.j.v.ly nc.ss.ry] Iw Ix ) ).

- 272 -

d.21.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively forbidden third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀Q0 , ∀Q ∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( ¯ 0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix Q ↔ Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] QIw Ix ) ).

d.22.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly deterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → 0 p ¯ R[p0 : d.sj.v.ly impl.c.tly d.t.c, p] p ¯↔( p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ∨ p ¯0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ). - 273 -

d.22.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly indeterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p0 : d.sj.v.ly impl.c.tly i.d.t.c, p] p ¯↔( ¯ [p0 : d.sj.v.ly nc.ss.ry, p] p p ¯0 R ¯ ∧ ¯ [p0 : d.sj.v.ly frb.dd.n, p] p p ¯0 R ¯ ) ).

d.22.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly deterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → p0 R[p0 : c.j.v.ly impl.c.tly d.t.c, p] p ↔ ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ∨ p0 R[p0 : c.j.v.ly frb.dd.n, p] p ) ).

- 274 -

d.22.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly indeterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → p0 R[p0 : c.j.v.ly impl.c.tly i.d.t.c, p] p ↔ ( ¯ [p0 : c.j.v.ly nc.ss.ry, p] p p0 R ∧ ¯ [p0 : c.j.v.ly frb.dd.n, p] p p0 R ) ).

d.23.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly deterministic third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly d.t.c, Q] Q ¯ Iw Ix ↔ ( Q ¯ 0 R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q ¯ Iw Ix Q ∨ ¯ 0 R[Q0 : Q

¯

d.sj.v.ly frb.dd.n, Q] Q Iw Ix

) ).

- 275 -

d.23.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly indeterministic third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly i.d.t.c, Q] QI ¯ w Ix ↔ ( Q ¯ [Q0 : d.sj.v.ly nc.ss.ry, Q] Q ¯ 0R ¯ Iw Ix Q ∧ ¯ [Q0 : d.sj.v.ly frb.dd.n, Q] Q ¯ 0R ¯ Iw Ix Q ) ).

d.23.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly deterministic third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → Q0 R[Q0 : c.j.v.ly impl.c.tly d.t.c, Q] Q Iw Ix ↔ ( Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix ∨ Q0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix ) ). - 276 -

d.23.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly indeterministic third unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → Q0 R[Q0 : c.j.v.ly impl.c.tly i.d.t.c, Q] Q Iw Ix ↔ ( ¯ [Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix Q0 R ∧ ¯ [Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix Q0 R ) ).

d.24.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly underdeterministic third propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p0 : d.sj.v.ly impl.c.tly u.d.t.c, p] p ¯↔( ¯ [p0 : d.sj.v.ly nc.ss.ry, p] p p ¯0 R ¯ ∨ ¯ [p0 : d.sj.v.ly frb.dd.n, p] p p ¯0 R ¯ ) ).

- 277 -

d.24.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly overdeterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p0 : d.sj.v.ly impl.c.tly o.d.t.c, p] p ¯↔( p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ∧ p ¯0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ).

d.24.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly underdeterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → p0 R[p0 : c.j.v.ly impl.c.tly u.d.t.c, p] p ↔ ( ¯ [p0 : c.j.v.ly nc.ss.ry, p] p p0 R ∨ ¯ [p0 : c.j.v.ly frb.dd.n, p] p p0 R ) ).

- 278 -

d.24.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly overdeterministic third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → p0 R[p0 : c.j.v.ly impl.c.tly o.d.t.c, p] p ↔ ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ∧ p0 R[p0 : c.j.v.ly frb.dd.n, p] p ) ).

d.25.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly underdeterministic third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly u.d.t.c, Q] Q ¯ Iw Ix ↔ ( Q ¯ [Q0 : d.sj.v.ly nc.ss.ry, Q] Q ¯ 0R ¯ Iw Ix Q ∨ ¯ [Q0 : ¯ 0R Q

¯

d.sj.v.ly frb.dd.n, Q] QIw Ix

) ).

- 279 -

d.25.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly overdeterministic third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly o.d.t.c, Q] Q ¯ Iw Ix ↔ ( Q ¯ 0 R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q ¯ Iw Ix Q ∧ ¯ 0 R[Q0 : d.sj.v.ly frb.dd.n, Q] Q ¯ Iw Ix Q ) ).

d.25.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly underdeterministic third unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → Q0 R[Q0 : c.j.v.ly impl.c.tly u.d.t.c, Q] Q Iw Ix ↔ ( ¯ [Q0 : c.j.v.ly nc.ss.ry, Q] QIw Ix Q0 R ∨ ¯ [Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix Q0 R ) ). - 280 -

d.25.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly overdeterministic third unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → Q0 R[Q0 : c.j.v.ly impl.c.tly o.d.t.c, Q] Q Iw Ix ↔ ( Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix ∧ Q0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix ) ).

d.26.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly complete second propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯R[p: d.sj.v.ly expl.c.tly c.pl.te, p0 ] p ¯0 ↔ ( 0 p ¯ R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ∨ p ¯0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ).

- 281 -

d.26.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly incomplete second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯R[p: d.sj.v.ly expl.c.tly i.c.pl.te, p0 ] p ¯0 ↔ ( ¯ [p0 : d.sj.v.ly nc.ss.ry, p] p p ¯0 R ¯ ∨ ¯ [p0 : d.sj.v.ly frb.dd.n, p] p p ¯0 R ¯ ) ).

d.26.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly complete second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → pR[p: c.j.v.ly expl.c.tly c.pl.te, p0 ] p0 ↔ ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ∨ p0 R[p0 : c.j.v.ly frb.dd.n, p] p ) ).

- 282 -

d.26.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly incomplete second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → pR[p: c.j.v.ly expl.c.tly i.c.pl.te, p0 ] p0 ↔ ( ¯ [p0 : c.j.v.ly nc.ss.ry, p] p p0 R ∨ ¯ [p0 : c.j.v.ly frb.dd.n, p] p p0 R ) ).

d.27.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly complete second unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ [Q: d.sj.v.ly expl.c.tly c.pl.te, Q0 ] Q ¯ 0 Iw Ix ↔ ( QR 0 ¯ ¯ Q R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q Iw Ix ∨ ¯ 0 R[Q0 : Q

¯

d.sj.v.ly frb.dd.n, Q] Q Iw Ix

) ).

- 283 -

d.27.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly incomplete second unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ¯ [Q: d.sj.v.ly expl.c.tly i.c.pl.te, Q0 ] Q ¯ 0 Iw Ix ↔ ( QR ¯ [Q0 : d.sj.v.ly nc.ss.ry, Q] QI ¯ 0R ¯ w Ix Q ∧ ¯ [Q0 : ¯ 0R Q

¯

d.sj.v.ly frb.dd.n, Q] QIw Ix

) ).

d.27.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly complete second unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QR[Q: c.j.v.ly expl.c.tly c.pl.te, Q0 ] Q0 Iw Ix ↔ ( Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix ∨ Q0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix ) ). - 284 -

d.27.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly incomplete second unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QR[Q: c.j.v.ly expl.c.tly i.c.pl.te, Q0 ] Q0 Iw Ix ↔ ( ¯ [Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix Q0 R ∧ ¯ [Q0 : c.j.v.ly frb.dd.n, Q] QIw Ix Q0 R ) ).

d.28.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly consistent second propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p,(¯ p0 ∈ L) → (¯ p ∈ L) → p ¯R[p: d.sj.v.ly expl.c.tly c.ss.tnt, p0 ] p ¯0 ↔ ( 0¯ p ¯ R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ∨ ¯ [p0 : d.sj.v.ly frb.dd.n, p] p p ¯0 R ¯ ) ).

- 285 -

d.28.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly inconsistent second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯R[p: d.sj.v.ly expl.c.tly i.c.ss.tnt, p0 ] p ¯0 ↔ ( p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ∧ p ¯0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ).

d.28.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly consistent second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → pR[p: c.j.v.ly expl.c.tly c.ss.tnt, p0 ] p0 ↔ ( ¯ [p0 : c.j.v.ly nc.ss.ry, p] p p0 R ∨ ¯ [p0 : c.j.v.ly frb.dd.n, p] p p0 R ) ).

- 286 -

d.28.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly inconsistent second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → pR[p: c.j.v.ly expl.c.tly i.c.ss.tnt, p0 ] p0 ↔ ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ∧ p0 R[p0 : c.j.v.ly frb.dd.n, p] p ) ).

d.29.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly consistent second unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ [Q: d.sj.v.ly expl.c.tly c.ss.tnt, Q0 ] Q ¯ 0 Iw Ix ↔ ( QR 0¯ ¯ ¯ Q R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q Iw Ix ∨ ¯ [Q0 : ¯ 0R Q

¯

d.sj.v.ly frb.dd.n, Q] Q Iw Ix

) ).

- 287 -

d.29.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element disjunctively explicitly inconsistent second unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ¯ 0 ,∀ Q, ¯ ∀Q (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ¯ [Q: d.sj.v.ly expl.c.tly i.c.ss.tnt, Q0 ] Q ¯ 0 Iw I x ↔ ( QR ¯ 0 R[Q0 : d.sj.v.ly nc.ss.ry, Q] QI ¯ w Ix Q ∧ ¯ 0 R[Q0 : Q

¯

d.sj.v.ly frb.dd.n, Q] Q Iw Ix

) ).

d.29.3 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly consistent second unary relations , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QR[Q: c.j.v.ly expl.c.tly c.ss.tnt, Q0 ] Q0 Iw Ix ↔ ( ¯ [Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix Q0 R ∨ ¯ [Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix Q0 R ) ). - 288 -

d.29.4 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the element conjunctively explicitly inconsistent second unary relations , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀Iw ,∀Ix , ∀Q0 ,∀Q, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QR[Q: c.j.v.ly expl.c.tly i.c.ss.tnt, Q0 ] Q0 Iw Ix ↔ ( Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] Q Iw Ix ∧ Q0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Iw Ix ) ).

d.30.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively implicitly G¨ odel like third propositions , given as LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pG ,∀¯ p,(pG ∈ L) → (¯ p ∈ L) → ( pG R[pG : d.sj.v.ly impl.c.tly G. l.ke, ] p ¯ ↔ pG R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ).

- 289 -

d.30.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively implicitly G¨ odel like third propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ pG ,∀p, (¯ pG ∈ L) → (p ∈ L) → ( p ¯G R[pG : c.j.v.ly impl.c.tly G. l.ke, ] p ↔ p ¯G R[p0 : c.j.v.ly frb.dd.n, p] p ) ).

d.31.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctively G¨ odel like second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ pG ,(¯ pG ∈ L) → ( R[pG : d.sj.v.ly G. l.ke] p ¯G ↔ p ¯G R[pG : d.sj.v.ly impl.c.tly G. l.ke, ] p ¯G ) ).

- 290 -

d.31.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctively G¨ odel like second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀pG ,(pG ∈ L) → ( R[pG : c.j.v.ly G. l.ke] pG ↔ pG R[pG : c.j.v.ly impl.c.tly G. l.ke, ] pG ) ).

d.32.1 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the disjunctive G¨ odel second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ pG ,∀pG , (¯ pG ∈ L) → (pG ∈ L) → p ¯G Rp¯ pG → R[pG : d.sj.ve G. ] p ¯G ↔ ( R[pG : d.sj.v.ly G. l.ke] p ¯G ∧ R[pG : d.sj.v.ly G. l.ke] pG ) ).

- 291 -

d.32.2 There will be a member of the set of elements of the language underlying a natural language constituting a derivative Law of the conjunctive G¨ odel second propositions , given as

LR(nt.l) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → ( ∀¯ pG ,∀pG , (¯ pG ∈ L) → (pG ∈ L) → p ¯0 Rp¯ p0 → R[pG : c.j.ve G. ] p ¯G ↔ ( R[pG : c.j.v.ly G. l.ke] p ¯G ∧ R[pG : c.j.v.ly G. l.ke] pG ) ).

- 292 -

A.1. Axiomatisation Unary Languages

∀S, ∀L,∀Le ,∀Lp , ∀LD ,∀LI , ∀S0 R ,∀S0 p , ∀S0 D ,∀S0 I ,∀S0 Iw , (S ∈ Scd.te ) → (L ∈ S) → (Le ∈ S) → (Lp ∈ S) → (LD ∈ S) → (LI ∈ S) → (S0 R ∈ S) → (S0 p ∈ S) → (S0 D ∈ S) → (S0 I ∈ S) → (S0 Iw ∈ S) → R[Scd.te ] S → ( LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw → (1)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ).

Anchor Law basic unary natural languages

- 293 -

(1)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (2)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: ext.d.d c.p.sed prp.s.n] p ) ).

(1): Law of the propositions’ extended composition

(2)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (3)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: ext.d.d c.p.sed u.ry r.l.n] Q ) ).

(2): Law of the unary relations’ extended composition

- 294 -

(3)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (4)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,(L(pwr) ∈ S) → L(pwr) R[S(pwr) ] L → (Le ∈ L(pwr) ) ) ).

(3): first Law of the composition of the set of elemental propositions

(4)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (5)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( (p ∈ Le ) ↔ R[p: el.l pr.p.s.n] p ) ) ).

(4): second Law of the composition of the set of elemental propositions

- 295 -

(5)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (6)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,(L(pwr) ∈ S) → L(pwr) R[S(pwr) ] L → (Lp ∈ L(pwr) ) ) ).

(5): first Law of the composition of the set of primary propositions

(6)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (7)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( (p ∈ Lp ) ↔ R[p: pr.ry prp.s.n] p ) ) ).

(6): second Law of the composition of the set of primary propositions

- 296 -

(7)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (8)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr) ,(S0 R ∈ S) (pwr) S0 R R[S(pwr) ] S0 R →

∀S0 R

(S0 p ∈ S0 R

(pwr)

→

)

) ).

(7): first Law of the composition of the set of primary unary relations

(8)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (9)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( (Q ∈ S0 p ) ↔ R[Q: pr.ry u.ry r.l.n] Q ) ) ).

(8): second Law of the composition of the set of primary unary relations

- 297 -

(9)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (10)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,(L(pwr) ∈ S) → L(pwr) R[S(pwr) ] L → (LD ∈ L(pwr) ) ) ).

(9): first Law of the composition of the set of Dirac propositions

(10)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (11)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( (p ∈ LD ) ↔ R[p: Dirac prp.s.n] p ) ) ).

(10): second Law of the composition of the set of Dirac propositions

- 298 -

(11)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (12)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr) ,(S0 R ∈ S) (pwr) S0 R R[S(pwr) ] S0 R →

∀S0 R

(S0 D ∈ S0 R

(pwr)

→

)

) ).

(11): first Law of the composition of the set of Dirac unary relations

(12)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (13)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( (Q ∈ S0 D ) ↔ R[Q: Dirac u.ry r.l.n] Q ) ) ).

(12): second Law of the composition of the set of Dirac unary relations

- 299 -

(13)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (14)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,(L(pwr) ∈ S) → L(pwr) R[S(pwr) ] L → (LI ∈ L(pwr) ) ) ).

(13): first Law of the composition of the set of reduced Dirac propositions

(14)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (15)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( (p ∈ LI ) ↔ R[p: r.d.c.ed Dirac prp.s.n] p ) ) ).

(14): second Law of the composition of the set of reduced Dirac propositions

- 300 -

(15)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (16)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr) ,(S0 R ∈ S) (pwr) S0 R R[S(pwr) ] S0 R →

∀S0 R

(S0 I ∈ S0 R

(pwr)

→

)

) ).

(15): first Law of the composition of the set of particular element identifying unary relations

(16)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (17)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( (Q ∈ S0 I ) ↔ R[Ix ] Q ) ) ).

(16): second Law of the composition of the set of particular element identifying unary relations

- 301 -

(17)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (18)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr) ,(S0 I ∈ S) (pwr) S0 I R[S(pwr) ] S0 I →

∀S0 I

(S0 Iw ∈ S0 I

(pwr)

→

)

) ).

(17): first Law of the composition of the set of particular world identifying unary relations

(18)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (19)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Ix ,(Ix ∈ S0 I ) → ( (Ix ∈ S0 Iw ) ↔ R[Iw ] Ix ) ) ).

(18): second Law of the composition of the set of particular world identifying unary relations

- 302 -

(19)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (20)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: j.nt el.l pr.p.s.n] p → R[p: b.c r.d.c.d prp.s.n] p ) ) ).

(19): Law of the joint elemental propositions’ basic reduced composition

(20)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (21)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: b.c c.p.sed u.ry r.l.n] Q ∧ R[Q: pr.ry u.ry r.l.n] Q )→ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q ) ).

(20): Law of the basic composed and primary unary relations

- 303 -

(21)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (22)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: pr.ry u.ry r.l.n] Q → R[Q: ext.d.d c.p.sed, pr.ry u.ry r.l.n] Q ) ) ).

(21): Law of the extended composed and primary unary relations

(22)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (23)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: a.l.c prp.s.n] p ↔ ( R[p¯∨ ] p T

∨ R[p∨ ] p F

) ) ).

(22): Law of the analytic propositions

- 304 -

(23)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (24)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: s.th.c prp.s.n] p ↔ ¯ [p: a.l.c prp.s.n] p R ) ) ).

(23): Law of the synthetic propositions

(24)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (25)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: a.l.c u.ry r.l.n] Q ↔ ( R[Q ¯ ∨] Q T

∨ R[Q∨ ] Q F

) ) ).

(24): Law of the analytic unary relations

- 305 -

(25)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (26)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: s.th.c u.ry r.l.n] Q ↔ ¯ [Q: a.l.c u.ry r.l.n] Q R ) ) ).

(25): Law of the synthetic unary relations

(26)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (27)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: a pr. prp.s.n] p ↔ R[p: pr.ry prp.s.n] p ) ) ).

(26): Law of the a priori propositions

- 306 -

(27)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (28)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: a p.st prp.s.n] p ↔ R[p: s.cd.ry prp.s.n] p ) ) ).

(27): Law of the a posteriori propositions

(28)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (29)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: a pr. u.ry r.l.n] Q ↔ R[Q: pr.ry u.ry r.l.n] Q ) ) ).

(28): Law of the a priori unary relations

- 307 -

(29)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (30)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: a p.st u.ry r.l.n] Q ↔ R[Q: s.cd.ry u.ry r.l.n] Q ) ) ).

(29): Law of the a posteriori unary relations

(30)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (31)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: pr.ry prp.s.n] p ∨ R[p: s.cd.ry prp.s.n] p ) ) ).

(30): Law of the propositions’ primary or secondary composition

- 308 -

(31)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (32)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: pr.ry u.ry r.l.n] Q ∨ R[Q: s.cd.ry u.ry r.l.n] Q ) ) ).

(31): Law of the unary relations’ primary or secondary composition

(32)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (33)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: pr.ry prp.s.n] p ∧ R[p: s.cd.ry prp.s.n] p )→ R[p: a.l.c prp.s.n] p ) ).

(32): Law of the primary and secondary propositions

- 309 -

(33)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (34)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: pr.ry u.ry r.l.n] Q ∧ R[Q: s.cd.ry u.ry r.l.n] Q )→ R[Q: a.l.c u.ry r.l.n] Q ) ).

(33): Law of the primary and secondary unary relations

(34)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (35)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ ∃p∨ F ,(pF ∈ L) ∧ R[p∨ ] p∨ F F

) ).

(34): Law of the existence of the basic disjunctively forbidden propositions

- 310 -

(35)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (36)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ ∀p∨ F,b ,∀pF,a , ∨ (p∨ F,b ∈ L) → (pF,a ∈ L) → ( R[p∨ ] p∨ F,b F

∧ R[p∨ ] p∨ F,a F

)→ ∨ p∨ F,b Rid pF,a ) ).

(35): Law of the uniqueness of the basic disjunctively forbidden propositions

(36)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (37)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∃¯ p∨ p∨ T ,(¯ T ∈ L) ∧ R[p¯∨ ] p ¯∨ T T

) ).

(36): Law of the existence of the basic disjunctively necessary propositions

- 311 -

(37)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (38)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∨ p∨ T,b ,∀¯ T,a , (¯ p∨ p∨ T,b ∈ L) → (¯ T,a ∈ L) → ( ¯∨ R[p¯∨ ] p T,b T

∧ R[p¯∨ ] p ¯∨ T,a T

)→ p ¯∨ ¯∨ T,b Rid p T,a ) ).

(37): Law of the uniqueness of the basic disjunctively necessary propositions

(38)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (39)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∃¯ p∧ p∧ F ,(¯ F ∈ L) ∧ R[p¯∧ ] p ¯∧ F F

) ).

(38): Law of the existence of the basic conjunctively forbidden propositions

- 312 -

(39)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (40)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∧ p∧ F,b ,∀¯ F,a , (¯ p∧ p∧ F,b ∈ L) → (¯ F,a ∈ L) → ( ¯∧ R[p¯∧ ] p F,b F

∧ R[p¯∧ ] p ¯∧ F,a F

)→ p ¯∧ ¯∧ F,b Rid p F,a ) ).

(39): Law of the uniqueness of the basic conjunctively forbidden propositions

(40)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (41)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ ∃p∧ T ,(pT ∈ L) ∧ R[p∧ ] p∧ T T

) ).

(40): Law of the existence of the basic conjunctively necessary propositions

- 313 -

(41)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (42)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ ∀p∧ T,b ,∀pT,a , ∧ (p∧ T,b ∈ L) → (pT,a ∈ L) → ( R[p∧ ] p∧ T,b T

∧ R[p∧ ] p∧ T,a T

)→ ∧ p∧ T,b Rid pT,a ) ).

(41): Law of the uniqueness of the basic conjunctively necessary propositions

(42)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (43)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ 0 ∃Q∨ F ,(QF ∈ S R ) ∧ ∨ R[Q∨ ] QF F

) ).

(42): Law of the existence of the basic disjunctively forbidden unary relations

- 314 -

(43)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (44)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ ∀Q∨ F,b ,∀QF,a , 0 ∨ 0 (Q∨ F,b ∈ S R ) → (QF,a ∈ S R ) → ( R[Q∨ ] Q∨ F,b F

∧ R[Q∨ ] Q∨ F,a F

)→ ∨ Q∨ F,b Rid QF,a ) ).

(43): Law of the uniqueness of the basic disjunctively forbidden unary relations

(44)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (45)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( 0 ¯∨ ¯∨ ∃Q T ,(QT ∈ S R ) ∧ ∨ ¯ R[Q ¯ ∨ ] QT T

) ).

(44): Law of the existence of the basic disjunctively necessary unary relations

- 315 -

(45)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (46)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∨ ¯∨ ∀Q T,b ,∀ QT,a , 0 0 ¯∨ ¯∨ (Q ∈ S R ) → (QT,a ∈ S R ) → T,b ( ¯∨ R[Q ¯ ∨ ] QT,b T ∧ ¯∨ R[Q ¯ ∨ ] QT,a T

)→ ¯∨ ¯∨ Q T,b Rid QT,a ) ).

(45): Law of the uniqueness of the basic disjunctively necessary unary relations

(46)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (47)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( 0 ¯∧ ¯∧ ∃Q F ,(QF ∈ S R ) ∧ ¯∧ R[Q ¯ ∧ ] QF F

) ).

(46): Law of the existence of the basic conjunctively forbidden unary relations

- 316 -

(47)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (48)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∧ ¯∧ ∀Q F,b ,∀ QF,a , 0 0 ¯∧ ¯∧ (Q ∈ S R ) → (QF,a ∈ S R ) → F,b ( ¯∧ R[Q ¯ ∧ ] QF,b F ∧ ¯∧ R[Q ¯ ∧ ] QF,a F

)→ ¯∧ ¯∧ Q F,b Rid QF,a ) ).

(47): Law of the uniqueness of the basic conjunctively forbidden unary relations

(48)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (49)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ 0 ∃Q∧ T ,(QT ∈ S R ) ∧ R[Q∧ ] Q∧ T T

) ).

(48): Law of the existence of the basic conjunctively necessary unary relations

- 317 -

(49)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (50)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ ∀Q∧ T,b ,∀QT,a , 0 ∧ 0 (Q∧ T,b ∈ S R ) → (QT,a ∈ S R ) → ( R[Q∧ ] Q∧ T,b T

∧ R[Q∧ ] Q∧ T,a T

)→ ∧ Q∧ T,b Rid QT,a ) ).

(49): Law of the uniqueness of the basic conjunctively necessary unary relations

- 318 -

(50)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (51)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,∀p,∀p∨ F, (¯ p ∈ L) → (p ∈ L) → (p∨ F ∈ L) → p ¯ Rp¯ p → ( R[p∨ ] p∨ F → F

p∨ ¯p F S[p: p00 !∧ p0 ] p ) ) ).

(50): Law of the composition of the basic disjunctively forbidden propositions

(51)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (52)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ ∀¯ p∨ T ,∀pF , ∨ (¯ p∨ T ∈ L) → (pF ∈ L) → ∨ ∨ p ¯T Rp¯ pF ( R[p∨ ] p∨ F → F

R[p¯∨ ] p ¯∨ T T

) ) ).

(51): Law of the composition of the basic disjunctively necessary propositions

- 319 -

(52)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (53)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p∨ p∧ F ,∀¯ F, (p∨ p∧ F ∈ L) → (¯ F ∈ L) → ∧ ∨ p ¯F Rp¯ pF ( R[p∨ ] p∨ F → F

R[p¯∧ ] p ¯∧ F F

) ) ).

(52): Law of the composition of the basic conjunctively forbidden propositions

- 320 -

(53)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (54)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ ∀¯ p∧ F ,∀pT , ∧ (¯ p∧ F ∈ L) → (pT ∈ L) → ∧ ∧ pT Rp¯ p ¯F → ( ¯∧ R[p¯∧ ] p F → F

R[p∧ ] p∧ T T

) ) ).

(53): Law of the composition of the basic conjunctively necessary propositions

- 321 -

(54)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (55)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ ¯ ∀ Q,∀Q,∀Q F, 0 ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q∨ (Q F ∈ S R) → ¯ QRQ ¯Q→ ( R[Q∨ ] Q∨ F → F

¯ Q∨ F S[Qx: Q00 x !∧ Q0 x] Q Q ) ) ).

(54): tions

Law of the composition of the basic disjunctively forbidden unary rela-

- 322 -

(55)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (56)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨ ¯∨ ∀Q T ,∀QF , 0 ∨ 0 ¯∨ (Q T ∈ S R ) → (QF ∈ S R ) → ∨ ∨ ¯ QT RQ ¯ QF →

( R[Q∨ ] Q∨ F → F

¯∨ R[Q ¯ ∨ ] QT T

) ) ).

(55): Law of the composition of the basic disjunctively necessary unary relations

- 323 -

(56)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (57)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∧ ∀Q∨ F ,∀ QF , 0 0 ¯∧ (Q∨ F ∈ S R ) → (QF ∈ S R ) → ∧ ∨ ¯ QF RQ ¯ QF → ( R[Q∨ ] Q∨ F → F

¯∧ R[Q ¯ ∧ ] QF F

) ) ).

(56): Law of the composition of the basic conjunctively forbidden unary relations

- 324 -

(57)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (58)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ ¯∧ ∀Q F ,∀QT , 0 ∧ 0 ¯∧ (Q F ∈ S R ) → (QT ∈ S R ) → ∧ ∧ ¯ QT RQ ¯ QF → ( ¯∧ R[Q ¯ ∧ ] QF → F

R[Q∧ ] Q∧ T T

) ) ).

(57): Law of the composition of the basic conjunctively necessary unary relations

(58)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (59)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∨ p∨ T ,(¯ T ∈ L) → ( R[p¯∨ ] p ¯∨ T → T

T∨ p ¯∨ T ) ) ).

(58): Law of the resolution of the basic disjunctively necessary propositions

- 325 -

(59)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (60)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∧ ∀p∧ T ,(pT ∈ L) → ( R[p∧ ] p∧ T → T

T∧ p∧ T ) ) ).

(59): Law of the resolution of the basic conjunctively necessary propositions

- 326 -

(60)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (61)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀Ix , ∨ ∀Q∨ F ,∀pF , (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → 0 ∨ (Q∨ F ∈ S R ) → (pF ∈ L) → R[Iw I → w cd.te ] cd.te

p∨ F R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Ix Q∨ F →( R[Q∨ ] Q∨ → F F

R[p∨ ] p∨ F F

) ) ).

(60): Law of the resolution of the basic disjunctively forbidden unary relations

(61)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (62)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,∃¯ p, (p ∈ L) → (¯ p ∈ L) ∧ p ¯ Rp¯ p ) ).

(61): Law of the existence of the propositions’ complements

- 327 -

(62)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (63)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pb ,∀¯ pa ,∀p, (¯ pb ∈ L) → (¯ pa ∈ L) → (p ∈ L) → ( p ¯b Rp¯ p ∧ p ¯a Rp¯ p )→ p ¯b Rid p ¯a ) ).

(62): Law of the uniqueness of the propositions’ complements

(63)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (64)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀Q,∃ Q, ¯ ∈ S0 R ) ∧ (Q ∈ S0 R ) → (Q ¯ QRQ ¯Q ) ).

(63): Law of the existence of the unary relations’ complements

- 328 -

(64)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (65)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ b ,∀ Q ¯ a ,∀Q, ∀Q ¯ a ∈ S0 R ) → (Q ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q (Q ( ¯ b RQ Q ¯Q ∧ ¯ a RQ Q ¯Q )→ ¯ b Rid Q ¯a Q ) ).

(64): Law of the uniqueness of the unary relations’ complements

(65)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (66)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → ( p ¯ Rp¯ p → pRp¯ p ¯ ) ) ).

(65): Law of the reciprocity of the propositions’ complementarity

- 329 -

(66)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (67)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀ Q,∀Q, ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ Q QR ¯Q→ ¯ QRQ ¯Q ) ) ).

(66): Law of the reciprocity of the unary relations’ complementarity

(67)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (68)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → ( p ¯ Rp¯ p → p ¯ S[p: p00 !∧ p0 ] pp ) ) ).

(67): Law of the composition of the propositions’ complements

- 330 -

(68)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (69)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀ Q,∀Q, ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ Q QR ¯Q→ ¯ [Qx: Q00 x !∧ Q0 x] Q Q QS ) ) ).

(68): Law of the composition of the unary relations’ complements

(69)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (70)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∃¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) ∧ p ¯ S[p: p00 !∧ p0 ] p00 p0 ) ).

(69): Law of the existence of the negatively conjunctive propositions

- 331 -

(70)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (71)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ pb ,∀¯ pa , (p00 ∈ L) → (p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b S[p: p00 !∧ p0 ] p00 p0 ∧ p ¯a S[p: p00 !∧ p0 ] p00 p0 )→ p ¯b Rid p ¯a ) ).

(70): Law of the uniqueness of the negatively conjunctive propositions

- 332 -

(71)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (72)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ ∃ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) ∧ (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 QS ) ).

(71): Law of the existence of the negatively conjunctive unary relations

- 333 -

(72)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (73)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ b ,∀ Q ¯ a, ∀Q (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b S[Qx: Q00 x !∧ Q0 x] Q00 Q0 Q ∧ ¯ a S[Qx: Q00 x !∧ Q0 x] Q00 Q0 Q )→ ¯ b Rid Q ¯a Q ) ).

(72): Law of the uniqueness of the negatively conjunctive unary relations

- 334 -

(73)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (74)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( R[p: b.c c.p.sed prp.s.n] p00 ∧ R[p: b.c c.p.sed prp.s.n] p0 )→ R[p: b.c c.p.sed prp.s.n] p ¯ ) ).

(73): Law of the basic composed negatively conjunctive propositions

- 335 -

(74)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (75)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS ( R[Q: b.c c.p.sed u.ry r.l.n] Q00 ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q0 )→ ¯ R[Q: b.c c.p.sed u.ry r.l.n] Q ) ).

(74): Law of the basic composed negatively conjunctive unary relations

- 336 -

(75)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (76)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( R[p: pr.ry prp.s.n] p00 ∧ R[p: pr.ry prp.s.n] p0 )→ R[p: pr.ry prp.s.n] p ¯ ) ).

(75): Law of the primary negatively conjunctive propositions

- 337 -

(76)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (77)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( ¯ [p: pr.ry prp.s.n] p00 R ∨ ¯ [p: pr.ry prp.s.n] p0 R )→ R[p: s.cd.ry prp.s.n] p ¯ ) ).

(76): Law of the secondary negatively conjunctive propositions

- 338 -

(77)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (78)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) ¯ ∈ S0 R ) → → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS ( R[Q: pr.ry u.ry r.l.n] Q00 ∧ R[Q: pr.ry u.ry r.l.n] Q0 )→ ¯ R[Q: pr.ry u.ry r.l.n] Q ) ).

(77): Law of the primary negatively conjunctive unary relations

- 339 -

(78)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (79)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS ( ¯ [Q: pr.ry u.ry r.l.n] Q00 R ∨ ¯ [Q: pr.ry u.ry r.l.n] Q0 R )→ ¯ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

(78): Law of the secondary negatively conjunctive unary relations

- 340 -

(79)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (80)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∃¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) ∧ p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 ) ).

(79): Law of the existence of the disjunctive propositions

- 341 -

(80)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (81)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∀¯ pb ,∀¯ pa , (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b S[p: p00 ∨ p0 ] p ¯00 p ¯0 ∧ p ¯a S[p: p00 ∨ p0 ] p ¯00 p ¯0 )→ p ¯b Rid p ¯a ) ).

(80): Law of the uniqueness of the disjunctive propositions

- 342 -

(81)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (82)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∃ Q, ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) ∧ (Q ¯ [Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 QS ) ).

(81): Law of the existence of the disjunctive unary relations

- 343 -

(82)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (83)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ b ,∀ Q ¯ a, ∀Q ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 Q ∧ ¯ a S[Qx: Q00 x ∨ Q0 x] Q ¯ 00 Q ¯0 Q )→ ¯ b Rid Q ¯a Q ) ).

(82): Law of the uniqueness of the disjunctive unary relations

- 344 -

(83)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (84)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀p00 , ∀¯ p0 ,∀p0 , ∀¯ p, (¯ p00 ∈ L) → (p00 ∈ L) → (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯00 Rp¯ p00 → p ¯00 Rp¯ p00 → p ¯0 Rp¯ p0 → ( p ¯ S[p: p00 ∨ p0 ] p ¯00 p ¯0 → p ¯ S[p: p00 !∧ p0 ] p00 p0 ) ) ).

(83): Law of the composition of the disjunctive propositions

- 345 -

(84)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (85)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀Q00 , ∀Q ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, ¯ 00 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 00 ¯ 00 RQ Q ¯Q → 0 0 ¯ Q RQ ¯Q → ( ¯ 00 Q ¯0 → ¯ [Qx: Q00 x ∨ Q0 x] Q QS 00 0 ¯ QS[Qx: Q00 x !∧ Q0 x] Q Q ) ) ).

(84): Law of the composition of the disjunctive unary relations

- 346 -

(85)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (86)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∃p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) ∧ pS[p: p00 ∧ p0 ] p00 p0 ) ).

(85): Law of the existence of the conjunctive propositions

- 347 -

(86)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (87)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀pb ,∀pa , (p00 ∈ L) → (p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → ( pb S[p: p00 ∧ p0 ] p00 p0 ∧ pa S[p: p00 ∧ p0 ] p00 p0 ) )→ pb Rid pa ) ).

(86): Law of the uniqueness of the conjunctive propositions

- 348 -

(87)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (88)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∃Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ QS[Qx: Q00 x ∧ Q0 x] Q00 Q0 ) ).

(87): Law of the existence of the conjunctive unary relations

- 349 -

(88)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (89)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∀Qb ,∀Qa , (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( Qb S[Qx: Q00 x ∧ Q0 x] Q00 Q0 ∧ Qa S[Qx: Q00 x ∧ Q0 x] Q00 Q0 )→ Qb Rid Qa ) ).

(88): Law of the uniqueness of the conjunctive unary relations

- 350 -

(89)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (90)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ p,∀p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → (p ∈ L) → p ¯ S[p: p00 !∧ p0 ] p00 p0 → ( pS[p: p00 ∧ p0 ] p00 p0 → p ¯ Rp¯ p ) ) ).

(89): Law of the composition of the conjunctive propositions

- 351 -

(90)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (91)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ ∀ Q,∀Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ [Qx: Q00 x !∧ Q0 x] Q00 Q0 → QS ( QS[Qx: Q00 x ∧ Q0 x] Q00 Q0 → ¯ Q QR ¯Q ) ) ).

(90): Law of the composition of the conjunctive unary relations

(91)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (92)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀¯ p0 , ∃p, (p00 ∈ L) → (¯ p0 ∈ L) → (p ∈ L) ∧ pS[p: p0 → p00 ] p ¯0 p00 ) ).

(91): Law of the existence of the implicative propositions

- 352 -

(92)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (93)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀¯ p0 , ∀pb ,∀pa , (p00 ∈ L) → (¯ p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → ( pb S[p: p0 → p00 ] p ¯0 p00 ∧ pa S[p: p0 → p00 ] p ¯0 p00 )→ pb Rid pa ) ).

(92): Law of the uniqueness of the implicative propositions

- 353 -

(93)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (94)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Q00 ,∀ Q ∃Q, ¯ 0 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q (Q ∈ S0 R ) ∧ ¯ 0 Q00 QS[Qx: Q0 x → Q00 x] Q ) ).

(93): Law of the existence of the implicative unary relations

- 354 -

(94)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (95)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Q00 ,∀ Q ∀Qb ,∀Qa , ¯ 0 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( ¯ 0 Q00 Qb S[Qx: Q0 x → Q00 x] Q ∧ ¯ 0 Q00 Qa S[Qx: Q0 x → Q00 x] Q )→ Qb Rid Qa ) ).

(94): Law of the uniqueness of the implicative unary relations

- 355 -

(95)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (96)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 , ∀¯ p0 ,∀p0 , ∀p, (p00 ∈ L) → (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( pS[p: p0 → p00 ] p ¯0 p00 → pS[p: p00 ∨ p0 ] p00 p0 ) ) ).

(95): Law of the composition of the implicative propositions

- 356 -

(96)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (97)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 , ¯ 0 ,∀Q0 , ∀Q ∀Q, (Q00 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( ¯ 0 Q00 → QS[Qx: Q0 x → Q00 x] Q QS[Qx: Q00 x ∨ Q0 x] Q00 Q0 ) ) ).

(96): Law of the composition of the implicative unary relations

- 357 -

(97)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (98)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀¯ p0 , ∃¯ p, (p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) ∧ p ¯ S[p: p0 !→ p00 ] p ¯0 p00 ) ).

(97): Law of the existence of the prohibitive propositions

- 358 -

(98)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (99)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀¯ p0 , ∀¯ pb ,∀¯ pa , (p00 ∈ L) → (¯ p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b S[p: p0 !→ p00 ] p ¯0 p00 ∧ p ¯a S[p: p0 !→ p00 ] p ¯0 p00 )→ p ¯b Rid p ¯a ) ).

(98): Law of the uniqueness of the prohibitive propositions

- 359 -

(99)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (100)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Q00 ,∀ Q ¯ ∃ Q, ¯ 0 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q ¯ ∈ S0 R ) ∧ (Q ¯ [Qx: Q0 x !→ Q00 x] Q ¯ 0 Q00 QS ) ).

(99): Law of the existence of the prohibitive unary relations

- 360 -

(100)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (101)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Q00 ,∀ Q ¯ b ,∀ Q ¯ a, ∀Q ¯ 0 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b S[Qx: Q0 x !→ Q00 x] Q ¯ 0 Q00 Q ∧ ¯ a S[Qx: Q0 x !→ Q00 x] Q ¯ 0 Q00 Q )→ ¯ b Rid Q ¯a Q ) ).

(100): Law of the uniqueness of the prohibitive unary relations

- 361 -

(101)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (102)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀p00 , ∀¯ p0 , ∀¯ p, (¯ p00 ∈ L) → (p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ Le ) → p ¯00 Rp¯ p00 → ( p ¯ S[p: p0 !→ p00 ] p ¯0 p00 p ¯ S[p: p00 ∧ p0 ] p ¯00 p ¯0 ) ) ).

(101): Law of the composition of the prohibitive propositions

- 362 -

(102)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (103)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀Q00 , ∀Q ¯ 0, ∀Q ¯ ∀ Q, ¯ 00 ∈ S0 R ) → (Q00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 00 ¯ 00 RQ Q ¯Q → ( ¯ [Qx: Q0 x !→ Q00 x] Q ¯ 0 Q00 → QS ¯ 00 Q ¯0 ¯ [Qx: Q00 x ∧ Q0 x] Q QS ) ) ).

(102): Law of the composition of the prohibitive unary relations

- 363 -

(103)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (104)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∃p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) ∧ pS[p: p00 ↔ p0 ] p00 p0 ) ).

(103): Law of the existence of the bijective propositions

- 364 -

(104)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (105)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀pb ,∀pa , (p00 ∈ L) → (p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → ( pb S[p: p00 ↔ p0 ] p00 p0 ∧ pa S[p: p00 ↔ p0 ] p00 p0 )→ pb Rid pa ) ).

(104): Law of the uniqueness of the bijective propositions

- 365 -

(105)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (106)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∃Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ QS[Qx: Q00 x ↔ Q0 x] Q00 Q0 ) ).

(105): Law of the existence of the bijective unary relations

- 366 -

(106)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (107)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∀Qb ,∀Qa , (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( Qb S[Qx: Q00 x ↔ Q0 x] Q00 Q0 ∧ Qa S[Qx: Q00 x ↔ Q0 x] Q00 Q0 )→ Qb Rid Qa ) ).

(106): Law of the uniqueness of the bijective unary relations

- 367 -

(107)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (108)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀pb ,∀pa , ∀p, (p00 ∈ L) → (p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → (p ∈ L) → pb S[p: p0 → p00 ] p00 p0 → pa S[p: p0 → p00 ] p0 p00 → ( pS[p: p00 ↔ p0 ] p00 p0 → pS[p: p00 ∧ p0 ] pb pa ) ) ).

(107): Law of the composition of the bijective propositions

- 368 -

(108)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (109)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∀Qb ,∀Qa , ∀Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → (Q ∈ S0 R ) → Qb S[Qx: Q0 x → Q00 x] Q00 Q0 → Qa S[Qx: Q0 x → Q00 x] Q0 Q00 → ( QS[Qx: Q00 x ↔ Q0 x] Q00 Q0 → QS[Qx: Q00 x ∧ Q0 x] Qb Qa ) ) ).

(108): Law of the composition of the bijective unary relations

- 369 -

(109)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (110)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∃¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) ∧ p ¯ S[p: p00 !↔ p0 ] p00 p0 ) ).

(109): Law of the existence of the exclusively disjunctive propositions

- 370 -

(110)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (111)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ pb ,∀¯ pa , (p00 ∈ L) → (p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b S[p: p00 !↔ p0 ] p00 p0 ∧ p ¯a S[p: p00 !↔ p0 ] p00 p0 )→ p ¯b Rid p ¯a ) ).

(110): Law of the uniqueness of the exclusively disjunctive propositions

- 371 -

(111)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (112)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ ∃ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) ∧ (Q ¯ [Qx: Q00 x !↔ Q0 x] Q00 Q0 QS ) ).

(111): Law of the existence of the exclusively disjunctive unary relations

- 372 -

(112)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (113)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ b ,∀ Q ¯ a, ∀Q (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b S[Qx: Q00 x !↔ Q0 x] Q00 Q0 Q ∧ ¯ a S[Qx: Q00 x !↔ Q0 x] Q00 Q0 Q )→ ¯ b Rid Q ¯a Q ) ).

(112): Law of the uniqueness of the exclusively disjunctive unary relations

- 373 -

(113)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (114)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀¯ pb ,∀¯ pa , ∀¯ p, (p00 ∈ L) → (p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → (¯ p ∈ L) → p ¯b S[p: p0 !→ p00 ] p00 p0 → p ¯a S[p: p0 !→ p00 ] p0 p00 → ( p ¯ S[p: p00 !↔ p0 ] p00 p0 → p ¯ S[p: p00 ∨ p0 ] p ¯b p ¯a ) ) ).

(113): Law of the composition of the exclusively disjunctive propositions

- 374 -

(114)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (115)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ¯ b ,∀ Q ¯ a, ∀Q ¯ ∀ Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q 0 ¯ (Q ∈ S R ) → ¯ b S[Qx: Q0 x !→ Q00 x] Q00 Q0 → Q ¯ a S[Qx: Q0 x !→ Q00 x] Q0 Q00 → Q ( ¯ [Qx: Q00 x !↔ Q0 x] Q00 Q0 → QS ¯ [Qx: Q00 x ∨ Q0 x] Q ¯bQ ¯a QS ) ) ).

(114): Law of the composition of the exclusively disjunctive unary relations

- 375 -

(115)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (116)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∃¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) ∧ L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ ) ).

(115): Law of the existence of the disjoint propositions

- 376 -

(116)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (117)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ pb ,∀¯ pa , (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ pb ∈ L) → (¯ pa ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → ( p ¯b S[L∨ ] L∨ ∧ p ¯a S[L∨ ] L∨ )→ p ¯b Rid p ¯a ) ).

(116): Law of the uniqueness of the disjoint propositions

- 377 -

(117)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (118)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∧ , ∃p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) ∧ L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → pS[L∧ ] L∧ ) ).

(117): Law of the existence of the joint propositions

- 378 -

(118)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (119)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∧ , ∀pb ,∀pa , (L(pwr) ∈ S) → (L∧ ∈ S) → (pb ∈ L) → (pa ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → ( pb S[L∧ ] L∧ ∧ pa S[L∧ ] L∧ )→ pb Rid pa ) ).

(118): Law of the uniqueness of the joint propositions

- 379 -

(119)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (120)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ¯ ∃ Q,

(pwr)

∨

,∀S0 R , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 ¯ (Q ∈ S R ) ∧ S0 R

(pwr)

∨ (S0 R

R[S(pwr) ] S0 R → (pwr)

∈ S0 R ¯ [S0 ∨ ] S0 R ∨ QS R

)→

) ).

(119): Law of the existence of the disjoint unary relations

- 380 -

(120)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (121)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ¯ b ,∀ Q ¯ a, ∀Q ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 ¯ a ∈ S0 R ) → ¯ (Qb ∈ S R ) → (Q S0 R

(pwr)

∨ (S0 R

R[S(pwr) ] S0 R →

∈ S0 R

(pwr)

)→

( ¯ b S[S0 ∨ ] S0 R ∨ Q R ∧ ¯ a S[S0 ∨ ] S0 R ∨ Q R )→ ¯ b Rid Q ¯a Q ) ).

(120): Law of the uniqueness of the disjoint unary relations

- 381 -

(121)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (122)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ∃Q,

(pwr)

∧

,∀S0 R , ∧

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 (Q ∈ S R ) ∧ S0 R

(pwr)

∧ (S0 R

R[S(pwr) ] S0 R → (pwr)

∈ S0 R ∧ QS[S0 R ∧ ] S0 R

)→

) ).

(121): Law of the existence of the joint unary relations

- 382 -

(122)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (123)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∧

,∀S0 R , ∀S0 R ∀Qb ,∀Qa , ∧

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 (Qb ∈ S R ) → (Qa ∈ S0 R ) → S0 R

(pwr)

∧ (S0 R

R[S(pwr) ] S0 R →

∈ S0 R

(pwr)

)→

( ∧

Qb S[S0 R ∧ ] S0 R ∧ ∧ Qa S[S0 R ∧ ] S0 R )→ Qb Rid Qa ) ).

(122): Law of the uniqueness of the joint unary relations

- 383 -

(123)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (124)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) , ∀L∨ ,∀L∧ , ∀¯ p,∀p, (L(pwr) ∈ S) → (L∨ ∈ S) → (L∧ ∈ S) → (¯ p ∈ L) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → (L∧ ∈ L(pwr) ) → ( ∀¯ p∨ ,∀p∧ , (¯ p∨ ∈ L) → (p∧ ∈ L) → p ¯∨ Rp¯ p∧ → ( (¯ p∨ ∈ L∨ ) ↔ (p∧ ∈ L∧ ) ) )→ p ¯ Rp¯ p → ( p ¯S[L∨ ] L∨ → pS[L∧ ] L∧ ) ) ).

(123): Law of the complementarity of the disjoint and the joint propositions

- 384 -

(124)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (125)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

, ∀S0 R ∨ ∧ ∀S0 R ,∀S0 R , ¯ ∀ Q,∀Q, (pwr)

∈ S) → (S0 R ∧ ∨ (S0 R ∈ S) → (S0 R ∈ S) → 0 ¯ (Q ∈ S R ) → (Q ∈ S0 R ) → S0 R

(pwr) ∨

R[S(pwr) ] S0 R →

(S0 R ∈ S0 R ∧ (S0 R

∈

(pwr)

)→

(pwr) S0 R )

→

( ¯ ∨ ,∀Q∧ , ∀Q ¯ (Q∨ ∈ S0 R ) → (Q∧ ∈ S0 R ) → ∧ ¯ ∨ RQ Q ¯Q → ( ¯ ∨ ∈ S0 R ∨ ) (Q ↔ ∧ (Q∧ ∈ S0 R ) ) )→ ¯ Q QR ¯Q→ ( ¯ [S0 ∨ ] S0 R ∨ → QS R QS[S0 R ∧ ] S0 R

∧

) ) ).

(124): Law of the complementarity of the disjoint and the joint unary relations

- 385 -

(125)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (126)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) , ∨ ∀L∨ b ,∀La , ∨ ∀L , ∀¯ pb ,∀¯ pa , ∀¯ p, (L(pwr) ∈ S) → ∨ (L∨ b ∈ S)(La ∈ S) ∨ (L ∈ S) → (¯ pb ∈ L)(¯ pa ∈ L) (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (pwr) (L∨ )→ b ∈L ∨ (pwr) (La ∈ L )→ (L∨ ∈ L(pwr) ) → p ¯b S[L∨ ] L∨ b → p ¯a S[L∨ ] L∨ a → p ¯S[L∨ ] L∨ → ( ∨ L∨ S[S: S00 ∪ S0 ] L∨ b La → p ¯ S[p: p00 ∨ p0 ] p ¯b p ¯a )

) ).

(125): third Law of the composition of the disjoint propositions

- 386 -

(126)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (127)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → ( L∨ R[S: s.ngle el.t s.t, x] p ¯→ p ¯S[L∨ ] L∨ ) ) ).

(126): second Law of the composition of the disjoint propositions

- 387 -

(127)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (128)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ p∨ T, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p∨ T ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → ( R{} L∨ → R[p¯∨ ] p ¯∨ T T

) ) ).

(127): first Law of the composition of the disjoint propositions

- 388 -

(128)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (129)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

, ∀S0 R ∨ ∨ ∀S0 R,b ,∀S0 R,a , ∨

∀S0 R , ¯ b ,∀ Q ¯ a, ∀Q ¯ ∀ Q, (pwr)

∈ S) → (S0 R ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → ∨

(S0 R ∈ S) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q 0 ¯ (Q ∈ S R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(S0 R,b ∈ S0 R ∨

(pwr)

)→

(S R,a ∈

(pwr) S0 R )

∨

(pwr)

0

→

(S0 R ∈ S0 R )→ ¯ b S[S0 ∨ ] S0 R,b ∨ → Q R

¯ a S[S0 ∨ ] S0 R,a ∨ → Q R ¯ [S0 ∨ ] S0 R ∨ → QS R

( ∨

∨

∨

S0 R S[S: S00 ∪ S0 ] S0 R,b S0 R,a → ¯bQ ¯a ¯ [Qx: Q00 x ∨ Q0 x] Q QS ) ) ).

(128): third Law of the composition of the disjoint unary relations

- 389 -

(129)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (130)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ¯ ∀ Q,

(pwr)

∨

,∀S0 R , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 ¯ (Q ∈ S R ) → S0 R

(pwr)

∨ (S0 R

R[S(pwr) ] S0 R →

∈ S0 R

(pwr)

)→

( ∨ ¯ → S0 R R[S: s.ngle el.t s.t, x] Q 0 ∨ ¯ QS[S0 ∨ ] S R R

) ) ).

(129): second Law of the composition of the disjoint unary relations

- 390 -

(130)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (131)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∀S0 R ¯∨ ∀Q T,

∨

,∀S0 R , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ∨ 0 ¯ (QT ∈ S R ) → S0 R

(pwr)

∨ (S0 R

R[S(pwr) ] S0 R →

∈ S0 R

(pwr)

)→

( ∨

R{} S0 R → ¯∨ R[Q ¯ ∨ ] QT T

) ) ).

(130): first Law of the composition of the disjoint unary relations

- 391 -

(131)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (132)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

,∀L(pwr) , ∀L(pwr) ∀Sb ,∀Sa , ∨ ∀L∨ b ,∀La , ∀¯ p, (pwr)

∈ S) → (L(pwr) ∈ S) → (L(pwr) (Sb ∈ S) → (Sa ∈ S) → ∨ (L∨ b ∈ S) → (La ∈ S) → (¯ p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) →

L(pwr) R[S(pwr) ] L → (Sb ∈ L(pwr)

(pwr)

)→

(pwr) (pwr)

(Sa ∈ L )→ (pwr) (L∨ )→ b ∈L (pwr) (L∨ )→ a ∈L ( ∀¯ pb ,(¯ pb ∈ L) → (¯ pb ∈ L∨ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ Sb ) ∧ p ¯b S[L∨ ] S0 b )→ )→ ( ∀¯ pa ,(¯ pa ∈ L) → (¯ pa ∈ L∨ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ Sa ) ∧ p ¯a S[L∨ ] S0 a )→ )→

- 392 -

( ∀¯ p∨ ,(¯ p∨ ∈ L) → ( ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ Sb ) ∧ (¯ p∨ ∈ S0 b ) ) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ Sa ) ∧ (¯ p∨ ∈ S0 a ) ) ) )→ ( p ¯S[L∨ ] L∨ b → p ¯S[L∨ ] L∨ a ) ) ).

(131): Law of the associativity of the disjoint propositions

- 393 -

(132)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (133)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

,∀L(pwr) , ∀L(pwr) ∀Sb ,∀Sa , ∧ ∀L∨ b ,∀La , ∀¯ p, (pwr)

∈ S) → (L(pwr) ∈ S) → (L(pwr) (Sb ∈ S) → (Sa ∈ S) → ∧ (L∨ b ∈ S) → (La ∈ S) → (¯ p ∈ L) → L(pwr)

(pwr)

R[S(pwr) ] L(pwr) →

L(pwr) R[S(pwr) ] L → (Sb ∈ L(pwr)

(pwr)

)→

(pwr) (pwr)

(Sa ∈ L )→ (pwr) (L∨ )→ b ∈L (pwr) (L∧ )→ a ∈L ( ∀¯ pb ,(¯ pb ∈ L) → (¯ pb ∈ L∨ b) ↔ ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ Sb ) ∧ p ¯b S[L∧ ] S0 b ) )→ ( ∀¯ pa ,(¯ pa ∈ L) → (¯ pa ∈ L∧ a) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ Sa ) ∧ p ¯a S[L∨ ] S0 a ) )→

- 394 -

Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( p ¯S[L∨ ] L∨ b → p ¯S[L∧ ] L∧ a ) ) ).

(132): Law of the joint distributivity of the disjoint propositions

- 395 -

(133)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (134)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∨ ∨ ∀S0 R,b ,∀S0 R,a , ¯ ∀ Q,

(pwr)

,

(pwr) (pwr)

(pwr)

∈ S) → ∈ S) → (S0 R (S0 R (Sb ∈ S) → (Sa ∈ S) → ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 ¯ ∈ S R) → (Q S0 R

(pwr) (pwr)

R[S(pwr) ] S0 R

(pwr) S0 R R[S(pwr) ] S0 R

(Sb ∈ S0 R (Sa ∈ S0 R ∨

(pwr) (pwr) (pwr) (pwr)

(S0 R,b ∈ S0 R ∨

(pwr)

→

)→ )→

)→

(pwr) S0 R )

(S0 R,a ∈ → ( ¯ b ,(Q ¯ b ∈ S0 R ) → ∀Q ¯ b ∈ S0 R,b ∨ ) ↔ ( (Q ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ Sb ) ∧ ¯ b S[S0 ∨ ] S0 b Q R )→ )→ ( ¯ a ,(Q ¯ a ∈ S0 R ) → ∀Q ¯ a ∈ S0 R,a ∨ ) ↔ ( (Q ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ Sa ) ∧ ¯ a S[S0 ∨ ] S0 a Q R )→ )→

- 396 -

(pwr)

→

( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ( ∀Q ( ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ Sb ) ∧ ¯ ∨ ∈ S0 b ) (Q ) ↔ ( ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ Sa ) ∧ ¯ ∨ ∈ S0 a ) (Q ) ) )→ ( ¯ [S0 ∨ ] S0 R,b ∨ → QS R ¯ [S0 ∨ ] S0 R,a ∨ QS R

) ) ).

(133): Law of the associativity of the disjoint unary relations

- 397 -

(134)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (135)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr)

∀S0 R ,∀S0 R ∀Sb ,∀Sa , ∨ ∧ ∀S0 R,b ,∀S0 R,a , ¯ ∀ Q,

(pwr)

,

(pwr) (pwr)

(pwr)

∈ S) → ∈ S) → (S0 R (S0 R (Sb ∈ S) → (Sa ∈ S) → ∨ ∧ (S0 R,b ∈ S) → (S0 R,a ∈ S) → 0 ¯ ∈ S R) → (Q S0 R

(pwr) (pwr)

R[S(pwr) ] S0 R

(pwr) S0 R R[S(pwr) ] S0 R

(Sb ∈ S0 R (Sa ∈ S0 R ∨

(pwr) (pwr) (pwr) (pwr)

(S0 R,b ∈ S0 R ∧

(pwr)

→

)→ )→

)→

(pwr) S0 R )

(S0 R,a ∈ → ( ¯ b ,(Q ¯ b ∈ S0 R ) → ∀Q ¯ b ∈ S0 R,b ∨ ) ↔ ( (Q ∃S0 b ,(S0 b ∈ S) ∧ (S0 b ∈ Sb ) ∧ ¯ b S[S0 ∧ ] S0 b Q R ) )→ ( ¯ a ,(Q ¯ a ∈ S0 R ) → ∀Q ¯ a ∈ S0 R,a ∧ ) ↔ ( (Q ∃S0 a ,(S0 a ∈ S) ∧ (S0 a ∈ Sa ) ∧ ¯ a S[S0 ∨ ] S0 a Q R ) )→

- 398 -

(pwr)

→

Sb R[S0 : r.arr.g.mn.t s.t, S] Sa → ( ¯ [S0 ∨ ] S0 R,b ∨ → QS R ¯ [S0 ∧ ] S0 R,a ∧ QS R

) ) ).

(134): Law of the joint distributivity of the disjoint unary relations

- 399 -

(135)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (136)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ → ( ∀¯ p∨ ,(¯ p∨ ∈ L) → (¯ p∨ ∈ L∨ ) → T∨ p ¯∨ )→ T∨ p ¯ ) ).

(135):

Law of the resolution of the disjoint propositions

- 400 -

(136)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (137)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∧ , ∀p, (L(pwr) ∈ S) → (L∧ ∈ S) → (p ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → pS[L∧ ] L∧ → ( ∀p∧ ,(p∧ ∈ L) → (p∧ ∈ L∧ ) → T∧ p∧ )→ T∧ p ) ).

(136):

Law of the resolution of the joint propositions

- 401 -

(137)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (138)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀S0 R ∨ ∀L∨ ,∀S0 R , ¯ ∀¯ p,∀ Q, ∀Iwcd.te ,∀I0 x ,

(pwr)

,

(pwr)

∈ S) → (L(pwr) ∈ S) → (S0 R ∨ (L∨ ∈ S) → (S0 R ∈ S) → ¯ ∈ S0 R ) → (¯ p ∈ L) → (Q (Iwcd.te ∈ S0 I ) → (I0 x ∈ S0 I ) → L(pwr) R[S(pwr) ] L → S0 R ∨

(pwr)

∨ (S0 R

(L ∈ L )→ R[Iw ] Iwcd.te → (

(pwr)

∈

R[S(pwr) ] S0 R →

(pwr) S0 R )

→

cd.te

∀¯ p∨ ,(¯ p∨ ∈ L) → ∨ (¯ p ∈ L∨ ) ↔ ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) ∧ ∃Q ∨ ∨ ¯ (Q ∈ S0 R ) ∧ p ¯∨ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯∨ (Ix x ∧ Qx)] Iwcd.te I0 x Q ) )→ p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯ →( (Ix x ∧ Qx)] Iwcd.te I0 x Q ¯ [S0 ∨ ] S0 R ∨ → QS R p ¯S[L∨ ] L∨ ) ) ).

(137): Law of the resolution of the disjoint unary relations

- 402 -

(138)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (139)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ → ( ∀¯ p∨ ,(¯ p∨ ∈ L) → (¯ p∨ ∈ L∨ ) → R[p: pr.ry prp.s.n] p ¯∨ )→ R[p: pr.ry prp.s.n] p ¯ ) ).

(138): Law of the primary disjoint propositions

- 403 -

(139)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (140)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ p, (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯S[L∨ ] L∨ → ( ∃¯ p∨ ,(¯ p∨ ∈ L) ∧ (¯ p∨ ∈ L∨ ) ∧ ¯ [p: pr.ry prp.s.n] p R ¯∨ )→ R[p: s.cd.ry prp.s.n] p ¯ ) ).

(139): Law of the secondary disjoint propositions

- 404 -

(140)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (141)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ¯ ∀ Q,

(pwr)

∨

,∀S0 R , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 ¯ (Q ∈ S R ) → S0 R

(pwr)

∨ (S0 R

R[S(pwr) ] S0 R → (pwr)

)→ ∈ S0 R ¯ [S0 ∨ ] S0 R ∨ → QS R ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ∀Q ¯ ∨ ∈ S0 R ∨ ) → (Q ¯∨ R[Q: pr.ry u.ry r.l.n] Q )→ ¯ R[Q: pr.ry u.ry r.l.n] Q

) ).

(140): Law of the primary disjoint unary relations

- 405 -

(141)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (142)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ¯ ∀ Q,

(pwr)

∨

,∀S0 R , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R 0 ¯ (Q ∈ S R ) → S0 R

(pwr)

∨ (S0 R

R[S(pwr) ] S0 R → (pwr)

)→ ∈ S0 R ¯ [S0 ∨ ] S0 R ∨ → QS R ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) ∧ ∃Q ¯ ∨ ∈ S0 R ∨ ) ∧ (Q ¯ [Q: pr.ry u.ry r.l.n] Q ¯∨ R )→ ¯ R[Q: s.cd.ry u.ry r.l.n] Q

) ).

(141): Law of the secondary disjoint unary relations

- 406 -

(142)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (143)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∃¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) ∧ R[p: b.c c.p.sed prp.s.n] p ¯0 → 00 0 00 0 p ¯ S[p: p |∨ p ] p ¯ p ¯ ) ).

(142): Law of the existence of the disjunctively conditional propositions

- 407 -

(143)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (144)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∀¯ pb ,∀¯ pa , (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b S[p: p00 |∨ p0 ] p ¯00 p ¯0 ∧ p ¯a S[p: p00 |∨ p0 ] p ¯00 p ¯0 )→ p ¯b Rid p ¯a ) ).

(143): Law of the uniqueness of the disjunctively conditional propositions

- 408 -

(144)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (145)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∃p, (p00 ∈ L) → (p0 ∈ L) → (p ∈ L) ∧ R[p: b.c c.p.sed prp.s.n] p0 → pS[p: p00 |∧ p0 ] p00 p0 ) ).

(144): Law of the existence of the conjunctively conditional propositions

- 409 -

(145)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (146)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 ,∀p0 , ∀pb ,∀pa , (p00 ∈ L) → (p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → ( pb S[p: p00 |∧ p0 ] p00 p0 ∧ pa S[p: p00 |∧ p0 ] p00 p0 )→ pb Rid pa ) ).

(145): Law of the uniqueness of the conjunctively conditional propositions

- 410 -

(146)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (147)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∃ Q, ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) ∧ (Q ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q 00 ¯ 0 ¯ ¯ 00 0 QS[Qx: Q x |∨ Q x] Q Q ) ).

(146): Law of the existence of the disjunctively conditional unary relations

- 411 -

(147)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (148)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ b ,∀ Q ¯ a, ∀Q ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 Q ∨ ∧ ¯ a S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 Q ∨ )→ ¯ b Rid Q ¯a Q ) ).

(147): Law of the uniqueness of the disjunctively conditional unary relations

- 412 -

(148)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (149)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∃Q, (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q0 → QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 ) ).

(148): Law of the existence of the conjunctively conditional unary relations

- 413 -

(149)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (150)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 ,∀Q0 , ∀Qb ,∀Qa , (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( Qb S[Qx: Q00 x |∧ Q0 x] Q00 Q0 ∧ Qa S[Qx: Q00 x |∧ Q0 x] Q00 Q0 )→ Qb Rid Qa ) ).

(149): Law of the uniqueness of the conjunctively conditional unary relations

- 414 -

(150)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (151)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0b ,∀¯ p0a , ∀¯ p00 ,∀¯ p0 , 000 ∀¯ p ,∀¯ p, (¯ p0b ∈ L) → (¯ p0a ∈ L) → (¯ p00 ∈ L) → (¯ p0 ∈ L) → 000 (¯ p ∈ L) → (¯ p ∈ L) → p ¯0 S[p: p00 ∨ p0 ] p ¯0b p ¯0a → 000 p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0a → ( p ¯ S[p: p00 |∨ p0 ] p ¯000 p ¯0b → 00 0 p ¯ S[p: p00 |∨ p0 ] p ¯ p ¯ ) ) ).

(150): Law of the double conditional disjunctively conditional propositions

- 415 -

(151)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (152)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0b ,∀ Q ¯ 0a , ∀Q ¯ 00 ,∀ Q ¯ 0, ∀Q 000 ¯ ¯ ∀ Q ,∀ Q, ¯ 0a ∈ S0 R ) → ¯ 0b ∈ S0 R ) → (Q (Q ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q 000 0 ¯ ¯ ∈ S0 R ) → (Q ∈ S R ) → (Q ¯ 0b Q ¯ 0a → ¯ 0 S[Qx: Q00 x ∨ Q0 x] Q Q 000 00 ¯ 0 ¯ ¯ Q S[Qx: Q00 x |∨ Q0 x] Q Qa → ( ¯ 000 Q ¯ 0b → ¯ [Qx: Q00 x | Q0 x] Q QS ∨ 00 ¯ 0 ¯ ¯ QS[Qx: Q00 x |∨ Q0 x] Q Q ) ) ).

(151): Law of the double conditional disjunctively conditional unary relations

- 416 -

(152)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (153)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 , ∀¯ p0 ,∀p0 , ∀p, (p00 ∈ L) → (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( pS[p: p00 |∧ p0 ] p00 p0 → pS[p: p00 |∨ p0 ] p00 p ¯0 ) ) ).

(152): Law of the composition of the conjunctively conditional propositions

- 417 -

(153)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (154)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 , ¯ 0 ,∀Q0 , ∀Q ∀Q, (Q00 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( QS[Qx: Q00 x |∧ Q0 x] Q00 Q0 → ¯0 QS[Qx: Q00 x |∨ Q0 x] Q00 Q ) ) ).

(153): Law of the composition of the conjunctively conditional unary relations

- 418 -

(154)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (155)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∀¯ p000 ,∀¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p000 ∈ L) → (¯ p ∈ L) → p ¯000 S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( p ¯ S[p: p00 ∨ p0 ] p ¯000 p ¯0 → 00 0 p ¯ S[p: p00 ∨ p0 ] p ¯ p ¯ ) ) ).

(154): Law of the disjunctively conditional propositions

- 419 -

(155)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (156)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ 000 ,∀ Q, ¯ ∀Q ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ 000 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ 000 S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → Q ∨

( ¯ [Qx: Q00 x ∨ Q0 x] Q ¯ 000 Q ¯0 → QS 00 ¯ 0 ¯ ¯ QS[Qx: Q00 x ∨ Q0 x] Q Q ) ) ).

(155): Law of the disjunctively conditional unary relations

- 420 -

(156)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (157)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p00 b ,∀p00 a , ∀¯ p00 , ∀¯ p0 , ∀pb ,∀pa , ∀¯ p, (p00 b ∈ L) → (p00 a ∈ L) → (¯ p00 ∈ L) → (¯ p0 ∈ L) → (pb ∈ L) → (pa ∈ L) → (¯ p ∈ L) → pb S[p: p00 |∨ p0 ] p00 b p ¯0 → pa S[p: p00 |∨ p0 ] p00 a p ¯0 → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( p ¯00 S[p: p00 !∧ p0 ] p00 b p00 a → p ¯ S[p: p00 !∧ p0 ] pb pa ) ) ).

(156): Law of the negatively conjunctive disjunctivly conditional propositions

- 421 -

(157)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (158)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q00 b ,∀Q00 a , ¯ 00 , ∀Q ¯ 0, ∀Q ∀Qb ,∀Qa , ¯ ∀ Q, (Q00 b ∈ S0 R ) → (Q00 a ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯0 → Qb S[Qx: Q00 x |∨ Q0 x] Q00 b Q ¯0 → Qa S[Qx: Q00 x |∨ Q0 x] Q00 a Q ¯ 00 Q ¯0 → ¯ [Qx: Q00 x | Q0 x] Q QS ∨ ( ¯ 00 S[Qx: Q00 x !∧ Q0 x] Q00 b Q00 a → Q ¯ [Qx: Q00 x !∧ Q0 x] Qb Qa QS ) ) ).

(157): Law of the negatively conjunctive disjunctivly conditional unary relations

- 422 -

(158)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (159)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) , ∨ ∀L∨ b ,∀La , 00 0 ∀¯ p ,∀¯ p, ∀¯ p, (L(pwr) ∈ S) → ∨ (L∨ b ∈ S) → (La ∈ S) → (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) → L(pwr) R[S(pwr) ] L → (pwr) (pwr) (L∨ ) → (L∨ )→ b ∈L a ∈L 0 R[p: b.c c.p.sed prp.s.n] p ¯ → ( ∀¯ p∨ ,(¯ p∨ ∈ L) → (¯ p∨ ∈ L∨ b) ↔ ( ∃¯ p00∨ ,(¯ p00∨ ∈ L) ∧ (¯ p00∨ ∈ L∨ a)∧ p ¯∨ S[p: p00 |∨ p0 ] p ¯00∨ p ¯0 ) )→ p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( p ¯00 S[L∨ ] L∨ a → p ¯S[L∨ ] L∨ b )

) ).

(158): Law of the disjoint disjunctively conditional propositions

- 423 -

(159)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (160)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

, ∀S0 R ∨ ∨ ∀S0 R,b ,∀S0 R,a , 00 0 ¯ ¯ ∀ Q ,∀ Q , ¯ ∀ Q, (pwr)

∈ S) → (S0 R ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(pwr)

∨

) → (S0 R,a ∈ S0 R (S0 R,b ∈ S0 R ¯0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ∀Q ¯ ∨ ∈ S0 R,b ∨ ) ↔ ( (Q ¯ 00∨ ,(Q ¯ 00∨ ∈ S0 R ) ∧ ∃Q ∨ 00 ¯ (Q∨ ∈ S0 R,a ) ∧ ¯ 00∨ Q ¯0 ¯ ∨ S[Qx: Q00 x | Q0 x] Q Q

(pwr)

)→

∨

) )→ ¯ [Qx: Q00 x | QS

0 ∨ Q x]

¯ 00 Q ¯0 → Q

( ¯ 00 S[S0 ∨ ] S0 R,a ∨ → Q R ∨ ¯ QS[S0 ∨ ] S0 R,b R

) ) ).

(159): Law of the disjoint disjunctively conditional unary relations

- 424 -

(160)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (161)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 , ∀¯ p0b ,∀¯ p0a , 0 ∀¯ p, ∀¯ pb ,∀¯ pa , ∀¯ p, (¯ p00 ∈ L) → (¯ p0b ∈ L) → (¯ p0a ∈ L) → 0 (¯ p ∈ L) → (¯ pb ∈ L) → (¯ pa ∈ L) → (¯ p ∈ L) → ( 0 ¯ ∨ p R [p ] ¯ F

∧ R[p: b.c c.p.sed prp.s.n] p ¯0 ) p ¯0 S[p: p00 ∨ p0 ] p ¯0b p ¯0a → 00 0 p ¯b S[p: p00 |∨ p0 ] p ¯ p ¯ → p ¯a S[p: p00 |∨ p0 ] p ¯00 p ¯0a → ( p ¯ S[p: p00 ∨ p0 ] p ¯a p ¯0b → p ¯ S[p: p00 ∨ p0 ] p ¯b p ¯0b ) ) ).

(160): tions

Law of the disjunctively conditioned disjunctively conditional proposi-

- 425 -

(161)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (162)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 , ∀Q ¯ 0b ,∀ Q ¯ 0a , ∀Q 0 ¯ ∀Q , ¯ b ,∀ Q ¯ a, ∀Q ¯ ∀ Q, ¯ 00 ∈ S0 R ) → (Q ¯ 0a ∈ S0 R ) → ¯ 0b ∈ S0 R ) → (Q (Q 0 0 ¯ (Q ∈ S R ) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ ∨ Q ¯0 R [Q ] F

∧ ¯0 R[Q: b.c c.p.sed u.ry r.l.n] Q )→ ¯ 0 S[Qx: Q00 x ∨ Q0 x] Q ¯0 Q ¯0 Q b a → 00 ¯ 0 ¯ ¯ Qb S[Qx: Q00 x |∨ Q0 x] Q Q → ¯ a S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → Q a ∨ ( ¯ [Qx: Q00 x ∨ Q0 x] Q ¯aQ ¯ 0b → QS ¯ [Qx: Q00 x ∨ Q0 x] Q ¯bQ ¯ 0b QS ) ) ).

(161): Law of the disjunctively conditioned disjunctively conditional unary relations

- 426 -

(162)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (163)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∨ p0 , T ,∀¯ (¯ p∨ p0 ∈ L) → T ∈ L) → (¯ 0 ¯ ¯ → R[p∨ ] p F

( p ¯∨ ¯0 p ¯0 → T S[p: p00 |∨ p0 ] p ¯∨ R[p¯∨ ] p T T

) ) ).

(162): Law of the self conditional disjunctively conditional propositions

- 427 -

(163)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (164)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∨ ¯0 ∀Q T ,∀ Q , 0 0 ¯0 ¯∨ (Q T ∈ S R ) → (Q ∈ S R ) → 0 ¯ ¯ R[Q∨ ] Q → F ( ¯∨ ¯0 ¯0 Q T S[Qx: Q00 x |∨ Q0 x] Q Q → ¯∨ R[Q ¯ ∨ ] QT T ) ) ).

(163): Law of the self conditional disjunctively conditional unary relations

(164)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (165)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p∨ p0 , F ,∀¯ ∨ (pF ∈ L) → (¯ p0 ∈ L) → ( R[p∨ ] p∨ F → F

∨ 0 p∨ ¯ F S[p: p00 |∨ p0 ] pF p

) ) ).

(164): Law of the basic disjunctively forbidden disjunctively conditional propositions

- 428 -

(165)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (166)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯0 ∀Q∨ F ,∀ Q , 0 0 ¯0 (Q∨ F ∈ S R ) → (Q ∈ S R ) → ( R[Q∨ ] Q∨ F → F

∨ ¯0 Q∨ F S[Qx: Q00 x |∨ Q0 x] QF Q

) ) ).

(165): Law of the basic disjunctively forbidden disjunctively conditional unary relations

(166)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (167)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p∨ p00 , F ,∀¯ (p∨ ∈ L) → (¯ p00 ∈ L) → F ( R[p∨ ] p∨ F → F

p ¯00 S[p: p00 |∨ p0 ] p ¯00 p∨ F ) ) ).

(166): Law of the negatively conditional disjunctively conditional propositions

- 429 -

(167)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (168)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ∀Q∨ F ,∀ Q , 0 0 ¯ 00 (Q∨ F ∈ S R ) → (Q ∈ S R ) → ( R[Q∨ ] Q∨ F → F

¯ 00 S[Qx: Q00 x | Q0 x] Q ¯ 00 Q∨ Q F ∨ ) ) ).

(167): tions

Law of the negatively conditional disjunctively conditional unary rela-

(168)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (169)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∨ p00 , T ,∀¯ (¯ p∨ ∈ L) → (¯ p00 ∈ L) → T ( R[p¯∨ ] p ¯∨ T → T

p ¯00 S[p: p00 |∨ p0 ] p ¯00 p ¯∨ T ) ) ).

(168): Law of the positively conditional disjunctively conditional propositions

- 430 -

(169)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (170)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∨ ¯ 00 ∀Q T ,∀ Q , 0 0 ¯ 00 ¯∨ (Q T ∈ S R ) → (Q ∈ S R ) → ( ¯∨ R[Q ¯ ∨ ] QT → T

¯ 00 S[Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯∨ Q T ∨ ) ) ).

(169): Law of the positively conditional disjunctively conditional unary relations

- 431 -

(170)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (171)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀Ix , ¯ 00 ,∀ Q ¯ 0 ,∀¯ ∀Q p00 ,∀¯ p0 , ¯ ∀ Q,∀¯ p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → ¯ 00 ∈ S0 R ) → (Q ¯ 0 ∈ S0 R ) → (¯ (Q p00 ∈ L) → (¯ p0 ∈ L) → 0 ¯ p ∈ L) → (Q ∈ S R ) → (¯ R[Iw ] Iwcd.te → cd.te

p ¯00 R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯ 00 → (Ix x ∧ Qx)] Iwcd.te Ix Q 0 p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯0 → (Ix x ∧ Qx)] Iw Ix Q cd.te

p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯ → (Ix x ∧ Qx)] Iwcd.te Ix Q ( ¯ 00 Q ¯0 → ¯ [Qx: Q00 x | Q0 x] Q QS ∨ 00 0 00 0 p ¯ S[p: p |∨ p ] p ¯ p ¯ ) ) ).

(170): Law of the resolution of the disjunctively conditional unary relations

- 432 -

(171)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (172)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∀¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( R[p: pr.ry prp.s.n] p ¯00 ∧ R[p: pr.ry prp.s.n] p ¯0 )→ R[p: pr.ry prp.s.n] p ¯ ) ).

(171): Law of the primary disjunctively conditional propositions

- 433 -

(172)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (173)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p00 ,∀¯ p0 , ∀¯ p, (¯ p00 ∈ L) → (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 → ( ¯ [p: pr.ry prp.s.n] p R ¯00 ∨ ¯ [p: pr.ry prp.s.n] p R ¯0 )→ R[p: s.cd.ry prp.s.n] p ¯ ) ).

(172): Law of the secondary disjunctively conditional propositions

- 434 -

(173)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (174)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∀ Q, ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨ ( ¯ 00 R[Q: pr.ry u.ry r.l.n] Q ∧ ¯0 R[Q: pr.ry u.ry r.l.n] Q )→ ¯ R[Q: pr.ry u.ry r.l.n] Q ) ).

(173): Law of the primary disjunctively conditional unary relations

- 435 -

(174)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (175)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 00 ,∀ Q ¯ 0, ∀Q ¯ ∀ Q, ¯ 0 ∈ S0 R ) → ¯ 00 ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 → QS ∨ ( ¯ [Q: pr.ry u.ry r.l.n] Q ¯ 00 R ∨ ¯ [Q: pr.ry u.ry r.l.n] Q ¯0 R )→ ¯ R[Q: s.cd.ry u.ry r.l.n] Q ) ).

(174): Law of the secondary disjunctively conditional unary relations

- 436 -

(175)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (176)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: el.l ex.st.l pr.p.s.n] p ↔ ( ∃I0 w ,∃I0 x , ∃Q∧ T, (I0 w ∈ S0 I ) ∧ (I0 x ∈ S0 I ) ∧ 0 (Q∧ T ∈ S R) ∧ R[Q∧ ] Q∧ T∧ T

pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] I0 w I0 x Q∧ T ) ) ).

(175): Law of the elemental existential propositions

- 437 -

(176)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (177)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → R[p: el.l u.l pr.p.s.n] p ¯↔( ∃I0 w ,∃I0 x , ¯∧ ∃Q F, (I0 w ∈ S0 I ) ∧ (I0 x ∈ S0 I ) ∧ 0 ¯∧ (Q F ∈ S R) ∧ ¯∧ R[Q ¯ ∧ ] QF ∧ F

p ¯ R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯∧ (Ix x → Qx)] I0 w I0 x Q F ) ) ).

(176): Law of the elemental universal propositions

- 438 -

(177)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (178)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: el.l pr.p.s.n] p ↔ ( R[p: el.l ex.st.l pr.p.s.n] p ∨ R[p: el.l u.l pr.p.s.n] p ) ) ).

(177): Law of the elemental propositions

(178)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (179)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: j.nt el.l pr.p.s.n] p ↔ ( ∃Le (pwr) ,∃L∧ , (Le (pwr) ∈ S) ∧ (L∧ ∈ S) ∧ Le (pwr) R[S(pwr) ] Le ∧ (L∧ ∈ Le (pwr) ) ∧ pS[L∧ ] L∧ ) ) ).

(178): Law of the joint elemental propositions

- 439 -

(179)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (180)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → R[p: b.c ex.st.l prp.s.n] p ¯↔( ∃I0 w ,∃I0 x , ¯ ∃ Q, (I0 w ∈ S0 I ) ∧ (I0 x ∈ S0 I ) ∧ ¯ ∈ S0 R ) ∧ (Q ¯∧ R[Q: b.c c.p.sed u.ry r.l.n] Q p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯ (Ix x ∧ Qx)] I0 w I0 x Q ) ) ).

(179): Law of the basic existential propositions

- 440 -

(180)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (181)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: b.c u.l prp.s.n] p ↔ ( ∃I0 w ,∃I0 x , ∃Q, (I0 w ∈ S0 I ) ∧ (I0 x ∈ S0 I ) ∧ (Q ∈ S0 R ) ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q ∧ pR[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ (Ix x → Qx)] I0 w I0 x Q ) ) ).

(180): Law of the basic universal propositions

- 441 -

(181)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (182)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: b.c prp.s.n] p ↔ ( R[p: b.c ex.st.l prp.s.n] p ∨ R[p: b.c u.l prp.s.n] p ) ) ).

(181): Law of the basic propositions

- 442 -

(182)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (183)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: j.nt b.c prp.s.n] p ↔ ( ∃L(pwr) ,∃L∧ , (L(pwr) ∈ S) ∧ (L∧ ∈ S) ∧ L(pwr) R[S(pwr) ] L ∧ (L∧ ∈ L(pwr) ) ∧ ( ∀p∧ ,(p∧ ∈ L) → R[p: b.c prp.s.n] p∧ )∧ p∧ S[L∧ ] L∧ ) ) ).

(182): Law of the joint basic propositions

- 443 -

(183)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (184)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: Dirac prp.s.n] p → R[p: c.j.v.ly c.t.t c.pl.te ] p ) ) ).

(183): Law of the Dirac propositions’ conjunctive content completeness

(184)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (185)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: Dirac prp.s.n] p → R[p: c.j.v.ly c.t.t c.ss.tnt ] p ) ) ).

(184): Law of the Dirac propositions’ conjunctive content consistency

- 444 -

(185)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (186)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: Dirac prp.s.n] p → R[p: j.nt b.c prp.s.n] p ) ) ).

(185): Law of the Dirac propositions’ joint basic composition

(186)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (187)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∧ p∧ F ,(¯ F ∈ L) → ( ∀p,(p ∈ L) → R[p: Dirac prp.s.n] p → p ¯∧ F R[p0 : c.j.v.ly c.t.t frb.dd.n, p] p )→ R[p¯∧ ] p ¯∧ F F

) ).

(186): Law of the existence of the particular Dirac propositions

- 445 -

(187)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (188)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: Dirac u.ry r.l.n] Q → R[Q: c.j.v.ly c.t.t c.pl.te ] Q ) ) ).

(187): Law of the Dirac unary relations’ conjunctive content completeness

(188)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (189)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Q: Dirac u.ry r.l.n] Q → R[Q: c.j.v.ly c.t.t c.ss.tnt ] Q ) ) ).

(188): Law of the Dirac unary relations’ conjunctive content consistency

- 446 -

(189)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (190)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( 0 ¯∧ ¯∧ ∀Q F ,(QF ∈ S R ) → ( ∀Q,(Q ∈ S0 R ) → R[Q: Dirac u.ry r.l.n] Q → ¯∧ Q F R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Q )→ ¯∧ R[Q ¯ ∧ ] QF F

) ).

(189): Law of the existence of the particular Dirac unary relations

(190)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (191)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: r.d.c.ed Dirac prp.s.n] p → pR[p: s.t c.j.v.ly c.t.t c.pl.te , La ] Le ) ) ).

(190): Law of the elemental conjunctive content completeness of the reduced Dirac propositions

- 447 -

(191)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (192)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: r.d.c.ed Dirac prp.s.n] p → pR[p: s.t c.j.v.ly c.t.t c.ss.tnt , La ] Le ) ) ).

(191): Law of the elemental conjunctive content consistency of the reduced Dirac propositions

(192)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (193)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: r.d.c.ed Dirac prp.s.n] p → R[p: j.nt el.l pr.p.s.n] p ) ) ).

(192): Law of the joint elemental composition of the reduced Dirac propositions

- 448 -

(193)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (194)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p∧ F ,(p ∈ L) → R[p: j.nt el.l pr.p.s.n] p ¯∧ F → ( ∀p,(p ∈ L) → R[p: r.d.c.ed Dirac prp.s.n] p → p ¯∧ F R[p0 : c.j.v.ly c.t.t frb.dd.n, p] p )→ R[p¯∧ ] p ¯∧ F F

) ).

(193): Law of the existence of the particular reduced Dirac propositions

(194)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (195)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Ix ] Q → QR[Q: s.t c.j.v.ly c.t.t c.pl.te , S0 R,a ] S0 p ) ) ).

(194): Law of the primary content completeness of the particular element identifying unary relations

- 449 -

(195)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (196)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Ix ] Q → QR[Q: s.t c.j.v.ly c.t.t c.ss.tnt , S0 R,a ] S0 p ) ) ).

(195): Law of the primary content consistency of the particular element identifying unary relations

(196)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (197)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Ix ] Q → ¯ [Q0 : c.j.v.ly c.t.t r.crs.ve] Q R ) ) ).

(196): Law of the conjunctive content non recursion of the particular element identifying unary relations

- 450 -

(197)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (198)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( 0 ¯∧ ¯∧ ∀Q F ,(QF ∈ S R ) → ¯∧ R[Q: pr.ry u.ry r.l.n] Q F → ( ∀Q,(Q ∈ S0 R ) → R[Ix ] Q → ¯∧ Q F R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Q )→ ¯∧ R[Q ¯ ∧ ] QF F

) ).

(197): Law of the existence of the particular element identifying unary relations

(198)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (199)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Ix ] Q → R[Q: b.c c.p.sed u.ry r.l.n] Q ) ) ).

(198): Law of the basic composition of the particular element identifying unary relations

- 451 -

(199)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (200)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ( R[Ix ] Q → R[Q: pr.ry u.ry r.l.n] Q ) ) ).

(199): Law of the primary composition of the particular element identifying unary relations

(200)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (201)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀Ix , ∀Q ¯ 0 ∈ S0 R ) → (Ix ∈ S0 I ) → (Q ¯0 → ¯ 0 R[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q Q ( ¯ 0 R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix → Q R[Iw ] Ix ) ) ).

(200): Law of the particular non empty world identifying unary relations

- 452 -

(201)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (202)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∃Ix , ∃Q ¯ 0 ∈ S0 R ) ∧ (Ix ∈ S0 I ) ∧ (Q ¯ 0 R[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q ¯0 ∧ Q ( ¯ 0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix ∧ Q R[Iw ] Ix ) ) ).

(201): Law of the existence of the particular empty world identifying unary relations

- 453 -

(202)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (203)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Q ∀Ix,b ,∀Ix,a , ¯ 0 ∈ S0 R ) → (Q (Ix,b ∈ S0 I ) → (Ix,a ∈ S0 I ) → ¯ 0 R[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q ¯0 → Q ( ¯ 0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix,b Q ∧ ¯ 0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Ix,a Q ∧ R[Iw ] Ix,b ∧ R[Iw ] Ix,a )→ Ix,b Rid Ix,a ) ).

(202): Law of the uniqueness of the particular empty world identifying unary relations

- 454 -

(203)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (204)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀Ix , ∀Q ¯ 0 ∈ S0 R ) → (Ix ∈ S0 I ) → (Q (Ix ∈ S0 Iw ) → ¯ 0 R[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q ¯0 → Q ( ¯ 0 R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Ix → Q ¯ [Q: joint basic world attribute ] Ix R ) ) ).

(203): Law of the joint basic composition of the particular non empty world identifying unary relations

- 455 -

(204)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (205)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te , ∀Q∧ T, ∀Q,∃Iw , (Iwcd.te ∈ S0 I ) → 0 (Q∧ T ∈ S R) → 0 (Q ∈ S R ) → (Iw ∈ S0 I ) ∧ R[Iw ] Iwcd.te → cd.te

R[Q∧ ] Q∧ T → T

R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q → ( ∀I0 x , ∀Q00 ,∀Q0 , (I0 x ∈ S0 I ) → (Q00 ∈ S0 R ) → (Q0 ∈ S0 R ) → Q00 R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q → Q0 R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q∧ T → ( Q00 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Iwcd.te ↔ Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Iw ) ) ) ).

(204): Law of the existence of the deterministic worlds

- 456 -

(205)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (206)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∃Iwcd.te ,(Iwcd.te ∈ S0 I ) ∧ R[Iw ] Iwcd.te cd.te

) ).

(205): Law of the existence of the candidate world identifying unary relations

(206)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (207)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: b.c c.p.sed prp.s.n] p ↔ ( (pwr)

∃LD

(pwr)

(LD

(pwr)

LD

,∃L∨ , ∈ S) ∧ (L∨ ∈ S) ∧

R[S(pwr) ] LD ∧ (pwr)

(L∨ ∈ LD pS[L∨ ] L∨

)∧

) ) ).

(206): Law of the basic composed propositions

- 457 -

(207)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (208)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → R[p: ext.d.d c.p.sed prp.s.n] p ¯↔( ∃¯ p00 ,∃¯ p0 , (¯ p00 ∈ L) ∧ (¯ p0 ∈ L) ∧ R[p: b.c c.p.sed prp.s.n] p ¯00 ∧ R[p: b.c c.p.sed prp.s.n] p ¯0 ∧ p ¯ S[p: p00 |∨ p0 ] p ¯00 p ¯0 ) ) ).

(207): Law of the extended composed propositions

- 458 -

(208)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (209)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: b.c c.p.sed u.ry r.l.n] Q ↔ ( ∨ (pwr) ,∃S0 R , (pwr) ∨ (S0 D ∈ S) ∧ (S0 R ∈ (pwr) S0 D R[S(pwr) ] S0 D ∧

∃S0 D

∨

(pwr)

(S0 R ∈ S0 D

S) ∧

)∧

∨ QS[S0 R ∨ ] S0 R

) ) ).

(208): Law of the basic composed unary relations

- 459 -

(209)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (210)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ Q ¯ ∈ S0 R ) → ∀ Q,( ¯ ↔( R[Q: ext.d.d c.p.sed u.ry r.l.n] Q ¯ 00 ,∃ Q ¯ 0, ∃Q ¯ 0 ∈ S0 R ) ∧ ¯ 00 ∈ S0 R ) ∧ (Q (Q ¯ 00 ∧ R[Q: b.c c.p.sed u.ry r.l.n] Q ¯ R[Q: b.c c.p.sed u.ry r.l.n] Q0 ∧ ¯ [Qx: Q00 x | Q0 x] Q ¯ 00 Q ¯0 QS ∨

) ) ).

(209): Law of the extended composed unary relations

- 460 -

(210)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (211)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: b.c r.d.c.d prp.s.n] p ↔ ( ∃LI (pwr) ,∃L∨ , (LI (pwr) ∈ S) ∧ (L∨ ∈ S) ∧ LI (pwr) R[S(pwr) ] LI ∧ (L∨ ∈ LI (pwr) ) ∧ pS[L∨ ] L∨ ) ) ).

(210): Law of the basic reduced propositions

- 461 -

(211)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (212)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: ext.d.d r.d.c.d prp.s.n] p ↔ ( ∃p00 ,∃p0 , (p00 ∈ L) ∧ (p0 ∈ L) ∧ R[p: b.c r.d.c.d prp.s.n] p00 ∧ R[p: b.c r.d.c.d prp.s.n] p0 ∧ pS[p: p00 |∨ p0 ] p00 p0 ) ) ).

(211): Law of the extended reduced propositions

- 462 -

(212)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (213)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q ↔ ( ∨ (pwr) ,∃S0 R , ∨ 0 (pwr) (S I ∈ S) ∧ (S0 R 0 (pwr) 0 SI R[S(pwr) ] S I ∧

∃S0 I

∨

∈ S) ∧

(pwr)

(S0 R ∈ S0 I )∧ ∨ QS[S0 R ∨ ] S0 R ) ) ).

(212): Law of the basic composed, primary unary relations

- 463 -

(213)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (214)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: ext.d.d c.p.sed, pr.ry u.ry r.l.n] Q ↔ ( ∃Q00 ,∃Q0 , (Q00 ∈ S0 R ) ∧ (Q0 ∈ S0 R ) ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q00 ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q0 ∧ QS[Qx: Q00 x |∨ Q0 x] Q00 Q0 ) ) ).

(213): Law of the extended composed, primary unary relations

(214)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (215)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Qx ,∀Q0 , ¯ ∃ Q, (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) ∧ (Q ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I QR

x,

0 Q0 ] Qx Q

) ).

(214): Law of the existence of the disjunctively deterministic generalised element existential world attributes

- 464 -

(215)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (216)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Qx ,∀Q0 , ¯ b ,∀ Q ¯ a, ∀Q (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b R[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Qx Q0 Q x ∧ ¯ a R[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Qx Q0 Q x )→ ¯ b Rid Q ¯a Q ) ).

(215): Law of the uniqueness of the disjunctively deterministic generalised element existential world attributes

- 465 -

(216)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (217)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Qx ,∀ Q ∃Q, ¯ 0 ∈ S0 R ) → (Qx ∈ S0 R ) → (Q (Q ∈ S0 R ) ∧ ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Qx Q ) ).

(216): Law of the existence of the disjunctively deterministic generalised element universal world attributes

- 466 -

(217)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (218)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Qx ,∀ Q ∀Qb ,∀Qa , (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( ¯0 Qb R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Qx Q ∧ ¯0 Qa R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Qx Q )→ Q0 b Rid Q0 a ) ).

(217): Law of the uniqueness of the disjunctively deterministic generalised element universal world attributes

- 467 -

(218)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (219)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Qx ,∀Q0 , ∃Q, (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) ∧ QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qx Q0 ) ).

(218): Law of the existence of the conjunctively deterministic generalised element existential world attributes

- 468 -

(219)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (220)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Qx ,∀Q0 , ∀Qb ,∀Qa , (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( Qb R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qx Q0 ∧ Qa R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qx Q0 )→ Qb Rid Qa ) ).

(219): Law of the uniqueness of the conjunctively deterministic generalised element existential world attributes

- 469 -

(220)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (221)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Qx ,∀ Q ¯ ∃ Q, ¯ 0 ∈ S0 R ) → (Qx ∈ S0 R ) → (Q ¯ ∈ S0 R ) ∧ (Q ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I QR

x,

¯0 Q0 ] Qx Q

) ).

(220): Law of the existence of the conjunctively deterministic generalised element universal world attributes

- 470 -

(221)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (222)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0, ∀Qx ,∀ Q ¯ b ,∀ Q ¯ a, ∀Q ¯ 0 ∈ S0 R ) → (Qx ∈ S0 R ) → (Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b R[Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] Qx Q ¯0 Q x ∧ ¯ a R[Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] Qx Q ¯0 Q x )→ ¯ b Rid Q ¯a Q ) ).

(221): Law of the uniqueness of the conjunctively deterministic generalised element universal world attributes

- 471 -

(222)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (223)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Qx , ¯ 0 ,∀Q0 , ∀Q ∀Q, (Qx ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qx Q0 → ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Qx Q ) ) ).

(222): Law of the composition of the conjunctively deterministic generalised element existential world attributes

- 472 -

(223)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (224)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Qx , ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, (Qx ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ( ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] Qx Q ¯0 → QR x ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Qx Q0 QR x

) ) ).

(223): Law of the composition of the conjunctively deterministic generalised element universal world attributes

- 473 -

(224)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (225)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

, ∀S0 R ∧ ∨ ∀S0 R,b ,∀S0 R,a , 0 ∀Qx ,∀Q , ¯ ∀ Q, (pwr)

∈ S) → (S0 R ∨ ∧ (S0 R,b ∈ S) → (S0 R,a ∈ S) → (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q S0 R

(pwr)

R[S(pwr) ] S0 R →

∧

(S0 R,b ∈ S0 R ∨

0

(pwr)

)→

(pwr) S0 R )

(S R,a ∈ → ( ¯ ∧ ,(Q ¯ ∧ ∈ S0 R ) → ∀Q b b ¯ ∧ ∈ S0 R,b ∧ ) ↔ ( (Q b

∃Qa ∨ ,(Qa ∨ ∈ S0 R ) ∧ ∨ (Qa ∨ ∈ S0 R,a ) ∧ ∧ ¯ Qb R[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qa ∨ Q0 ) )→ ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Qx Q0 → QR x ( ∨ Qx S[S0 R ∨ ] S0 R,a → ∧ 0 ¯ [S0 ∧ ] S R,b QS R

) ) ).

(224): first Law of the distributivity of the deterministic world attributes

- 474 -

(225)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (226)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

, ∀S0 R ∨ ∨ ∀S0 R,b ,∀S0 R,a , 0 ∀Qx ,∀Q , ∀Q, (pwr)

∈ S) → (S0 R ∨ ∨ (S0 R,b ∈ S) → (S0 R,a ∈ S) → (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → S0 R

(pwr) ∨

R[S(pwr) ] S0 R →

(S0 R,b ∈ S0 R 0

∨

(pwr)

)→

(pwr) S0 R )

(S R,a ∈ → ( ∀Qb ∨ ,(Qb ∨ ∈ S0 R ) → ∨ (Qb ∨ ∈ S0 R,b ) ↔ ( ∨ ∨ ∃Qa ,(Qa ∈ S0 R ) ∧ ∨ (Qa ∨ ∈ S0 R,a ) ∧ ∨ Qb R[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qa ∨ Q0 ) )→ QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qx Q0 ( ∨ Qx S[S0 R ∨ ] S0 R,a → QS[S0 R ∨ ] S0 R,b

∨

) ) ).

(225): second Law of the distributivity of the deterministic world attributes

- 475 -

(226)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (227)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q∧ T, ∀Qx ,∀Q0 , ¯ ∀Q0 x ,∀ Q, 0 (Q∧ ∈ S R) → T (Qx ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q R[Q∧ ] Q∧ → T T

Q0 x S[Qx: Q00 x ∧ Q0 x] Qx Q0 → ( QR[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Qx Q0 → 0 QR[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Q∧ TQ x ) ) ).

(226): Law of the deterministic world attributes’ conjunctive attribute transfer

- 476 -

(227)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (228)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q∧ T, ¯ 0, ∀Qx ,∀ Q 0 ¯ ∀ Qx ,∀Q, 0 (Q∧ T ∈ S R) → ¯ 0 ∈ S0 R ) → (Qx ∈ S0 R ) → (Q 0 0 ¯ (Qx ∈ S R ) → (Q ∈ S0 R ) → R[Q∧ ] Q∧ T → T

¯ 0x S[Qx: Q0 x → Q00 x] Qx Q ¯0 → Q ( ¯0 → QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Qx Q ¯0 QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] Q∧ T Qx x

) ) ).

(227): Law of the deterministic wold attributes’ disjunctive attribute transfer

- 477 -

(228)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (229)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 x , ¯∨ ∀Q T, ∀Q0 , ¯ 00 , ∀Qp ,∀ Q ¯ ∀ Q, (I0 x ∈ S0 I ) → 0 ¯∨ (Q T ∈ S R) → 0 0 (Q ∈ S R ) → ¯ 00 ∈ S0 R ) → (Qp ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯∨ R[Q ¯ ∨ ] QT → T

¯∨ Qp R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q T → 00 0 ¯ Q R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I x Q0 → ( ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q0 → QR x ¯ [Qx: Q0 x → Q00 x] Qp Q ¯ 00 QS ) ) ).

(228): Law of the deterministic world attributes’ disjunctive separation

- 478 -

(229)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (230)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 x , ¯∧ ∀Q F, ¯ 0 ,∀Qp , ∀Q ∀Q, (I0 x ∈ S0 I ) → 0 ¯∨ (Q T ∈ S R) → ¯ 0 ∈ S0 R ) → (Qp ∈ S0 R ) → (Q (Q ∈ S0 R ) → ¯∨ R[Q ¯ ∨ ] QT → T

¯∨ Qp R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q T → ( ¯0 → QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q QS[Qx: Q00 x ∧ Q0 x] Q Qp ) ) ).

(229): Law of the deterministic world attributes’ conjunctive separation

- 479 -

(230)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (231)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∨ ∀I0 x ,∀ Q T, ∀Q0 b ,∀Q0 a , ∀Q0 , ¯ 00b ,∀ Q ¯ 00a , ∀Q ¯ 00 , ∀Q ¯ ∀Qp ,∀ Q, 0 ¯∨ (I0 x ∈ S0 I ) → (Q T ∈ S R) → 0 0 0 (Q b ∈ S R ) → (Q a ∈ S0 R ) → (Q0 ∈ S0 R ) → ¯ 00b ∈ S0 R ) → (Q ¯ 00a ∈ S0 R ) → (Q 00 0 ¯ (Q ∈ S R ) → ¯ ∈ S0 R ) → (Qp ∈ S0 R ) → (Q ¯∨ R[Q ¯ ∨ ] QT → T

¯∨ Qp R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q T → ¯ 00b R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] I0 x Q0 b → Q x ¯ 00a R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,I , Q0 ] I0 x Q0 a → Q x Q0 S[Qx: Q00 x |∨ Q0 x] Q0 b Q0 a → ¯ 00b Q ¯ 00a → ¯ 00 S[Qx: Q00 x | Q0 x] Q Q ∨ ( ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q0 → QR x ¯ ¯ 00 QS[Qx: Q0 x → Q00 x] Qp Q ) ) ).

(230): first Law of the conditional deterministic world attributes

- 480 -

(231)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (232)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∨ ∀I0 x ,∀ Q T, ¯ 0b ,∀ Q ¯ 0a , ∀Q ¯ 0, ∀Q ∀Q00 b ,∀Q00 a , ∀Q00 , ∀Qp ,∀Q, 0 ¯∨ (I0 x ∈ S0 I ) → (Q T ∈ S R) → 0 0 0 ¯ ¯ (Qb ∈ S R ) → (Qa ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q00 b ∈ S0 R ) → (Q00 a ∈ S0 R ) → (Q00 ∈ S0 R ) → (Qp ∈ S0 R ) → (Q ∈ S0 R ) → ¯∨ R[Q ¯ ∨ ] QT → T

¯∨ → Qp R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q T ¯0 → Q00 b R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q b ¯0 → Q00 a R[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q a ¯ 0 S[Qx: Q00 x | Q0 x] Q ¯0 Q ¯0 Q b a → ∨ 00 00 Q S[Qx: Q00 x |∨ Q0 x] Q b Q00 a → ( ¯0 → QR[Q: d.sj.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] I0 x Q 00 QS[Qx: Q00 x ∧ Q0 x] Q Qp ) ) ).

(231): second Law of the conditional deterministic world attributes

- 481 -

(232)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (233)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 x ,∀Q0 , ¯ ∀ Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q0 → QR x ¯ R[Q: pr.ry u.ry r.l.n] Q ) ) ).

(232): first Law of the deterministic world attributes’ primary composition

- 482 -

(233)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (234)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 x ,∀Q0 , ∀Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → R[Q: pr.ry u.ry r.l.n] Q ) ) ).

(233): second Law of the deterministic world attributes’ primary composition

- 483 -

(234)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (235)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 x ,∀Q0 , ¯ ∀ Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] I0 x Q0 → QR x ( R[Q: b.c c.p.sed u.ry r.l.n] Q0 → ¯ R[Q: b.c c.p.sed u.ry r.l.n] Q ) ) ).

(234): first Law of the basic composed deterministic world attributes

- 484 -

(235)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (236)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 x ,∀Q0 , ∀Q, (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → ( R[Q: b.c c.p.sed u.ry r.l.n] Q0 → R[Q: b.c c.p.sed u.ry r.l.n] Q ) ) ).

(235): second Law of the basic composed deterministic world attributes

- 485 -

(236)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (237)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 w ,∀I0 x , ¯ 0 ,∀Q, ∀Q ∀¯ p0 , (I0 w ∈ S0 I ) → (I0 x ∈ S0 I ) → ¯ 0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q (¯ p0 ∈ L) → ¯0 → QR[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q 0 p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯0 → (Ix x ∧ Qx)] I0 w I0 x Q ( T∨ p ¯0 → QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 w ) ) ).

(236): first Law of the resolution of the deterministic world attributes

- 486 -

(237)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (238)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 w ,∀I0 x , ∀Q0 ,∀Q, ∀p0 , (I0 w ∈ S0 I ) → (I0 x ∈ S0 I ) → (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (p0 ∈ L) → QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 → p0 R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] I0 w I0 x Q0 → ( T∧ p0 → QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] I0 w ) ) ).

(237): second Law of the resolution of the deterministic world attributes

- 487 -

(238)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (239)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀Q0 x ,∃ Q, ¯ ∈ S0 R ) ∧ (Q0 x ∈ S0 R ) → (Q ¯ QR[Q: prt el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ,Ix ] Q0 x ) ).

(238): Law of the existence of the generalised element disjunctively deterministically empty element world attributes

(239)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (240)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 x , ¯ b ,∀ Q ¯ a, ∀Q 0 (Q x ∈ S0 R ) → ¯ a ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q (Q ( ¯ b R[Q: prt el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q0 x Q x ∧ ¯ a R[Q: prt el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q0 x Q x )→ ¯ b Rid Q ¯a Q ) ).

(239): Law of the uniqueness of the generalised element disjunctively deterministically empty element world attributes

- 488 -

(240)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (241)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀Q0 x ,∃ Q, ¯ ∈ S0 R ) ∧ (Q0 x ∈ S0 R ) → (Q ¯ QR[Q: prt el.t c.j.v.ly d.t.c.lly .mpt.el.t world attribute ,Ix ] Q0 x ) ).

(240): Law of the existence of the generalised element conjunctively deterministically empty element world attributes

(241)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (242)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 x , ¯ b ,∀ Q ¯ a, ∀Q 0 (Q x ∈ S0 R ) → ¯ a ∈ S0 R ) → ¯ b ∈ S0 R ) → (Q (Q ( ¯ b R[Q: prt el.t c.j.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q0 x Q x ∧ ¯ a R[Q: prt el.t c.j.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q0 x Q x )→ ¯ b Rid Q ¯a Q ) ).

(241): Law of the uniqueness of the generalised element conjunctively deterministically empty element world attributes

- 489 -

(242)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (243)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ Q ¯ ∈ S0 R ) ∧ ∃ Q,( ¯ [Q: g.l el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ] QR ) ).

(242): Law of the existence of the general element disjunctively deterministically empty element world attributes

(243)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (244)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ b ,∀ Q ¯ a, ∀Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b R[Q: g.l el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ] Q ∧ ¯ a R[Q: g.l el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ] Q )→ ¯ b Rid Q ¯a Q ) ).

(243): Law of the uniqueness of the general element disjunctively deterministically empty element world attributes

- 490 -

(244)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (245)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ Q ¯ ∈ S0 R ) ∧ ∃ Q,( ¯ [Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q ¯ QR ) ).

(244): Law of the existence of the general element conjunctively deterministically empty element world attributes

(245)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (246)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ b ,∀ Q ¯ a, ∀Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯b ¯ b R[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q Q ∧ ¯a ¯ a R[Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q Q )→ ¯ b Rid Q ¯a Q ) ).

(245): Law of the uniqueness of the general element conjunctively deterministically empty element world attributes

- 491 -

(246)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (247)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q∨ F, ¯ ∀Q0 x ,∀ Q, 0 (Q∨ F ∈ S R) → ¯ ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q R[Q∨ ] Q∨ F → F

¯ [Q: prt el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q0 x → QR x ¯ QR[Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Q0 x Q∨ F ) ) ).

(246): Law of the composition of the generalised element disjunctively deterministically empty element world attributes

- 492 -

(247)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (248)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯∧ ∀Q F, ¯ ∀Q0 x ,∀ Q, 0 ¯∧ (Q F ∈ S R) → ¯ ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ¯∧ R[Q∧ ] Q F → T

¯ [Q: prt el.t c.j.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q0 x → QR x ¯ ¯∧ QR[Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,Ix , Q0 ] Q0 x Q F ) ) ).

(247): Law of the composition of the generalised element conjunctively deterministically empty element world attributes

- 493 -

(248)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (249)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀Q∨ F ,∀ Q, 0 0 ¯ (Q∨ F ∈ S R ) → (Q ∈ S R ) → ∨ R[Q∨ ] QF → F

( ¯ [Q: g.l el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ] → QR ¯ [Q: prt el.t d.sj.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q∨ QR F x ) ) ).

(248): Law of the composition of the general element disjunctively deterministically empty element world attributes

- 494 -

(249)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (250)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀Q∧ T ,∀ Q, 0 0 ¯ (Q∧ T ∈ S R ) → (Q ∈ S R ) → ∧ R[Q∧ ] QT → T

( ¯ [Q: g.l el.t c.j.v.ly d.t.c.lly empty el.t world attribute ] Q ¯ → QR ¯ [Q: prt el.t c.j.v.ly d.t.c.lly .mpt.el.t world attribute ,I ] Q∧ QR T x

) ) ).

(249): Law of the composition of the general element conjunctively deterministically empty element world attributes

- 495 -

(250)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (251)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: b.c d.t.c ex.st.l w.rld attr.te] Q ↔ ( ∃I0 x ,∃Q0 , (I0 x ∈ S0 I ) ∧ (Q0 ∈ S0 R ) ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q0 ∧ QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] I0 x Q0 ) ) ).

(250): Law of the basic deterministic existential world attributes

(251)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (252)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ Q ¯ ∈ S0 R ) → ∀ Q,( ¯ ↔( R[Q: b.c d.t.c u.l w.rld attr.te] Q ¯ 0, ∃I0 x ,∃ Q ¯ 0 ∈ S0 R ) ∧ (I0 x ∈ S0 I ) ∧ (Q ¯0 ∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q ¯ [Q: c.j.v.ly d.t.c prt el.t u.l world attribute ,I QR

x,

0 ¯0 Q0 ] I x Q

) ) ).

(251): Law of the basic deterministic universal world attributes

- 496 -

(252)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (253)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 ,(Q0 ∈ S0 R ) → R[Q: b.c d.t.c w.rld attr.te] Q0 ↔ ( R[Q: b.c d.t.c ex.st.l w.rld attr.te] Q0 ∨ R[Q: b.c d.t.c u.l w.rld attr.te] Q0 ) ) ).

(252): Law of the basic deterministic world attributes

(253)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (254)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 ,(Q0 ∈ S0 R ) → ( R[Q: b.c d.t.c w.rld attr.te] Q0 → ¯ [Q: b.c i.d.t.c w.rld attr.te] Q0 R ) ) ).

(253): Law of the determinism of the basic deterministic world attributes

- 497 -

(254)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (255)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 ,(Q0 ∈ S0 R ) → R[Q: b.c w.rld attr.te] Q0 ↔ ( R[Q: b.c d.t.c w.rld attr.te] Q0 ∨ R[Q: b.c i.d.t.c w.rld attr.te] Q0 ) ) ).

(254): Law of the basic world attributes

- 498 -

(255)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (256)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q,(Q ∈ S0 R ) → ¯ [Q: joint basic world attribute ] ↔ ( Q0 R ∧ (pwr) ,∃S0 R , ∧ 0 (pwr) ∈ S) ∧ (S0 R ∈ (S R 0 (pwr) 0 SR R[S(pwr) ] S R ∧

∃S0 R

∧

(pwr)

(S0 R ∈ S0 R )∧ ( ∀Q0 ∧ ,(Q0 ∧ ∈ S0 R ) → ∧ (Q0 ∧ ∈ S0 R ) → R[Q: b.c w.rld attr.te] Q0 ∧ )∧ ∧ QS[S0 R ∧ ] S0 R ) ) ).

(255): Law of the joint basic world attributes

- 499 -

S) ∧

(256)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (257)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ∀Q,∃p, (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) ∧ pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Q0 w Q0 x Q ) ).

(256): Law of the existence of the particular world existential, particular element existential basic propositions

- 500 -

(257)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (258)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ∀Q, ∀pb ,∀pa , (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (pb ∈ L) → (pa ∈ L) → ( pb R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Q0 w Q0 x Q ∧ pa R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Q0 w Q0 x Q )→ pb Rid pa ) ).

(257): Law of the uniqueness of the particular world existential, particular element existential basic propositions

- 501 -

(258)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (259)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ¯ p, ∀ Q,∃¯ (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) ∧ p ¯ R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯ (Ix x → Qx)] Q0 w Q0 x Q ) ).

(258): Law of the existence of the particular world existential, particular element universal basic propositions

- 502 -

(259)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (260)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ¯ ∀ Q, ∀¯ pb ,∀¯ pa , (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯ (Ix x → Qx)] Q0 w Q0 x Q ∧ p ¯a R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯ (Ix x → Qx)] Q0 w Q0 x Q )→ p ¯b Rid p ¯a ) ).

(259): Law of the uniqueness of the particular world existential, particular element universal basic propositions

- 503 -

(260)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (261)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ∀Q,∃p, (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) ∧ pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Q0 w Q0 x Q ) ).

(260): Law of the existence of the particular world universal, particular element existential basic propositions

- 504 -

(261)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (262)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ∀Q, ∀pb ,∀pa , (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (pb ∈ L) → (pa ∈ L) → ( pb R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Q0 w Q0 x Q ∧ pa R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Q0 w Q0 x Q )→ pb Rid pa ) ).

(261): Law of the uniqueness of the particular world universal, particular element existential basic propositions

- 505 -

(262)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (263)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ¯ p, ∀ Q,∃¯ (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) ∧ p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ (Ix x → Qx)] Q0 w Q0 x Q ) ).

(262): Law of the existence of the particular world universal, particular element universal basic propositions

- 506 -

(263)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (264)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ¯ ∀ Q, ∀¯ pb ,∀¯ pa , (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (¯ pb ∈ L) → (¯ pa ∈ L) → ( p ¯b R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ (Ix x → Qx)] Q0 w Q0 x Q ∧ p ¯a R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ (Ix x → Qx)] Q0 w Q0 x Q )→ p ¯b Rid p ¯a ) ).

(263): Law of the uniqueness of the particular world universal, particular element universal basic propositions

- 507 -

(264)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (265)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀I0 x , ∀Q,∀p, (Iwcd.te ∈ S0 I ) → (I0 x ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

( pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te I0 x Q → pR[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ (Ix x → Qx)] Iwcd.te I0 x Q ) ) ).

(264): propositional Law of the world of candidate elements

- 508 -

(265)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (266)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te , ∀Iw ,∀Qw ,∀Q0 w , ∀Q0 x ,∀Q, ∀pw ,∀px , ∀p, (Iwcd.te ∈ S0 I ) → (Iw ∈ S0 I ) → (Qw ∈ S0 R ) → (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (pw ∈ L) → (px ∈ L) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

Q0 w S[Qx: Q00 x ∧ Q0 x] Iw Qw → pw R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Iw Qw → px R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Q0 x Q → ( pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Q0 w Q0 x Q → pS[p: p00 ∧ p0 ] px pw ) ) ).

(265): Law of the basic propositions’ conjunctive parcelling

- 509 -

(266)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (267)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te , ∀Iw ,∀Qw ,∀Q0 w , ¯ 0x ,∀ Q, ¯ ∀Q ∀pw ,∀¯ px , ∀¯ p, (Iwcd.te ∈ S0 I ) → (Iw ∈ S0 I ) → (Qw ∈ S0 R ) → (Q0 w ∈ S0 R ) → ¯ ∈ S0 R ) → ¯ 0x ∈ S0 R ) → (Q (Q (pw ∈ L) → (¯ px ∈ L) → (¯ p ∈ L) → R[Iw ] Iwcd.te → cd.te

Q0 w S[Qx: Q00 x ∧ Q0 x] Iw Qw → pw R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Iw Qw → p ¯x R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯0 Q ¯ → (Ix x ∧ Qx)] Iw Q x ( p ¯ R[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → ¯0 Q ¯ → (Ix x ∧ Qx)] Q0 w Q x p ¯ S[p: p0 → p00 ] pw p ¯x ) ) ).

(266): Law of the basic propositions’ disjunctive parcelling

- 510 -

(267)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (268)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iw , ∀Q∧ T, ∀Qx ,∀Q, ∀Q0 ,∀p, (Iw ∈ S0 I ) → 0 (Q∧ T ∈ S R) → (Qx ∈ S0 R ) → (Q ∈ S0 R ) → (Q0 ∈ S0 R ) → (p ∈ L) → R[Q∧ ] Q∧ T → T

Q0 S[Qx: Q00 x ∧ Q0 x] Qx Q → ( pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Qx Q → pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ 0 (Ix x ∧ Qx)] Iw Q∧ TQ ) ) ).

(267): Law of the basic propositions’ conjunctive attribute transfer

- 511 -

(268)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (269)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀Ix , ∀Q,∀p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Ix Q → ( R[Q: b.c c.p.sed u.ry r.l.n] Q → R[p: b.c c.p.sed prp.s.n] p ) ) ).

(268): Law of the basic composed basic propositions

- 512 -

(269)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (270)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀Ix , ∀Q,∀p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Ix Q → ( R[Q: s.cd.ry u.ry r.l.n] Q → R[p: s.cd.ry prp.s.n] p ) ) ).

(269): Law of the secondary basic propositions

- 513 -

(270)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (271)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀Ix , ∀Q,∀p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → (Q ∈ S0 R ) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Ix Q → ( R[Q: pr.ry u.ry r.l.n] Q → R[p: pr.ry prp.s.n] p ) ) ).

(270): Law of the primary basic propositions

- 514 -

(271)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (272)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te ,∀Ix , ∀Q∧ T ,∀p, (Iwcd.te ∈ S0 I ) → (Ix ∈ S0 I ) → 0 (Q∧ T ∈ S R ) → (p ∈ L) → R[Q∧ ] Q∧ T → T

pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Ix Q∧ T → ( R[p: pr.ry prp.s.n] p → R[Iw ] Iwcd.te cd.te

) ) ).

(271): Law of the normal composition of the primary elemental propositions

- 515 -

(272)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (273)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iw ,∀Ix , ∀Q∧ T ,∀p, (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → 0 (Q∧ T ∈ S R ) → (p ∈ L) → R[Q∧ ] Q∧ T → T

( pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Ix Q∧ T → R[p: b.c c.p.sed prp.s.n] p ) ) ).

(272): Law of the elemental propositions’ basic composition

- 516 -

(273)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (274)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Iwcd.te , ∀Iw ,∀Ix , ∀Q∧ T ,∀Q, ∀pn ,∀pp , ∀p, (Iwcd.te ∈ S0 I ) → (Iw ∈ S0 I ) → (Ix ∈ S0 I ) → 0 0 (Q∧ T ∈ S R ) → (Q ∈ S R ) → (pn ∈ L) → (pp ∈ L) → (p ∈ L) → R[Iw ] Iwcd.te → cd.te

R[Q∧ ] Q∧ T → T

pn R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iwcd.te Ix Q → pp R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Ix Q∧ T → ( pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Ix Q → pS[p: p00 ∧ p0 ] pn pp ) ) ).

(273): Law of the basic propositions’ conjunctive normalisation

- 517 -

(274)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (275)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ¯ ∀ Q,∀Q, ∀¯ p,∀p, (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q (¯ p ∈ L) → (p ∈ L) → p ¯ R[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ → (Ix x → Qx)] Q0 w Q0 x Q pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Q0 w Q0 x Q → ( ¯ Q QR ¯Q→ p ¯ Rp¯ p ) ) ).

(274): Law of the complements of the generalised world existential, generalised element existential basic propositions

- 518 -

(275)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (276)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 w ,∀Q0 x , ¯ ∀ Q,∀Q, ∀¯ p,∀p, (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q (¯ p ∈ L) → (p ∈ L) → p ¯ R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯ → (Ix x → Qx)] Q0 w Q0 x Q pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] Q0 w Q0 x Q → ( ¯ Q QR ¯Q→ p ¯ Rp¯ p ) ) ).

(275): Law of the complements of the generalised world existential, generalised element universal basic propositions

- 519 -

(276)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (277)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀I0 w ,∀Q0 x , ∀Q,∀p, (I0 w ∈ S0 I ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) → ( pR[p : ∀w,∃x,(w ∈ wcd.te ) → (x ∈ w) ∧ Iw w → (Ix x ∧ Qx)] I0 w Q0 x Q → pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] I0 w Q0 x Q ) ) ).

(276): Law of the composition of the particular world universal, generalised element existential basic propositions

- 520 -

(277)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (278)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀S0 R ∨ ∀L∨ ,∀S0 R , 0 0 ∀Q w ,∀Q x , ∀Q,∀p,

(pwr)

,

(pwr)

∈ S) → (L(pwr) ∈ S) → (S0 R (L∨ ∈ S) → ∨ (S0 R ∈ S) → (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) → L(pwr) R[S(pwr) ] L → S0 R ∨

(pwr)

R[S(pwr) ] S0 R → (pwr)

(L∨ ∈ L(pwr) ) → (S0 R ∈ S0 R )→ ( ∀p∨ ,(p∨ ∈ L) → (p∨ ∈ L∨ ) ↔ ( ∃Qw ∨ ,(Qw ∨ ∈ S0 R ) ∧ ∨ Qw ∨ S0 R ∧ p∨ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Qw ∨ Q0 x Q ) )→ pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Q0 w Q0 x Q → ( ∨ Q0 w S[S0 R ∨ ] S0 R → pS[L∨ ] L∨ ) ) ).

(277): second Law of the distributivity of the basic propositions - 521 -

(278)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (279)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀S0 R ∨ ∀L∨ ,∀S0 R , 0 0 ∀Q w ,∀Q x , ¯ p, ∀ Q,∀¯

(pwr)

,

(pwr)

∈ S) → (L(pwr) ∈ S) → (S0 R ∨ (L∨ ∈ S) → (S0 R ∈ S) → (Q0 w ∈ S0 R ) → (Q0 x ∈ S0 R ) → ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → L(pwr) R[S(pwr) ] L → S0 R ∨

(pwr)

R[S(pwr) ] S0 R → (pwr)

)→ (L∨ ∈ L(pwr) ) → (S0 R ∈ S0 R ( ∀¯ p∨ ,(¯ p∨ ∈ L) → ∨ (p ∈ L∨ ) ↔ ( ∃Qw ∨ ,(Qw ∨ ∈ S0 R ) ∧ ∨ Qw ∨ S0 R ∧ ∨ p ¯ R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯ (Ix x → Qx)] Qw ∨ Q0 x Q ) )→ p ¯ R[p : ∃w,∀x,(w ∈ wcd.te ) ∧ (x ∈ w) → Iw w∧ ¯ → (Ix x → Qx)] Q0 w Q0 x Q ( ∨ Q0 w S[S0 R ∨ ] S0 R → ∨ p ¯S[L∨ ] L ) ) ).

(278): third Law of the distributivity of the basic propositions

- 522 -

(279)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (280)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀S0 R ∨ ∀L∨ ,∀S0 R , 0 ∀Iw ,∀Q x , ∀Q,∀p,

(pwr)

,

(pwr)

∈ S) → (L(pwr) ∈ S) → (S0 R ∨ (L∨ ∈ S) → (S0 R ∈ S) → (Iw ∈ S0 I ) → (Q0 x ∈ S0 R ) → (Q ∈ S0 R ) → (p ∈ L) → L(pwr) R[S(pwr) ] L → S0 R ∨

(pwr)

R[S(pwr) ] S0 R → (pwr)

)→ (L∨ ∈ L(pwr) ) → (S0 R ∈ S0 R ( ∀p∨ ,(p∨ ∈ L) → (p∨ ∈ L∨ ) ↔ ( ∃Qx ∨ ,(Qx ∨ ∈ S0 R ) ∧ ∨ Qx ∨ S0 R ∧ ∨ p R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Qx ∨ Q ) )→ pR[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ (Ix x ∧ Qx)] Iw Q0 x Q → ( ∨ Q0 x S[S0 R ∨ ] S0 R → ∨ pS[L∨ ] L ) ) ).

(279): first Law of the distributivity of the basic propositions

- 523 -

∀S, ∀L,∀Le ,∀Lp , ∀LD ,∀LI , ∀S0 R ,∀S0 p , ∀S0 D ,∀S0 I ,∀S0 Iw , (S ∈ Scd.te ) → (L ∈ S) → (Le ∈ S) → (Lp ∈ S) → (LD ∈ S) → (LI ∈ S) → (S0 R ∈ S) → (S0 p ∈ S) → (S0 D ∈ S) → (S0 I ∈ S) → (S0 Iw ∈ S) → R[Scd.te ] S → ( (280)

LR(nt.l,u.ry,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ 0 0 0 0 0 LR∧ T Le Lp LD LI S R S p S D S I S Iw ).

(280): final Law basic unary natural languages

- 524 -

A.2. Axiomatisation Unary Languages’ Dependencies

∀S, ∀L,∀Le ,∀Lp , ∀LD ,∀LI , ∀S0 R ,∀S0 p , ∀S0 D ,∀S0 I ,∀S0 Iw , (S ∈ Scd.te ) → (L ∈ S) → (Le ∈ S) → (Lp ∈ S) → (LD ∈ S) → (LI ∈ S) → (S0 R ∈ S) → (S0 p ∈ S) → (S0 D ∈ S) → (S0 I ∈ S) → (S0 Iw ∈ S) → R[Scd.te ] S → ( LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ (1) LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ).

Anchor Law dependency relations

- 525 -

(1)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (2)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ↔ p ¯ S[p: p00 ∨ p0 ] p ¯p ¯0 ) ) ).

(d1):

Law of the disjunctively content present propositions

(2)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (3)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀¯ p, (p0 ∈ L) → (¯ p ∈ L) → ( p0 R[p0 : d.sj.v.ly c.t.t abs.nt, p] p ¯ ↔ ¯ [p0 : d.sj.v.ly c.t.t pr.s.nt, p] p p0 R ¯ ) ) ).

(d2):

Law of the disjunctively content absent propositions

- 526 -

(3)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (4)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ↔ pS[p: p00 ∧ p0 ] p p0 ) ) ).

(d3):

Law of the conjunctively content present propositions

(4)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (5)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p, (¯ p0 ∈ L) → (p ∈ L) → ( p ¯0 R[p0 : c.j.v.ly c.t.t abs.nt, p] p ↔ ¯ [p0 : c.j.v.ly c.t.t pr.s.nt, p] p p ¯0 R ) ) ).

(d4):

Law of the conjunctively content absent propositions

- 527 -

(5)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (6)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀ Q, ¯ ∀Q ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ Q ↔ ¯ [Qx: Q00 x ∨ Q0 x] Q ¯Q ¯0 QS ) ) ).

(d5):

Law of the disjunctively content present unary relations

(6)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (7)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ ∀Q0 ,∀ Q, ¯ ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ( ¯ Q0 R[Q0 : d.sj.v.ly c.t.t abs.nt, Q] Q ↔ ¯ [Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ Q0 R ) ) ).

(d6):

Law of the disjunctively content absent unary relations

- 528 -

(7)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (8)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 ,∀Q, (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( Q0 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ↔ QS[Qx: Q00 x ∧ Q0 x] Q Q0 ) ) ).

(d7):

Law of the conjunctively content present unary relations

(8)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (9)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀Q, ∀Q ¯ 0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : c.j.v.ly c.t.t abs.nt, Q] Q Q ↔ ¯ [Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ¯ 0R Q ) ) ).

(d8):

Law of the conjunctively content absent unary relations

- 529 -

(9)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (10)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p0 , ∀¯ p,∀p, ∀¯ p00 , (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → (p ∈ L) → (¯ p00 ∈ L) → p ¯0 Rp¯ p0 → p ¯ Rp¯ p → p ¯00 S[p: p0 !→ p00 ] pp0 → ( p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ↔ p ¯00 R[p0 : d.sj.v.ly c.t.t pr.s.nt, p] p ¯ ) ) ).

(d9):

Law of the disjunctively content necessary propositions

- 530 -

(10)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (11)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p0 , ∀¯ p, (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯0 Rp¯ p0 → ( p0 R[p0 : d.sj.v.ly c.t.t frb.dd.n, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ) ) ).

(d10):

Law of the disjunctively content forbidden propositions

- 531 -

(11)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (12)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, ∀p00 , (p0 ∈ L) → (p ∈ L) → (p00 ∈ L) → p00 S[p: p0 → p00 ] p p0 → ( p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ↔ p00 R[p0 : c.j.v.ly c.t.t pr.s.nt, p] p ) ) ).

(d11):

Law of the conjunctively content necessary propositions

- 532 -

(12)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (13)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p0 , ∀p, (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( p ¯0 R[p0 : c.j.v.ly c.t.t frb.dd.n, p] p ↔ p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ) ) ).

(d12):

Law of the conjunctively content forbidden propositions

- 533 -

(13)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (14)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q,∀Q, ¯ 00 , ∀Q ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ 00 ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ¯ QRQ ¯Q→ ¯ 00 S[Qx: Q0 x !→ Q00 x] Q Q0 → Q ( ¯ ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q Q ↔ ¯ 00 R[Q0 : d.sj.v.ly c.t.t pr.s.nt, Q] Q ¯ Q ) ) ).

(d13):

Law of the disjunctively content necessary unary relations

- 534 -

(14)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (15)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ( ¯ Q0 R[Q0 : d.sj.v.ly c.t.t frb.dd.n, Q] Q ↔ ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q ¯ Q ) ) ).

(d14):

Law of the disjunctively content forbidden unary relations

- 535 -

(15)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (16)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 ,∀Q, ∀Q00 , (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q00 ∈ S0 R ) → Q00 S[Qx: Q0 x → Q00 x] Q Q0 → ( Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Q ↔ Q00 R[Q0 : c.j.v.ly c.t.t pr.s.nt, Q] Q ) ) ).

(d15):

Law of the conjunctively content necessary unary relations

- 536 -

(16)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (17)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀Q0 , ∀Q ∀Q, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( ¯ 0 R[Q0 : c.j.v.ly c.t.t frb.dd.n, Q] Q Q ↔ Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Q ) ) ).

(d16):

Law of the conjunctively content forbidden unary relations

- 537 -

(17)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (18)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t nc.ss.ry, p] p ¯ ) ) ).

(d17):

Law of the element disjunctively content recursive propositions

(18)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (19)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, p] p ↔ p0 R[p0 : c.j.v.ly c.t.t nc.ss.ry, p] p ) ) ).

(d18):

Law of the element conjunctively content recursive propositions

- 538 -

(19)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (20)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0 ,∀ Q, ¯ ∀Q ¯ ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q ( ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, Q] Q ¯ Q ↔ ¯ 0 R[Q0 : d.sj.v.ly c.t.t nc.ss.ry, Q] Q ¯ Q ) ) ).

(d19):

Law of the element disjunctively content recursive unary relations

(20)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (21)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀Q0 ,∀Q, (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( Q0 R[Q0 : c.j.v.ly c.t.t r.crs.ve, Q] Q ↔ Q0 R[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Q ) ) ).

(d20):

Law of the element conjunctively content recursive unary relations

- 539 -

(21)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (22)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ p0 , (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ p0 ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, La ] L∨ ↔ ( p ¯0 S[L∨ ] L∨ → ( ∃¯ p∨ ,(¯ p∨ ∈ L) ∧ (¯ p∨ ∈ L∨ ) ∧ p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, p] p ¯∨ ) ) ) ).

(d21):

Law of the set disjunctively content recursive propositions

- 540 -

(22)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (23)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∧ , ∀p0 , (L(pwr) ∈ S) → (L∧ ∈ S) → (p0 ∈ L) → L(pwr) R[S(pwr) ] L → (L∧ ∈ L(pwr) ) → p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, La ] L∨ ↔ ( p0 S[L∧ ] L∧ → ( ∃p∧ ,(p∧ ∈ L) ∧ (p∧ ∈ L∧ ) ∧ p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, p] p∧ ) ) ) ).

(d22):

Law of the set conjunctively content recursive propositions

- 541 -

(23)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (24)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ¯ 0, ∀Q

(pwr)

(pwr)

∨

,∀S0 R , ∨

∈ S) → (S0 R ∈ S) → (S0 R 0 0 ¯ (Q ∈ S R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

∨ (pwr) )→ (S0 R ∈ S0 R 0 ¯ R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S0 ] S0 R ∨ Q R,a ¯ 0 S[S0 ∨ ] S0 R ∨ → ( Q R ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) ∧ ∃Q ¯ ∨ ∈ S0 R ∨ ) ∧ (Q ¯∨ ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, Q] Q Q

↔(

) ) ) ).

(d23):

Law of the set disjunctively content recursive unary relations

- 542 -

(24)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (25)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ∀Q0 ,

(pwr)

(pwr)

∧

,∀S0 R , ∧

∈ S) → (S0 R ∈ S) → (S0 R 0 0 (Q ∈ S R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

∧ (pwr) )→ (S0 R ∈ S0 R ∧ 0 Q R[Q0 : c.j.v.ly c.t.t r.crs.ve, S0 R,a ] S0 R 0 0 ∧ Q S[S0 R ∧ ] S R → ( ∃Q∧ ,(Q∧ ∈ S0 R ) ∧ ∧ (Q∧ ∈ S0 R ) ∧ 0 Q R[Q0 : c.j.v.ly c.t.t r.crs.ve, Q] Q∧

↔(

) ) ) ).

(d24):

Law of the set conjunctively content recursive unary relations

- 543 -

(25)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (26)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∀L(pwr) ∀Sb ,∀¯ p0 ,

,∀L(pwr) ,

(pwr)

∈ S) → (L(pwr) ∈ S) → (L(pwr) (Sb ∈ S) → (¯ p0 ∈ L) → L(pwr) (pwr)

L

(pwr)

R[S(pwr) ] L(pwr) →

R[S(pwr) ] L → (pwr)

(Sb ∈ L(pwr) )→ p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, Sa ] Sb ↔ ( ∀L∨ ,(L∨ ∈ S) → (L∨ ∈ Sb ) → p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, La ] L∨ ) ) ).

(d25):

Law of the set set disjunctively content recursive propositions

- 544 -

(26)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (27)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∀L(pwr) ∀Sb ,∀p0 ,

,∀L(pwr) ,

(pwr)

∈ S) → (L(pwr) ∈ S) → (L(pwr) (Sb ∈ S) → (p0 ∈ L) → L(pwr) (pwr)

L

(pwr)

R[S(pwr) ] L(pwr) →

R[S(pwr) ] L → (pwr)

(Sb ∈ L(pwr) )→ p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, Sa ] Sb ↔ ( ∀L∧ ,(L∧ ∈ S) → (L∧ ∈ Sb ) → p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, La ] L∧ ) ) ).

(d26):

Law of the set set conjunctively content recursive propositions

- 545 -

(27)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (28)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr)

∀S0 R ¯ ∀Sb ,∀ Q,

,∀S0 R

(pwr)

,

(pwr) (pwr)

∈ S) → (S0 R (S0 R ¯ (Sb ∈ S) → (Q0 ∈ S0 R ) → S0 R

(pwr) (pwr)

R[S(pwr) ] S0 R

(pwr) S0 R R[S(pwr) ] S0 R

(pwr)

(pwr)

∈ S) →

→

→

(pwr) (pwr) S0 R )

(Sb ∈ → ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S ] Sb ↔ ( Q a ∨

∨

∀S0 R ,(S0 R ∈ S) → ∨ (S0 R ∈ Sb ) → ∨ 0 ¯ Q R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S0 R,a ] S0 R ) ) ).

(d27):

Law of the set set disjunctively content recursive unary relations

- 546 -

(28)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (29)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr) (pwr)

∀S0 R ∀Sb ,∀Q0 ,

,∀S0 R

(pwr)

,

(pwr) (pwr)

(S0 R ∈ S) → (S0 R (Sb ∈ S) → (Q0 ∈ S0 R ) → S0 R

(pwr) (pwr)

R[S(pwr) ] S0 R

(pwr) S0 R R[S(pwr) ] S0 R

(pwr)

(pwr)

∈ S) →

→

→

(pwr) (pwr) S0 R )

(Sb ∈ → Q0 R[Q0 : c.j.v.ly c.t.t r.crs.ve, Sa ] Sb ↔ ( ∧

∧

∀S0 R ,(S0 R ∈ S) → ∧ (S0 R ∈ Sb ) → ∧ 0 Q R[Q0 : c.j.v.ly c.t.t r.crs.ve, S0 R,a ] S0 R ) ) ).

(d28):

Law of the set set conjunctively content recursive unary relations

- 547 -

(29)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (30)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) , ∀¯ p0 , (L(pwr) ∈ S) → (¯ p0 ∈ L) → L(pwr) R[S(pwr) ] L → ( R[p0 : d.sj.v.ly c.t.t r.crs.ve] p ¯0 ↔ p ¯0 R[p0 : d.sj.v.ly c.t.t r.crs.ve, Sa ] L(pwr) ) ) ).

(d29):

Law of the disjunctively content recursive propositions

- 548 -

(30)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (31)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) , ∀p0 , (L(pwr) ∈ S) → (p0 ∈ L) → L(pwr) R[S(pwr) ] L → ( R[p0 : c.j.v.ly c.t.t r.crs.ve] p0 ↔ p0 R[p0 : c.j.v.ly c.t.t r.crs.ve, Sa ] L(pwr) ) ) ).

(d30):

Law of the conjunctively content recursive propositions

- 549 -

(31)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (32)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ¯ 0, ∀Q

(pwr)

,

(pwr) (pwr)

(S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → (Q S0 R

(pwr)

(pwr)

∈ S) →

R[S(pwr) ] S0 R →

( ¯0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve] Q ↔ ¯ 0 R[Q0 : d.sj.v.ly c.t.t r.crs.ve, S ] S0 R (pwr) Q a ) ) ).

(d31):

Law of the disjunctively content recursive unary relations

- 550 -

(32)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (33)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀S0 R ∀Q0 ,

(pwr)

,

(pwr)

∈ S) → (S0 R (Q0 ∈ S0 R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

( R[Q0 : c.j.v.ly c.t.t r.crs.ve] Q0 ↔ (pwr) Q0 R[Q0 : c.j.v.ly c.t.t r.crs.ve, Sa ] S0 R ) ) ).

(d32):

Law of the conjunctively content recursive unary relations

- 551 -

(33)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (34)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: c.j.v.ly nc.ss.ry] p ↔ T∧ p ) ) ).

(d33):

Law of the conjunctively necessary propositions

(34)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (35)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,∀p, (¯ p ∈ L) → (p ∈ L) → p ¯ Rp¯ p → ( R[p: c.j.v.ly frb.dd.n] p ¯ ↔ R[p: c.j.v.ly nc.ss.ry] p ) ) ).

(d34):

Law of the conjunctively forbidden propositions

- 552 -

(35)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (36)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ p, ∀ Q,∀¯ ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (¯ (Q p ∈ L) → p ¯ R[p : ∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ Iw w∧ ¯ 0w Q ¯ 0x Q ¯ → (Ix x ∧ Qx)] Q ( ¯ 0w Q ¯ 0x ¯ [Q: d.sj.v.ly nc.ss.ry] Q QR ↔ R[p: d.sj.v.ly nc.ss.ry] p ¯ ) ) ).

(d35):

Law of the disjunctively necessary unary relations

- 553 -

(36)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (37)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ ∀ Q,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ Q QR ¯Q→ ( ¯ 0w Q ¯ 0x QR[Q: d.sj.v.ly frb.dd.n] Q ↔ ¯ [Q: QR

¯0 ¯0 d.sj.v.ly nc.ss.ry] Qw Qx

) ) ).

(d36):

Law of the disjunctively forbidden unary relations

- 554 -

(37)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (38)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q,∀p, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q ∈ S0 R ) → (p ∈ L) → pR[p : ∀w,∀x,(w ∈ wcd.te ) → (x ∈ w) → Iw w → ¯ 0w Q ¯ 0x Q → (Ix x → Qx)] Q ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly nc.ss.ry] Q ↔ R[p: c.j.v.ly nc.ss.ry] p ) ) ).

(d37):

Law of the conjunctively necessary unary relations

- 555 -

(38)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (39)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ ∀ Q,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ Q QR ¯Q→ ( ¯ [Q: c.j.v.ly frb.dd.n] Q ¯ 0w Q ¯ 0x QR ↔ QR[Q:

¯0 ¯0 c.j.v.ly nc.ss.ry] Qw Qx

) ) ).

(d38):

Law of the conjunctively forbidden unary relations

- 556 -

(39)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (40)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → R[p: d.sj.v.ly d.t.c] p ¯↔( R[p: d.sj.v.ly nc.ss.ry] p ¯ ∨ R[p: d.sj.v.ly frb.dd.n] p ¯ ) ) ).

(d39):

Law of the disjunctively deterministic propositions

(40)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (41)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → ( R[p: d.sj.v.ly i.d.t.c] p ¯ ↔ ¯ [p: d.sj.v.ly d.t.c] p R ¯ ) ) ).

(d40):

Law of the disjunctively indeterministic propositions

- 557 -

(41)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (42)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: c.j.v.ly d.t.c] p ↔ ( R[p: c.j.v.ly nc.ss.ry] p ∨ R[p: c.j.v.ly frb.dd.n] p ) ) ).

(d41):

Law of the conjunctively deterministic propositions

(42)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (43)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: c.j.v.ly i.d.t.c] p ↔ ¯ [p: c.j.v.ly d.t.c] p R ) ) ).

(d42):

Law of the conjunctively indeterministic propositions

- 558 -

(43)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (44)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ ∀ Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ¯ [Q: d.sj.v.ly d.t.c] Q ¯ 0w Q ¯ 0x ↔ ( QR ¯ ¯ 0w Q ¯ 0x QR[Q: d.sj.v.ly nc.ss.ry] Q ∨ ¯ [Q: d.sj.v.ly frb.dd.n] Q ¯ 0w Q ¯ 0x QR ) ) ).

(d43):

Law of the disjunctively deterministic unary relations

- 559 -

(44)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (45)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ ∀ Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly i.d.t.c] Q ¯ 0w Q ¯ 0x QR ↔ ¯ [Q: d.sj.v.ly d.t.c] Q ¯R ¯ 0w Q ¯ 0x Q ) ) ).

(d44):

Law of the disjunctively indeterministic unary relations

- 560 -

(45)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (46)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q ∈ S0 R ) → ¯ 0w Q ¯ 0x ↔ ( QR[Q: c.j.v.ly d.t.c] Q ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly nc.ss.ry] Q ∨ ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly frb.dd.n] Q ) ) ).

(d45):

Law of the conjunctively deterministic unary relations

- 561 -

(46)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (47)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly i.d.t.c] Q ↔ ¯ [Q: c.j.v.ly d.t.c] Q ¯ 0w Q ¯ 0x QR ) ) ).

(d46):

Law of the conjunctively indeterministic unary relations

- 562 -

(47)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (48)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → R[p: d.sj.v.ly u.d.t.c] p ¯↔( ¯ [p: d.sj.v.ly nc.ss.ry] p R ¯ ∨ ¯ [p: d.sj.v.ly frb.dd.n] p R ¯ ) ) ).

(d47):

Law of the disjunctively underdeterministic propositions

(48)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (49)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p,(¯ p ∈ L) → ( R[p: d.sj.v.ly o.d.t.c] p ¯ ↔ ¯ [p: d.sj.v.ly u.d.t.c] p R ¯ ) ) ).

(d48):

Law of the disjunctively overdeterministic propositions

- 563 -

(49)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (50)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → R[p: c.j.v.ly u.d.t.c] p ↔ ( ¯ [p: c.j.v.ly nc.ss.ry] p R ∨ ¯ [p: c.j.v.ly frb.dd.n] p R ) ) ).

(d49):

Law of the conjunctively underdeterministic propositions

(50)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (51)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p,(p ∈ L) → ( R[p: c.j.v.ly o.d.t.c] p ↔ ¯ [p: c.j.v.ly u.d.t.c] p R ) ) ).

(d50):

Law of the conjunctively overdeterministic propositions

- 564 -

(51)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (52)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ ∀ Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ¯ [Q: d.sj.v.ly u.d.t.c] Q ¯ 0w Q ¯ 0x ↔ ( QR ¯ ¯ ¯ ¯ 0x QR[Q: d.sj.v.ly nc.ss.ry] Q0w Q ∨ ¯ [Q: d.sj.v.ly frb.dd.n] Q ¯R ¯ 0w Q ¯ 0x Q ) ) ).

(d51):

Law of the disjunctively underdeterministic unary relations

- 565 -

(52)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (53)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ ∀ Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly o.d.t.c] Q ¯ 0w Q ¯ 0x QR ↔ ¯ [Q: d.sj.v.ly u.d.t.c] Q ¯R ¯ 0w Q ¯ 0x Q ) ) ).

(d52):

Law of the disjunctively overdeterministic unary relations

- 566 -

(53)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (54)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q ∈ S0 R ) → ¯ 0w Q ¯ 0x ↔ ( QR[Q: c.j.v.ly u.d.t.c] Q ¯ ¯ ¯ 0x QR[Q: c.j.v.ly nc.ss.ry] Q0w Q ∨ ¯ [Q: c.j.v.ly frb.dd.n] Q ¯ 0w Q ¯ 0x QR ) ) ).

(d53):

Law of the conjunctively underdeterministic unary relations

- 567 -

(54)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (55)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly o.d.t.c] Q ↔ ¯ [Q: c.j.v.ly u.d.t.c] Q ¯ 0w Q ¯ 0x QR ) ) ).

(d54):

Law of the conjunctively overdeterministic unary relations

- 568 -

(55)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (56)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p0 , ∀¯ p,∀p, ∀¯ p00 , (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → (p ∈ L) → (¯ p00 ∈ L) → p ¯0 Rp¯ p0 → p ¯ Rp¯ p → p ¯00 S[p: p0 !→ p00 ] pp0 → ( p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ↔ R[p: d.sj.v.ly nc.ss.ry] p ¯00 ) ) ).

(d55):

Law of the disjunctively necessary third propositions

- 569 -

(56)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (57)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p0 , ∀¯ p, (¯ p0 ∈ L) → (p0 ∈ L) → (¯ p ∈ L) → p ¯0 Rp¯ p0 → ( p0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ↔ p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ) ) ).

(d56):

Law of the disjunctively forbidden third propositions

- 570 -

(57)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (58)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, ∀p00 , (p0 ∈ L) → (p ∈ L) → (p00 ∈ L) → p00 S[p: p0 → p00 ] p p0 → ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ↔ R[p: c.j.v.ly nc.ss.ry] p00 ) ) ).

(d57):

Law of the conjunctively necessary third propositions

- 571 -

(58)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (59)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀p0 , ∀p, (¯ p0 ∈ L) → (p0 ∈ L) → (p ∈ L) → p ¯0 Rp¯ p0 → ( p ¯0 R[p0 : c.j.v.ly frb.dd.n, p] p ↔ p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ) ) ).

(d58):

Law of the conjunctively forbidden third propositions

- 572 -

(59)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (60)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q,∀Q, ¯ 00 , ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q 0 0 ¯ (Q ∈ S R ) → (Q0 ∈ S0 R ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q ¯ 00 ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ¯ Q QR ¯Q→ ¯ 00 S[Qx: Q0 x !→ Q00 x] Q Q0 → Q ( ¯Q ¯ 0w Q ¯ 0x ¯ 0 R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q Q ↔ ¯ 00 R[Q: Q

¯0 ¯0 d.sj.v.ly nc.ss.ry] Qw Qx

) ) ).

(d59):

Law of the disjunctively necessary third unary relations

- 573 -

(60)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (61)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀Q0 , ∀Q ¯ ∀ Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q 0 ¯ 0 RQ Q ¯Q → ( ¯Q ¯ 0w Q ¯ 0x Q0 R[Q0 : d.sj.v.ly frb.dd.n, Q] Q ↔ ¯ 0 R[Q0 : Q

¯ ¯0 ¯0 d.sj.v.ly nc.ss.ry, Q] Q Qw Qx

) ) ).

(d60):

Law of the disjunctively forbidden third unary relations

- 574 -

(61)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (62)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ∀Q00 , ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → (Q00 ∈ S0 R ) → Q00 S[Qx: Q0 x → Q00 x] Q Q0 → ( ¯ 0w Q ¯ 0x Q0 R[Q0 : c.j.v.ly nc.ss.ry, Q] Q Q ↔ Q00 R[Q:

¯0 ¯0 c.j.v.ly nc.ss.ry] Qw Qx

) ) ).

(d61):

Law of the conjunctively necessary third unary relations

- 575 -

(62)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (63)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀Q0 , ∀Q ∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → ( ¯ 0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Q ¯ 0w Q ¯ 0x Q ↔ Q0 R[Q0 :

¯0 ¯0 c.j.v.ly nc.ss.ry, Q] Q Qw Qx

) ) ).

(d62):

Law of the conjunctively forbidden third unary relations

- 576 -

(63)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (64)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p0 : d.sj.v.ly impl.c.tly d.t.c, p] p ¯↔( p ¯0 R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯ ∨ p ¯0 R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ) ).

(d63):

Law of the disjunctively implicitly deterministic third propositions

(64)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (65)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly impl.c.tly i.d.t.c, p] p ¯ ↔ ¯ [p0 : d.sj.v.ly impl.c.tly d.t.c, p] p p ¯0 R ¯ ) ) ).

(d64):

Law of the disjunctively implicitly indeterministic third propositions

- 577 -

(65)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (66)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → p0 R[p0 : c.j.v.ly impl.c.tly d.t.c, p] p ↔ ( p0 R[p0 : c.j.v.ly nc.ss.ry, p] p ∨ p0 R[p0 : c.j.v.ly frb.dd.n, p] p ) ) ).

(d65):

Law of the conjunctively implicitly deterministic third propositions

(66)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (67)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly impl.c.tly i.d.t.c, p] p ↔ ¯ [p0 : c.j.v.ly impl.c.tly d.t.c, p] p p0 R ) ) ).

(d66):

Law of the conjunctively implicitly indeterministic third propositions

- 578 -

(67)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (68)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly d.t.c, Q] Q ¯Q ¯ 0w Q ¯ 0x ↔ ( Q 0 0 ¯0 ¯ ¯ ¯ 0 Q R[Q : d.sj.v.ly nc.ss.ry, Q] Q Qw Qx ∨ ¯ 0 R[Q0 : d.sj.v.ly frb.dd.n, Q] Q ¯ 0w Q ¯Q ¯ 0x Q ) ) ).

(d67):

Law of the disjunctively implicitly deterministic third unary relations

- 579 -

(68)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (69)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly i.d.t.c, Q] Q ¯Q ¯ 0w Q ¯ 0x Q ↔ ¯ [Q0 : d.sj.v.ly impl.c.tly d.t.c, Q] Q ¯ 0R ¯Q ¯ 0w Q ¯ 0x Q ) ) ).

(d68):

Law of the disjunctively implicitly indeterministic third unary relations

- 580 -

(69)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (70)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ¯ 0w Q ¯ 0x ↔ ( Q0 R[Q0 : c.j.v.ly impl.c.tly d.t.c, Q] Q Q 0 0 ¯0 ¯ 0 Q R[Q : c.j.v.ly nc.ss.ry, Q] Q Qw Qx ∨ ¯ 0w Q ¯ 0x Q0 R[Q0 : c.j.v.ly frb.dd.n, Q] Q Q ) ) ).

(d69):

Law of the conjunctively implicitly deterministic third unary relations

- 581 -

(70)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (71)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x Q0 R[Q0 : c.j.v.ly impl.c.tly i.d.t.c, Q] Q Q ↔ ¯ [Q0 : c.j.v.ly impl.c.tly d.t.c, Q] Q Q ¯ 0w Q ¯ 0x Q0 R ) ) ).

(d70): Law of the conjunctively implicitly indeterministic third unary relations

- 582 -

(71)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (72)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → p ¯0 R[p0 : d.sj.v.ly impl.c.tly u.d.t.c, p] p ¯↔( ¯ [p0 : d.sj.v.ly nc.ss.ry, p] p p ¯0 R ¯ ∨ ¯ [p0 : d.sj.v.ly frb.dd.n, p] p p ¯0 R ¯ ) ) ).

(d71): ons

Law of the disjunctively implicitly underdeterministic third propositi-

(72)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (73)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯0 R[p0 : d.sj.v.ly impl.c.tly o.d.t.c, p] p ¯ ↔ ¯ [p0 : d.sj.v.ly impl.c.tly u.d.t.c, p] p p ¯0 R ¯ ) ) ).

(d72):

Law of the disjunctively implicitly overdeterministic third propositions

- 583 -

(73)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (74)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → p0 R[p0 : c.j.v.ly impl.c.tly u.d.t.c, p] p ↔ ( ¯ [p0 : c.j.v.ly nc.ss.ry, p] p p0 R ∨ ¯ [p0 : c.j.v.ly frb.dd.n, p] p p0 R ) ) ).

(d73): ons

Law of the conjunctively implicitly underdeterministic third propositi-

(74)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (75)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( p0 R[p0 : c.j.v.ly impl.c.tly o.d.t.c, p] p ↔ ¯ [p0 : c.j.v.ly impl.c.tly u.d.t.c, p] p p0 R ) ) ).

(d74): Law of the conjunctively implicitly overdeterministic third propositions

- 584 -

(75)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (76)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly u.d.t.c, Q] Q ¯Q ¯ 0w Q ¯ 0x ↔ ( Q 0¯ 0 ¯0 ¯ ¯ ¯ 0 Q R[Q : d.sj.v.ly nc.ss.ry, Q] Q Qw Qx ∨ ¯ [Q0 : d.sj.v.ly frb.dd.n, Q] Q ¯ 0R ¯Q ¯ 0w Q ¯ 0x Q ) ) ).

(d75): Law of the disjunctively implicitly underdeterministic third unary relations

- 585 -

(76)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (77)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly o.d.t.c, Q] Q ¯Q ¯ 0w Q ¯ 0x Q ↔ ¯ [Q0 : d.sj.v.ly impl.c.tly u.d.t.c, Q] Q ¯ 0R ¯Q ¯ 0w Q ¯ 0x Q ) ) ).

(d76): tions

Law of the disjunctively implicitly overdeterministic third unary rela-

- 586 -

(77)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (78)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ¯ 0w Q ¯ 0x ↔ ( Q0 R[Q0 : c.j.v.ly impl.c.tly u.d.t.c, Q] Q Q 0¯ 0 ¯0 ¯ 0 Q R[Q : c.j.v.ly nc.ss.ry, Q] Q Qw Qx ∨ ¯ [Q0 : c.j.v.ly frb.dd.n, Q] Q Q ¯ 0w Q ¯ 0x Q0 R ) ) ).

(d77): Law of the conjunctively implicitly underdeterministic third unary relations

- 587 -

(78)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (79)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x Q0 R[Q0 : c.j.v.ly impl.c.tly o.d.t.c, Q] Q Q ↔ ¯ [Q0 : c.j.v.ly impl.c.tly u.d.t.c, Q] Q Q ¯ 0w Q ¯ 0x Q0 R ) ) ).

(d78): tions

Law of the conjunctively implicitly overdeterministic third unary rela-

- 588 -

(79)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (80)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯R[p: d.sj.v.ly expl.c.tly c.pl.te, p0 ] p ¯0 ↔ p ¯0 R[p0 : d.sj.v.ly impl.c.tly d.t.c, p] p ¯ ) ) ).

(d79): Law of the element disjunctively explicitly complete second propositions

(80)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (81)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯R[p: d.sj.v.ly expl.c.tly i.c.pl.te, p0 ] p ¯0 ↔ ¯ [p: d.sj.v.ly expl.c.tly c.pl.te, p0 ] p p ¯R ¯0 ) ) ).

(d80): tions

Law of the element disjunctively explicitly incomplete second proposi-

- 589 -

(81)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (82)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( pR[p: c.j.v.ly expl.c.tly c.pl.te, p0 ] p0 ↔ p0 R[p0 : c.j.v.ly impl.c.tly d.t.c, p] p ) ) ).

(d81): ons

Law of the element conjunctively explicitly complete second propositi-

- 590 -

(82)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (83)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( pR[p: c.j.v.ly expl.c.tly i.c.pl.te, p0 ] p0 ↔ ¯ [p: c.j.v.ly expl.c.tly c.pl.te, p0 ] p0 pR ) ) ).

(d82): sitions

Law of the element conjunctively explicitly incomplete second propo-

- 591 -

(83)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (84)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly expl.c.tly c.pl.te, Q0 ] Q ¯0Q ¯ 0w Q ¯ 0x QR ↔ ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly d.t.c, Q] Q ¯Q ¯ 0w Q ¯ 0x Q ) ) ).

(d83): Law of the element disjunctively explicitly complete second unary relations

- 592 -

(84)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (85)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly expl.c.tly i.c.pl.te, Q0 ] Q ¯0Q ¯ 0w Q ¯ 0x QR ↔ ¯ [Q: d.sj.v.ly expl.c.tly c.pl.te, Q0 ] Q ¯R ¯0Q ¯ 0w Q ¯ 0x Q ) ) ).

(d84): Law of the element disjunctively explicitly incomplete second unary relations

- 593 -

(85)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (86)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly expl.c.tly c.pl.te, Q0 ] Q0 Q ↔ ¯ 0w Q ¯ 0x Q0 R[Q0 : c.j.v.ly impl.c.tly d.t.c, Q] Q Q ) ) ).

(d85): Law of the element conjunctively explicitly complete second unary relations

- 594 -

(86)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (87)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly expl.c.tly i.c.pl.te, Q0 ] Q0 Q ↔ ¯ [Q: c.j.v.ly expl.c.tly c.pl.te, Q0 ] Q0 Q ¯ 0w Q ¯ 0x QR ) ) ).

(d86): Law of the element conjunctively explicitly incomplete second unary relations

- 595 -

(87)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (88)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯R[p: d.sj.v.ly expl.c.tly c.ss.tnt, p0 ] p ¯0 ↔ p ¯0 R[p0 : d.sj.v.ly impl.c.tly u.d.t.c, p] p ¯ ) ) ).

(d87): tions

Law of the element disjunctively explicitly consistent second proposi-

- 596 -

(88)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (89)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ p0 ,∀¯ p, (¯ p0 ∈ L) → (¯ p ∈ L) → ( p ¯R[p: d.sj.v.ly expl.c.tly i.c.ss.tnt, p0 ] p ¯0 ↔ ¯ [p: d.sj.v.ly expl.c.tly c.ss.tnt, p0 ] p p ¯R ¯0 ) ) ).

(d88): sitions

Law of the element disjunctively explicitly inconsistent second propo-

- 597 -

(89)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (90)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( pR[p: c.j.v.ly expl.c.tly c.ss.tnt, p0 ] p0 ↔ p0 R[p0 : c.j.v.ly impl.c.tly u.d.t.c, p] p ) ) ).

(d89): tions

Law of the element conjunctively explicitly consistent second proposi-

- 598 -

(90)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (91)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀p0 ,∀p, (p0 ∈ L) → (p ∈ L) → ( pR[p: c.j.v.ly expl.c.tly i.c.ss.tnt, p0 ] p0 ↔ ¯ [p: c.j.v.ly expl.c.tly c.ss.tnt, p0 ] p0 pR ) ) ).

(d90): sitions

Law of the element conjunctively explicitly inconsistent second propo-

- 599 -

(91)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (92)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly expl.c.tly c.ss.tnt, Q0 ] Q ¯0Q ¯ 0w Q ¯ 0x QR ↔ ¯ 0 R[Q0 : d.sj.v.ly impl.c.tly u.d.t.c, Q] Q ¯Q ¯ 0w Q ¯ 0x Q ) ) ).

(d91): Law of the element disjunctively explicitly consistent second unary relations

- 600 -

(92)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (93)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ 0 ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ 0 ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ [Q: d.sj.v.ly expl.c.tly i.c.ss.tnt, Q0 ] Q ¯0Q ¯ 0w Q ¯ 0x QR ↔ ¯ [Q: d.sj.v.ly expl.c.tly c.ss.tnt, Q0 ] Q ¯R ¯0Q ¯ 0w Q ¯ 0x Q ) ) ).

(d92): Law of the element disjunctively explicitly inconsistent second unary relations

- 601 -

(93)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (94)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly expl.c.tly c.ss.tnt, Q0 ] Q0 Q ↔ ¯ 0w Q ¯ 0x Q0 R[Q0 : c.j.v.ly impl.c.tly u.d.t.c, Q] Q Q ) ) ).

(d93): Law of the element conjunctively explicitly consistent second unary relations

- 602 -

(94)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (95)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Q0 ,∀Q, ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Q0 ∈ S0 R ) → (Q ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QR[Q: c.j.v.ly expl.c.tly i.c.ss.tnt, Q0 ] Q0 Q0 Q ↔ ¯ [Q: c.j.v.ly expl.c.tly c.ss.tnt, Q0 ] Q0 Q ¯ 0w Q ¯ 0x QR ) ) ).

(d94): Law of the element conjunctively explicitly inconsistent second unary relations

- 603 -

(95)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (96)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pG ,∀¯ p, (pG ∈ L) → (¯ p ∈ L) → ( pG R[pG : d.sj.v.ly impl.c.tly G. l.ke, ] p ¯ ↔ pG R[p0 : d.sj.v.ly frb.dd.n, p] p ¯ ) ) ).

(d95):

Law of the disjunctively implicitly G¨odel like third propositions

(96)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (97)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pG ,∀p, (¯ pG ∈ L) → (p ∈ L) → ( p ¯G R[pG : c.j.v.ly impl.c.tly G. l.ke, ] p ↔ p ¯G R[p0 : c.j.v.ly frb.dd.n, p] p ) ) ).

(d96):

Law of the conjunctively implicitly G¨odel like third propositions

- 604 -

(97)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (98)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ G ,∀ Q, ¯ ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ G ∈ S0 R ) → (Q ¯ ∈ S0 R ) → (Q ( ¯ G R[Q : d.sj.v.ly impl.c.tly G. l.ke, ] Q ¯Q ¯ 0w Q ¯ 0x Q G ↔ ¯ G R[Q0 : d.sj.v.ly frb.dd.n, Q] Q ¯Q ¯ 0w Q ¯ 0x Q ) ) ).

(d97):

Law of the disjunctively implicitly G¨odel like third unary relations

- 605 -

(98)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (99)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ G ,∀Q, ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ G ∈ S0 R ) → (Q ∈ S0 R ) → (Q ( ¯ G R[Q : c.j.v.ly impl.c.tly G. l.ke, Q] Q Q ¯ 0w Q ¯ 0x Q G ↔ ¯ G R[Q0 : c.j.v.ly frb.dd.n, Q] Q Q ¯ 0w Q ¯ 0x Q ) ) ).

(d98):

Law of the conjunctively implicitly G¨odel like third unary relations

- 606 -

(99)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (100)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pG , (¯ pG ∈ L) → ( R[pG : d.sj.v.ly G. l.ke] p ¯G ↔ p ¯G R[pG : d.sj.v.ly impl.c.tly G. l.ke, ] p ¯G ) ) ).

(d99):

Law of the disjunctively G¨odel like second propositions

(100)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (101)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pG , (pG ∈ L) → ( R[pG : c.j.v.ly G. l.ke] pG ↔ pG R[pG : c.j.v.ly impl.c.tly G. l.ke, ] pG ) ) ).

(d100):

Law of the conjunctively G¨odel like second propositions

- 607 -

(101)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (102)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ G, ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ G ∈ S0 R ) → (Q ( ¯ G R[Q : d.sj.v.ly G. l.ke] Q ¯ 0w Q ¯ 0x Q G ↔ ¯ G R[Q : d.sj.v.ly impl.c.tly G. l.ke, ] Q ¯Q ¯ 0w Q ¯ 0x Q G ) ) ).

(d101):

Law of the disjunctively G¨odel like second unary relations

- 608 -

(102)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (103)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀QG , ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (QG ∈ S0 R ) → ( ¯ 0w Q ¯ 0x QG R[QG : c.j.v.ly G. l.ke] Q ↔ ¯ 0w Q ¯ 0x QG R[QG : c.j.v.ly impl.c.tly G. l.ke, Q] Q Q ) ) ).

(d102):

Law of the conjunctively G¨odel like second unary relations

- 609 -

(103)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (104)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pG ,∀pG , (¯ pG ∈ L) → (pG ∈ L) → p ¯G Rp¯ pG → R[pG : d.sj.ve G. ] p ¯G ↔ ( R[pG : d.sj.v.ly G. l.ke] p ¯G ∧ R[pG : d.sj.v.ly G. l.ke] pG ) ) ).

(d103):

Law of the disjunctive G¨odel second propositions

- 610 -

(104)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (105)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pG ,∀pG , (¯ pG ∈ L) → (pG ∈ L) → p ¯G Rp¯ pG → R[pG : c.j.ve G. ] pG ↔ ( R[pG : c.j.v.ly G. l.ke] p ¯G ∧ R[pG : c.j.v.ly G. l.ke] pG ) ) ).

(d104):

Law of the conjunctive G¨odel second propositions

- 611 -

(105)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (106)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ G ,∀QG , ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ G ∈ S0 R ) → (QG ∈ S0 R ) → (Q ¯ G RQ Q ¯ QG → ¯ G R[Q : d.sj.ve G.] Q ¯ 0w Q ¯ 0x ↔ ( Q G ¯ 0w Q ¯ 0x ¯ G R[Q : d.sj.v.ly G. l.ke] Q Q G

∧ QG R[QG :

¯0 ¯0 d.sj.v.ly G. l.ke] Qw Qx

) ) ).

(d105):

Law of the disjunctive G¨odel second unary relations

- 612 -

(106)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (107)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ G ,∀QG , ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ G ∈ S0 R ) → (QG ∈ S0 R ) → (Q ¯ G RQ Q ¯ QG → ¯ 0w Q ¯ 0x ↔ ( QG R[QG : c.j.ve G.] Q ¯ G R[Q : c.j.v.ly G. l.ke] Q ¯ 0w Q ¯ 0x Q G

∧ QG R[QG :

¯0 ¯0 c.j.v.ly G. l.ke] Qw Qx

) ) ).

(d106):

Law of the conjunctive G¨odel second unary relations

- 613 -

(107)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (108)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pt ,∀¯ po , (pt ∈ L) → (¯ po ∈ L) → ( pt R[pt : type Ad.sj.v.ly sc.c, po ] p ¯o ↔ pt R[p0 : d.sj.v.ly frb.dd.n, p] p ¯o ) ) ).

(d107):

Law of the element type A disjunctively scientific third propositions

(108)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (109)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pt ,∀¯ po , (¯ pt ∈ L) → (¯ po ∈ L) → ( p ¯t R[pt : type B d.sj.v.ly sc.c, po ] p ¯o ↔ p ¯t R[p0 : d.sj.v.ly nc.ss.ry, p] p ¯o ) ) ).

(d108):

Law of the element type B disjunctively scientific third propositions

- 614 -

(109)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (110)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pt ,∀po , (¯ pt ∈ L) → (po ∈ L) → ( p ¯t R[pt : type A c.j.v.ly sc.c, po ] po ↔ p ¯t R[p0 : c.j.v.ly frb.dd.n, p] po ) ) ).

(d109):

Law of the element type A conjunctively scientific third propositions

(110)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (111)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pt ,∀po , (pt ∈ L) → (po ∈ L) → ( pt R[pt : type B c.j.v.ly sc.c, po ] po ↔ pt R[p0 : c.j.v.ly nc.ss.ry, p] po ) ) ).

(d110):

Law of the element type B conjunctively scientific third propositions

- 615 -

(111)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (112)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ o, ∀Qt ,∀ Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ o ∈ S0 R ) → (Qt ∈ S0 R ) → (Q ( ¯oQ ¯ 0w Q ¯ 0x Qt R[Qt : type A d.sj.v.ly sc.c, Qo ] Q ↔ ¯oQ ¯ 0w Q ¯ 0x Qt R[Q0 : d.sj.v.ly frb.dd.n, Q] Q ) ) ).

(d111): Law of the element type A disjunctively scientific third unary relations

- 616 -

(112)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (113)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t ,∀ Q ¯ o, ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ t ∈ S0 R ) → (Q ¯ o ∈ S0 R ) → (Q ( ¯ t R[Q : type B d.sj.v.ly sc.c, Q ] Q ¯oQ ¯ 0w Q ¯ 0x Q t o ↔ ¯ t R[Q0 : d.sj.v.ly nc.ss.ry, Q] Q ¯oQ ¯ 0w Q ¯ 0x Q ) ) ).

(d112): Law of the element type B disjunctively scientific third unary relations

- 617 -

(113)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (114)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t ,∀Qo , ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ t ∈ S0 R ) → (Qo ∈ S0 R ) → (Q ( ¯ t R[Q : type A c.j.v.ly sc.c, Q ] Qo Q ¯ 0w Q ¯ 0x Q t o ↔ ¯ t R[Q0 : c.j.v.ly frb.dd.n, Q] Qo Q ¯ 0w Q ¯ 0x Q ) ) ).

(d113): Law of the element type A conjunctively scientific third unary relations

- 618 -

(114)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (115)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Qt ,∀Qo , ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Qt ∈ S0 R ) → (Qo ∈ S0 R ) → ( ¯ 0w Q ¯ 0x Qt R[Qt : type B c.j.v.ly sc.c, Qo ] Qo Q ↔ ¯ 0w Q ¯ 0x Qt R[Q0 : c.j.v.ly nc.ss.ry, Q] Qo Q ) ) ).

(d114): Law of the element type B conjunctively scientific third unary relations

- 619 -

(115)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (116)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pt ,∀¯ po , (¯ pt ∈ L) → (¯ po ∈ L) → p ¯t R[pt : d.sj.v.ly sc.c, po ] p ¯o ↔ ( p ¯t R[pt : type B d.sj.v.ly sc.c, po ] p ¯o ∨ p ¯t R[pt : type Ad.sj.v.ly sc.c, po ] p ¯o ) ) ).

(d115):

Law of the element disjunctively scientific third propositions

(116)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (117)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pt ,∀po , (pt ∈ L) → (po ∈ L) → pt R[pt : c.j.v.ly sc.c, po ] po ↔ ( pt R[pt : type B c.j.v.ly sc.c, po ] po ∨ pt R[pt : type A c.j.v.ly sc.c, po ] po ) ) ).

(d116):

Law of the element conjunctively scientific third propositions

- 620 -

(117)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (118)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t ,∀ Q ¯ o, ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ t ∈ S0 R ) → (Q ¯ o ∈ S0 R ) → (Q ¯ t R[Q : d.sj.v.ly sc.c, Q ] Q ¯oQ ¯ 0w Q ¯ 0x ↔ ( Q t o ¯ ¯oQ ¯ 0w Q ¯ 0x Qt R[Q : type B d.sj.v.ly sc.c, Q ] Q t

∨ ¯ t R[Q : Q t

o

¯ ¯0 ¯0 type A d.sj.v.ly sc.c, Qo ] Qo Qw Qx

) ) ).

(d117):

Law of the element disjunctively scientific third unary relations

- 621 -

(118)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (119)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Qt ,∀Qo , ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Qt ∈ S0 R ) → (Qo ∈ S0 R ) → ¯ 0w Q ¯ 0x ↔ ( Qt R[Qt : c.j.v.ly sc.c, Qo ] Qo Q ¯ 0w Q ¯ 0x Qt R[Qt : type B c.j.v.ly sc.c, Qo ] Qo Q ∨ ¯ 0w Q ¯ 0x Qt R[Qt : type A c.j.v.ly sc.c, Qo ] Qo Q ) ) ).

(d118):

Law of the element conjunctively scientific third unary relations

- 622 -

(119)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (120)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀pt , (L(pwr) ∈ S) → (L∨ ∈ S) → (pt ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → pt R[pt : type A d.sj.v.ly sc.c, La ] L∨ ↔ ( ∃¯ p∨ p∨ o ,(¯ o ∈ L) ∧ (¯ p∨ ∈ L∨ ) ∧ o pt R[pt : type Ad.sj.v.ly sc.c, po ] p ¯∨ o ) ) ).

(d119):

Law of the set type A disjunctively scientific third propositions

- 623 -

(120)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (121)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ pt , (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ pt ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯t R[pt : type B d.sj.v.ly sc.c, La ] L∨ ↔ ( ∃¯ p∨ p∨ o ,(¯ o ∈ L) ∧ (¯ p∨ ∈ L∨ ) ∧ o p ¯t R[pt : type B d.sj.v.ly sc.c, po ] p ¯∨ o ) ) ).

(d120):

Law of the set type B disjunctively scientific third propositions

- 624 -

(121)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (122)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ pt , (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ pt ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯t R[pt : type A c.j.v.ly sc.c, La ] L∨ ↔ ( ∨ ∃p∨ o ,(po ∈ L) ∧ ∨ (p∨ ∈ L )∧ o p ¯t R[pt : type A c.j.v.ly sc.c, po ] p∨ o ) ) ).

(d121):

Law of the set type A conjunctively scientific third propositions

- 625 -

(122)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (123)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀pt , (L(pwr) ∈ S) → (L∨ ∈ S) → (pt ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → pt R[pt : type B c.j.v.ly sc.c, La ] L∨ ↔ ( ∨ ∃p∨ o ,(po ∈ L) ∧ ∨ (p∨ ∈ L )∧ o pt R[pt : type B c.j.v.ly sc.c, po ] p∨ o ) ) ).

(d122):

Law of the set type B conjunctively scientific third propositions

- 626 -

(123)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (124)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Qt , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q x w 0 (Qt ∈ S R ) → S0 R

(pwr) ∨

R[S(pwr) ] S0 R → (pwr)

)→ (S0 R ∈ S0 R ∨ ¯0 ¯0 Qt R[Qt : type A d.sj.v.ly sc.c, S0 R,a ] S0 R Q w Qx ↔ ( ∨ ¯∨ 0 ¯ ∃ Qo ,(Qo ∈ S R ) ∧ 0 ∨ ¯∨ (Q o ∈S R )∧ ¯∨ ¯0 ¯0 Qt R[Qt : type A d.sj.v.ly sc.c, Qo ] Q o Qw Qx ) ) ).

(d123):

Law of the set type A disjunctively scientific third unary relations

- 627 -

(124)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (125)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t, ∀Q ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q x w 0 ¯ (Qt ∈ S R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(pwr)

)→ (S0 R ∈ S0 R ¯ [Q : type B d.sj.v.ly sc.c, S0 ] S0 R ∨ Q ¯ 0w Q ¯ 0x ↔ ( QR t R,a 0 ¯∨ ¯∨ ∃Q o ,(Qo ∈ S R ) ∧ 0 ∨ ¯∨ (Q o ∈SR )∧ ¯∨ ¯0 ¯0 ¯ t R[Q : type B d.sj.v.ly sc.c, Q ] Q Q o Qw Qx t

o

) ) ).

(d124):

Law of the set type B disjunctively scientific third unary relations

- 628 -

(125)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (126)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∨

(pwr)

,∀S0 R , ∀S0 R ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t, ∀Q ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q x w 0 ¯ (Qt ∈ S R ) → S0 R

(pwr) ∨

R[S(pwr) ] S0 R → (pwr)

(S0 R ∈ S0 R ¯ t R[Q : type Q

)→

0 ∨ ¯0 ¯0 A c.j.v.ly sc.c, S0 R,a ] S R Qw Qx ∨ ∨ 0 ∃Qo ,(Qo ∈ S R ) ∧ 0 ∨ (Q∨ o ∈S R )∧ ¯ t R[Q : type A c.j.v.ly sc.c, Q ] Q∨ ¯0 ¯0 Q o Qw Qx t o t

↔(

) ) ).

(d125):

Law of the set type A conjunctively scientific third unary relations

- 629 -

(126)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (127)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Qt , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q x w 0 (Qt ∈ S R ) → S0 R

(pwr) ∨

R[S(pwr) ] S0 R → (pwr)

)→ (S0 R ∈ S0 R ∨ ¯0 ¯0 Qt R[Qt : type B c.j.v.ly sc.c, S0 R,a ] S0 R Q w Qx ↔ ( ∨ ∨ 0 ∃Qo ,(Qo ∈ S R ) ∧ 0 ∨ (Q∨ o ∈S R )∧ ¯0 ¯0 Qt R[Qt : type B c.j.v.ly sc.c, Qo ] Q∨ o Qw Qx ) ) ).

(d126):

Law of the set type B conjunctively scientific third unary relations

- 630 -

(127)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (128)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀¯ pt , (L(pwr) ∈ S) → (L∨ ∈ S) → (¯ pt ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → p ¯t R[pt : d.sj.v.ly sc.c, La ] L∨ ↔ ( p ¯t R[pt : type B d.sj.v.ly sc.c, La ] L∨ ∨ p ¯t R[pt : type A d.sj.v.ly sc.c, La ] L∨ ) ) ).

(d127):

Law of the set disjunctively scientific third propositions

- 631 -

(128)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (129)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀L(pwr) ,∀L∨ , ∀pt , (L(pwr) ∈ S) → (L∨ ∈ S) → (pt ∈ L) → L(pwr) R[S(pwr) ] L → (L∨ ∈ L(pwr) ) → pt R[pt : c.j.v.ly sc.c, La ] L∨ ↔ ( pt R[pt : type B c.j.v.ly sc.c, La ] L∨ ∨ pt R[pt : type A c.j.v.ly sc.c, La ] L∨ ) ) ).

(d128):

Law of the set conjunctively scientific third propositions

- 632 -

(129)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (130)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t, ∀Q ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q x w 0 ¯ (Qt ∈ S R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(pwr)

)→ (S0 R ∈ S0 R ¯ t R[Q : d.sj.v.ly sc.c, S0 Q t

¯ t R[Q : Q t ∨ ¯ t R[Q : Q t

∨ ¯0 ¯0 S0 R Q w Qx ↔ ( 0 ∨ ¯0 ¯0 0 sc.c, S ] S R Qw Qx

R,a ]

type B d.sj.v.ly

R,a

0 ∨ ¯0 ¯0 type A d.sj.v.ly sc.c, S0 R,a ] S R Qw Qx

) ) ).

(d129):

Law of the set disjunctively scientific third unary relations

- 633 -

(130)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (131)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Qt , ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R ¯ 0 ∈ S0 R ) → ¯ 0 ∈ S0 R ) → (Q (Q x w 0 (Qt ∈ S R ) → S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(pwr)

)→ (S0 R ∈ S0 R ∨ ¯0 ¯0 Qt R[Qt : c.j.v.ly sc.c, S0 R,a ] S0 R Q w Qx ↔ ( ∨ ¯0 ¯0 Qt R[Q : type B c.j.v.ly sc.c, S0 ] S0 R Q w Qx t

∨ Qt R[Qt :

R,a

0 ∨ ¯0 ¯0 type A c.j.v.ly sc.c, S0 R,a ] S R Qw Qx

) ) ).

(d130):

Law of the set conjunctively scientific third unary relations

- 634 -

(131)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (132)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pt ,(pt ∈ L) → ( R[pt : type A d.sj.v.ly sc.c] pt ↔ pt R[pt : type A d.sj.v.ly sc.c, La ] L ) ) ).

(d131):

Law of the type A disjunctively scientific third propositions

(132)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (133)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pt ,(¯ pt ∈ L) → ( R[pt : type B d.sj.v.ly sc.c] p ¯t ↔ p ¯t R[pt : type B d.sj.v.ly sc.c, La ] L ) ) ).

(d132):

Law of the type B disjunctively scientific third propositions

- 635 -

(133)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (134)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pt ,(¯ pt ∈ L) → ( R[pt : type A c.j.v.ly sc.c] p ¯t ↔ p ¯t R[pt : type A c.j.v.ly sc.c, La ] L ) ) ).

(d133):

Law of the type A conjunctively scientific third propositions

(134)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (135)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pt ,(pt ∈ L) → ( R[pt : type B c.j.v.ly sc.c] pt ↔ pt R[pt : type B c.j.v.ly sc.c, La ] L ) ) ).

(d134):

Law of the type B conjunctively scientific third propositions

- 636 -

(135)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (136)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀¯ pt ,(¯ pt ∈ L) → R[pt : d.sj.v.ly sc.c] p ¯t ↔ ( R[pt : type B d.sj.v.ly sc.c] p ¯t ∨ R[pt : type A d.sj.v.ly sc.c] p ¯t ) ) ).

(d135):

Law of the disjunctively scientific third propositions

(136)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (137)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ∀pt ,(pt ∈ L) → R[pt : c.j.v.ly sc.c] pt ↔ ( R[pt : type B c.j.v.ly sc.c] pt ∨ R[pt : type A c.j.v.ly sc.c] pt ) ) ).

(d136):

Law of the conjunctively scientific third propositions

- 637 -

(137)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (138)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ¯ t, ∀Q ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q ¯ t ∈ S0 R ) → (Q ¯ t R[Q : d.sj.v.ly sc.c] Q ¯ 0w Q ¯ 0x ↔ ( Q t ¯ ¯ 0w Q ¯ 0x Qt R[Qt : type B d.sj.v.ly sc.c] Q ∨ ¯ t R[Q : type A d.sj.v.ly sc.c] Q ¯ 0w Q ¯ 0x Q t ) ) ).

(d137):

Law of the disjunctively scientific third unary relations

- 638 -

(138)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ↔ ( (139)

LR(nt.l,u.ry,b.c,dp.c.s) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw ∧ ( ¯ 0w ,∀ Q ¯ 0x , ∀Q ∀Qt , ¯ 0x ∈ S0 R ) → ¯ 0w ∈ S0 R ) → (Q (Q (Qt ∈ S0 R ) → ¯ 0w Q ¯ 0x ↔ ( Qt R[Qt : c.j.v.ly sc.c] Q ¯ 0w Q ¯ 0x Qt R[Qt : type B c.j.v.ly sc.c] Q ∨ ¯ 0w Q ¯ 0x Qt R[Qt : type A c.j.v.ly sc.c] Q ) ) ).

(d138):

Law of the conjunctively scientific third unary relations

- 639 -

A.3. Axiomatisation Probabilistic Languages

∀S, ∀L,∀Le ,∀Lp , ∀LD ,∀LI , ∀S0 R ,∀S0 p , ∀S0 D ,∀S0 I ,∀S0 Iw , ∀SR ,∀Sq , (S ∈ Scd.te ) → (L ∈ S) → (Le ∈ S) → (Lp ∈ S) → (LD ∈ S) → (LI ∈ S) → (S0 R ∈ S) → (S0 p ∈ S) → (S0 D ∈ S) → (S0 I ∈ S) → (S0 Iw ∈ S) → (SR ∈ S) → (Sq ∈ S) → R[Scd.te ] S → ( LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq → (1)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ).

anchor Law basic probabilistic natural languages

- 640 -

(1)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (2)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( RR SR ) ).

(p1): Law of the real number reference sets

(2)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (3)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀SR (pwr) ,(SR (pwr) ∈ S) → SR (pwr) R[S(pwr) ] SR → (Sq ∈ SR (pwr) ) ) ).

(p2): Law of the probability sets

(3)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (4)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∃qT ,(qT ∈ Sq ) ∧ R[q: aff.t.ve prb.ty] qT ) ).

(p3): Law of the existence of the affirmative probabilities

- 641 -

(4)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (5)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qT,b ,∀qT,a , (qT,b ∈ Sq ) → (qT,a ∈ Sq ) → ( R[q: aff.t.ve prb.ty] qT,b ∧ R[q: aff.t.ve prb.ty] qT,a )→ qT,b Rid qT,a ) ).

(p4): Law of the uniqueness of the affirmative probabilities

(5)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (6)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∃qF ,(qF ∈ Sq ) ∧ R[q: d.sm.ss.ve prb.ty] qF ) ).

(p5): Law of the existence of the dismissive probabilities

- 642 -

(6)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (7)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qF,b ,∀qF,a , (qF,b ∈ Sq ) → (qF,a ∈ Sq ) → ( R[q: d.sm.ss.ve prb.ty] qF,b ∧ R[q: d.sm.ss.ve prb.ty] qF,a )→ qF,b Rid qF,a ) ).

(p6): Law of the uniqueness of the dismissive probabilities

(7)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (8)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qT ,(qT ∈ Sq ) → ( R[q: aff.t.ve prb.ty] qT → R[‘1’]qT ) ) ).

(p7): Law of the composition of the affirmative probabilities

- 643 -

(8)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (9)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qF ,(qF ∈ Sq ) → ( R[q: d.sm.ss.ve prb.ty] qF → R[‘0’]qF ) ) ).

(p8): Law of the composition of the dismissive probabilities

(9)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (10)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qT ,∀q, (qT ∈ Sq ) → (q ∈ Sq ) → R[q: aff.t.ve prb.ty] qT → qR≤ qT ) ).

(p9): Law of the probabilities’ upper bound

- 644 -

(10)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (11)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qF ,∀q, (qF ∈ Sq ) → (q ∈ Sq ) → R[q: d.sm.ss.ve prb.ty] qF → qF R≤ q ) ).

(p10): Law of the probabilities’ lower bound

(11)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (12)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀q,(q ∈ Sq ) → R[q: d.t.c prb.ty] q ↔ ( R[q: aff.t.ve prb.ty] q ∨ R[q: d.sm.ss.ve prb.ty] q ) ) ).

(p11): Law of the deterministic probabilities

- 645 -

(12)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (13)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀q,(q ∈ Sq ) → ( R[q: i.d.t.c prb.ty] q ↔ ¯ [q: d.t.c prb.ty] q R ) ) ).

(p12): Law of the indeterministic probabilities

(13)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (14)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀q,∃¯ q, (q ∈ Sq ) → (¯ q ∈ Sq ) ∧ q ¯Rq¯ q ) ).

(p13): Law of the existence of the complementary probabilities

- 646 -

(14)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (15)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀q, ∀¯ qb ,∀¯ qa , (q ∈ Sq ) → (¯ qb ∈ Sq ) → (¯ qa ∈ Sq ) → ( q ¯a Rq¯ q ∧ q ¯b Rq¯ q )→ q ¯b Rid q ¯a ) ).

(p14): Law of the uniqueness of the complementary probabilities

- 647 -

(15)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (16)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀qT , ∀¯ q,∀q, (qT ∈ Sq ) → (¯ q ∈ Sq ) → (q ∈ Sq ) → R[q: aff.t.ve prb.ty] qT → ( q ¯Rq¯ q → qT S[q:q00 +q0 ] q ¯q ) ) ).

(p15): Law of the composition of the complementary probabilities

(16)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (17)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀p,∃¯ q∨ , (p ∈ L) → (¯ q∨ ∈ Sq ) ∧ p T∨ q ¯∨ ) ).

(p16): Law of the existence of the disjunctive probabilities

- 648 -

(17)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (18)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀p, ∀¯ q∨ q∨ b ,∀¯ a, (p ∈ L) → (¯ q∨ q∨ b ∈ Sq ) → (¯ a ∈ Sq ) → ( p T∨ q ¯∨ b ∧ p T∨ q ¯∨ a )→ q ¯∨ ¯∨ b Rid q a ) ).

(p17): Law of the uniqueness of the disjunctive probabilities

(18)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (19)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀p,∃q∧ , (p ∈ L) → (q∧ ∈ Sq ) ∧ p T∧ q∧ ) ).

(p18): Law of the existence of the conjunctive probabilities

- 649 -

(19)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (20)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀p, ∀qb ∧ ,∀qa ∧ , (p ∈ L) → (qb ∧ ∈ Sq ) → (qa ∧ ∈ Sq ) → ( p T∧ qb ∧ ∧ p T∧ qa ∧ )→ qb ∧ Rid qa ∧ ) ).

(p19): Law of the uniqueness of the conjunctive probabilities

- 650 -

(20)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (21)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀p, ∀¯ q∨ ,∀q∧ , (p ∈ L) → (¯ q∨ ∈ Sq ) → (q∧ ∈ Sq ) → q ¯∨ Rq¯ q∧ → ( p T∨ q ¯∨ → p T∧ q∧ ) ) ).

(p20): Law of the conjunctive probabilities

(21)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (22)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀¯ p,∀qT , (¯ p ∈ L) → (qT ∈ Sq ) → ( T∨ p ¯ ↔ p ¯ T∨ qT ) ) ).

(p21): Law of the disjunctively certain propositions

- 651 -

(22)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (23)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀p,∀qT , (p ∈ L) → (qT ∈ Sq ) → ( T∧ p ↔ p T∧ qT ) ) ).

(p22): Law of the conjunctively certain propositions

- 652 -

(23)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (24)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀¯ p00 ,∀¯ p0 , 000 ∀¯ p ,∀¯ p, ∀q000 ,∀q0 ,∀q, (¯ p00 ∈ L) → (¯ p0 ∈ L) → 000 (¯ p ∈ L) → (¯ p ∈ L) → (q000 ∈ Sq ) → (q0 ∈ Sq ) → (q ∈ Sq ) → p ¯000 S[p: p00 |∨ p0 ] p ¯00 p ¯0 → 00 0 p ¯ S[p: p00 ∨ p0 ] p ¯ p ¯ → ( p ¯000 T∨ q000 ∧ p ¯0 T∨ q0 )→ ( p ¯ T∨ q → qS[q:q00 ∗q0 ] q000 q0 ) ) ).

(p23): first Law of the resolution of the disjunctively conditional propositions

- 653 -

(24)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (25)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀¯ p00 ,∀p00 ,∀¯ p0 , 000 ∀p ,∀¯ p, ∀q00 ,∀q0 ,∀¯ q000 , ∀q(iv) ,∀q, (¯ p00 ∈ L) → (p00 ∈ L) → (¯ p0 ∈ L) → (p000 ∈ L) → (¯ p ∈ L) → (q00 ∈ Sq ) → (q0 ∈ Sq ) → (¯ q000 ∈ Sq ) → (q(iv) ∈ Sq ) → (q ∈ Sq ) → p ¯00 Rp¯ p00 → p000 S[p: p00 |∨ p0 ] p00 p ¯0 → 00 0 p ¯ S[p: p00 ∧ p0 ] p ¯ p ¯ → ( p ¯00 T∨ q00 ∧ p ¯0 T∨ q0 ∧ p000 T∨ q ¯000 )→ q(iv) S[q:q00 ∗q0 ] q ¯000 q0 → ( p ¯ T∨ q → qS[q:q00 +q0 ] q(iv) q00 ) ) ).

(p24): second Law of the resolution of the disjunctively conditional propositions

- 654 -

(25)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (26)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀¯ q, ∃Q, q ∈ Sq ) → (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (¯ (Q ∈ S0 R ) ∧ QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯ ) ).

(p25): Law of the existence of the disjunctively probabilistic generalised element existential world attributes

(26)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (27)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀¯ q, ∀Qb ,∀Qa , (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (¯ q ∈ Sq ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( Qb R[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯ ∧ Qa R[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯ )→ Qb Rid Qa ) ).

(p26): Law of the uniqueness of the disjunctively probabilistic generalised element existential world attributes

- 655 -

(27)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (28)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ¯ 0 ,∀¯ ∀Q0 x ,∀ Q q, ¯ ∃ Q, ¯ 0 ∈ S0 R ) → (¯ q ∈ Sq ) → (Q0 x ∈ S0 R ) → (Q ¯ ∈ S0 R ) ∧ (Q ¯ [Q: d.sj.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 , Q0 , q] Q0 x Q ¯ 0q QR ¯ x

) ).

(p27): Law of the existence of the disjunctively probabilistic generalised element universal world attributes

(28)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (29)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ¯ 0 ,∀¯ ∀Q0 x ,∀ Q q, ¯ ¯ ∀ Qb ,∀ Qa , ¯ 0 ∈ S0 R ) → (¯ (Q0 x ∈ S0 R ) → (Q q ∈ Sq ) → ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b R[Q: d.sj.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 , Q0 , q] Q0 x Q ¯ 0q Q ¯ x ∧ ¯ a R[Q: d.sj.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 , Q0 , q] Q0 x Q ¯ 0q Q ¯ x )→ ¯ b Rid Q ¯a Q ) ).

(p28): Law of the uniqueness of the disjunctively probabilistic generalised element universal world attributes

- 656 -

(29)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (30)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀q, ∃Q, (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (q ∈ Sq ) → (Q ∈ S0 R ) ∧ QR[Q: c.j.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ) ).

(p29): Law of the existence of the conjunctively probabilistic generalised element existential world attributes

(30)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (31)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀q, ∀Qb ,∀Qa , (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (q ∈ Sq ) → (Qb ∈ S0 R ) → (Qa ∈ S0 R ) → ( Qb R[Q: c.j.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ∧ Qa R[Q: c.j.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q )→ Qb Rid Qa ) ).

(p30): Law of the uniqueness of the conjunctively probabilistic generalised element existential world attributes

- 657 -

(31)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (32)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ¯ 0 ,∀q, ∀Q0 x ,∀ Q ¯ ∃ Q, ¯ 0 ∈ S0 R ) → (q ∈ Sq ) → (Q0 x ∈ S0 R ) → (Q ¯ ∈ S0 R ) ∧ (Q ¯ [Q: c.j.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 , Q0 , q] Q0 x Q ¯ 0q QR x

) ).

(p31): Law of the existence of the conjunctively probabilistic generalised element universal world attributes

(32)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (33)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ¯ 0 ,∀q, ∀Q0 x ,∀ Q ¯ ¯ ∀ Qb ,∀ Qa , ¯ 0 ∈ S0 R ) → (q ∈ Sq ) → (Q0 x ∈ S0 R ) → (Q ¯ b ∈ S0 R ) → (Q ¯ a ∈ S0 R ) → (Q ( ¯ b R[Q: c.j.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 , Q0 , q] Q0 x Q ¯ 0q Q x ∧ ¯ a R[Q: c.j.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 , Q0 , q] Q0 x Q ¯ 0q Q x )→ ¯ b Rid Q ¯a Q ) ).

(p32): Law of the uniqueness of the conjunctively probabilistic generalised element universal world attributes

- 658 -

(33)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (34)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ¯ 0 ,∀Q0 , ∀Q0 x ,∀ Q ∀¯ q,∀q, ∀Q, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q (¯ q ∈ Sq ) → (q ∈ Sq ) → (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → q ¯Rq¯ q → ( ¯ 0q → QR[Q: d.sj.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 x , Q0 , q] Q0 x Q QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯ ) ) ).

(p33): Law of the composition of the disjunctively probabilistic generalised element universal world attributes

- 659 -

(34)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (35)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ¯ 0 ,∀Q0 , ∀Q0 x ,∀ Q ∀¯ q,∀q, ∀Q, ¯ 0 ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q0 x ∈ S0 R ) → (Q (¯ q ∈ Sq ) → (q ∈ Sq ) → (Q ∈ S0 R ) → 0 ¯ 0 RQ Q ¯Q → q ¯Rq¯ q → ( ¯ 0q QR[Q: c.j.v.ly prb.c g.l.sed el.t u.l world attribute ,Q0 x , Q0 , q] Q0 x Q ¯→ QR[Q: c.j.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ) ) ).

(p34): Law of the composition of the conjunctively probabilistic generalised element universal world attributes

- 660 -

(35)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (36)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 , ∀¯ q,∀q, ∀Q, (Q0 x ∈ S0 R ) → ((∈∈ Q0 ))S0 R → (¯ q ∈ Sq ) → (q ∈ Sq ) → (Q ∈ S0 R ) → q ¯Rq¯ q → ( QR[Q: c.j.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q → QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯ ) ) ).

(p35): Law of the composition of the conjunctively probabilistic generalised element existential world attributes

- 661 -

(36)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (37)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( (pwr)

∨

,∀S0 R , ∀S0 R ∀Q0 x ,∀Q0 ,∀¯ q, ¯ ∀ Q,∀Q, ∨

(pwr)

∈ S) → (S0 R ∈ S) → (S0 R (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (¯ q ∈ Sq ) → ¯ ∈ S0 R ) → (Q ∈ S0 R ) → (Q S0 R

(pwr)

R[S(pwr) ] S0 R →

∨

(pwr)

(S0 R ∈ S0 R )→ ¯ Q QR ¯Q→ ( ¯ ∨ ,(Q ¯ ∨ ∈ S0 R ) → ∀Q ¯ ∨ ∈ S0 R ∨ ) ↔ ( (Q ∃q0 ,(q0 ∈ Sq ) ∧ ¯ id q q0 R ¯∧ ¯ ∨ R[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 , Q0 , q] Q0 x Q0 q0 Q x ) )→ ( QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯→ ¯ [S0 ∨ ] S0 R ∨ QS R

) ) ).

(p36): Law of the basic probabilistic world attributes’ distributivity

- 662 -

(37)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (38)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Ix ,∀Q0 ,∀qT , ¯ ∀ Q, (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (qT ∈ Sq ) → ¯ ∈ S0 R ) → (Q R[q: aff.t.ve prb.ty] qT → ( ¯ [Q: d.sj.v.ly d.t.c prt el.t ex.st.l world attribute ,I , Q0 ] Ix Q0 → QR x ¯ QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Ix Q0 qT ) ) ).

(p37): first Law of the composition of the basic deterministic world attributes

- 663 -

(38)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (39)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Ix ,∀Q0 ,∀qT , ∀Q, (Ix ∈ S0 I ) → (Q0 ∈ S0 R ) → (qT ∈ Sq ) → (Q ∈ S0 R ) → R[q: aff.t.ve prb.ty] qT → ( QR[Q: c.j.v.ly d.t.c prt el.t ex.st.l world attribute ,Ix , Q0 ] Ix Q0 → QR[Q: c.j.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Ix Q0 qT ) ) ).

(p38): second Law of the composition of the basic deterministic world attributes

- 664 -

(39)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (40)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀Q000 , ∀Q∧ q, T ,∀¯ ∀Q, (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (Q000 ∈ S0 R ) → 0 (Q∧ q ∈ Sq ) → T ∈ S R ) → (¯ (Q ∈ S0 R ) → R[Q∧ ] Q∧ T → T

Q000 S[Qx: Q00 x ∧ Q0 x] Q0 x Q0 → ( QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯→ 000 QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q∧ Q q ¯ T ) ) ).

(p39): Law of the basic probabilistic world attributes’ conjunctive attribute transfer

- 665 -

(40)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (41)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Iw , ∀Q0 x ,∀Q0 ,∀¯ q, ∀Q,∀p, (Iw ∈ S0 I ) → q ∈ Sq ) → (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (¯ (Q ∈ S0 R ) → (p ∈ L) → pR[p : ‘∃w,∃x,(w ∈ wcd.te ) ∧ (x ∈ w) ∧ ‘Iw ’w ∧ (‘Ix ’x ∧ ‘Q’x)’]Iw Q0 x Q0 → QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯→ ( p T∨ q ¯ QR[Q0 : c.j.v.ly c.t.t nc.ss.ry, Q] Iw ) ) ).

(p40): Law of the resolution of the basic probabilistic world attributes

- 666 -

(41)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (42)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀¯ q, ∀Q, q ∈ Sq ) → (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (¯ (Q ∈ S0 R ) → R[q: i.d.t.c prb.ty] q ¯→ ( QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯→ R[Q: pr.ry u.ry r.l.n] Q ) ) ).

(p41): Law of the primary composition of the basic world attributes

(42)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (43)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q0 x ,∀Q0 ,∀¯ q, ∀Q, (Q0 x ∈ S0 R ) → (Q0 ∈ S0 R ) → (¯ q ∈ Sq ) → (Q ∈ S0 R ) → R[q: i.d.t.c prb.ty] q ¯→ ( QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q0 x Q0 q ¯→ R[Q: b.c c.p.sed u.ry r.l.n] Q ) ) ).

(p42): Law of the basic composition of the indeterministic world attributes

- 667 -

(43)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ ( (44)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ∧ ( ∀Q,(Q ∈ S0 R ) → R[Q: b.c i.d.t.c w.rld attr.te] Q ↔ ( 000 ∃Q∧ q, T ,∃Q ,∃¯ 0 000 0 (Q∧ q ∈ Sq ) ∧ T ∈ S R ) ∧ (Q ∈ S R ) ∧ (¯ ∧ R[Q∧ ] QT ∧ T

R[q: i.d.t.c prb.ty] q ¯∧ R[Q: b.c c.p.sed, pr.ry u.ry r.l.n] Q000 ∧ 000 QR[Q: d.sj.v.ly prb.c g.l.sed el.t ex.st.l world attribute ,Q0 x , Q0 , q] Q∧ ¯ TQ q ) ) ).

(p43): Law of the basic indeterministic world attributes

- 668 -

∀S, ∀L,∀Le ,∀Lp , ∀LD ,∀LI , ∀S0 R ,∀S0 p , ∀S0 D ,∀S0 I ,∀S0 Iw , ∀SR ,∀Sq , (S ∈ Scd.te ) → (L ∈ S) → (Le ∈ S) → (Lp ∈ S) → (LD ∈ S) → (LI ∈ S) → (S0 R ∈ S) → (S0 p ∈ S) → (S0 D ∈ S) → (S0 I ∈ S) → (S0 Iw ∈ S) → (SR ∈ S) → (Sq ∈ S) → R[Scd.te ] S → ( (44)

LR(nt.l,u.ry,prb.c,b.c) Le Lp LD LI S0 R S0 p S0 D S0 I S0 Iw SR Sq ↔ 0 0 0 0 0 LR∧ T Le Lp LD LI S R S p S D S I S Iw SR Sq ).

(44): final Law probabilistic natural languages

- 669 -

(29)

I.1. Index Basic Laws Unary Languages (1): Law of the propositions’ extended composition (2): Law of the unary relations’ extended composition (3): first Law of the composition of the set of elemental propositions (4): second Law of the composition of the set of elemental propositions (5): first Law of the composition of the set of primary propositions (6): second Law of the composition of the set of primary propositions (7): first Law of the composition of the set of primary unary relations (8): second Law of the composition of the set of primary unary relations (9): first Law of the composition of the set of Dirac propositions (10): second Law of the composition of the set of Dirac propositions (11): first Law of the composition of the set of Dirac unary relations (12): second Law of the composition of the set of Dirac unary relations (13): first Law of the composition of the set of reduced Dirac propositions (14): second Law of the composition of the set of reduced Dirac propositions (15): first Law of the composition of the set of particular element identifying unary relations

- 670 -

(16): second Law of the composition of the set of particular element identifying unary relations (17): first Law of the composition of the set of particular world identifying unary relations (18): second Law of the composition of the set of particular world identifying unary relations (19): Law of the joint elemental propositions’ basic reduced composition (20): Law of the basic composed and primary unary relations (21): Law of the extended composed and primary unary relations (22): Law of the analytic propositions (23): Law of the synthetic propositions (24): Law of the analytic unary relations (25): Law of the synthetic unary relations (26): Law of the a priori propositions (27): Law of the a posteriori propositions (28): Law of the a priori unary relations (29): Law of the a posteriori unary relations (30): Law of the propositions’ primary or secondary composition (31): Law of the unary relations’ primary or secondary composition

- 671 -

(32): Law of the primary and secondary propositions (33): Law of the primary and secondary unary relations (34): Law of the existence of the basic disjunctively forbidden propositions (35): Law of the uniqueness of the basic disjunctively forbidden propositions (36): Law of the existence of the basic disjunctively necessary propositions (37): Law of the uniqueness of the basic disjunctively necessary propositions (38): Law of the existence of the basic conjunctively forbidden propositions (39): Law of the uniqueness of the basic conjunctively forbidden propositions (40): Law of the existence of the basic conjunctively necessary propositions (41): Law of the uniqueness of the basic conjunctively necessary propositions (42): Law of the existence of the basic disjunctively forbidden unary relations (43): Law of the uniqueness of the basic disjunctively forbidden unary relations (44): Law of the existence of the basic disjunctively necessary unary relations (45): Law of the uniqueness of the basic disjunctively necessary unary relations (46): Law of the existence of the basic conjunctively forbidden unary relations (47): Law of the uniqueness of the basic conjunctively forbidden unary relations (48): Law of the existence of the basic conjunctively necessary unary relations

- 672 -

(49): Law of the uniqueness of the basic conjunctively necessary unary relations (50): Law of the composition of the basic disjunctively forbidden propositions (51): Law of the composition of the basic disjunctively necessary propositions (52): Law of the composition of the basic conjunctively forbidden propositions (53): Law of the composition of the basic conjunctively necessary propositions (54): Law of the composition of the basic disjunctively forbidden unary relations (55): Law of the composition of the basic disjunctively necessary unary relations (56): Law of the composition of the basic conjunctively forbidden unary relations (57): Law of the composition of the basic conjunctively necessary unary relations (58): Law of the resolution of the basic disjunctively necessary propositions (59): Law of the resolution of the basic conjunctively necessary propositions (60): Law of the resolution of the basic disjunctively forbidden unary relations (61): Law of the existence of the propositions’ complements (62): Law of the uniqueness of the propositions’ complements (63): Law of the existence of the unary relations’ complements (64): Law of the uniqueness of the unary relations’ complements (65): Law of the reciprocity of the propositions’ complementarity

- 673 -

(66): Law of the reciprocity of the unary relations’ complementarity (67): Law of the composition of the propositions’ complements (68): Law of the composition of the unary relations’ complements (69): Law of the existence of the negatively conjunctive propositions (70): Law of the uniqueness of the negatively conjunctive propositions (71): Law of the existence of the negatively conjunctive unary relations (72): Law of the uniqueness of the negatively conjunctive unary relations (73): Law of the basic composed negatively conjunctive propositions (74): Law of the basic composed negatively conjunctive unary relations (75): Law of the primary negatively conjunctive propositions (76): Law of the secondary negatively conjunctive propositions (77): Law of the primary negatively conjunctive unary relations (78): Law of the secondary negatively conjunctive unary relations (79): Law of the existence of the disjunctive propositions (80): Law of the uniqueness of the disjunctive propositions (81): Law of the existence of the disjunctive unary relations (82): Law of the uniqueness of the disjunctive unary relations

- 674 -

(83): Law of the composition of the disjunctive propositions (84): Law of the composition of the disjunctive unary relations (85): Law of the existence of the conjunctive propositions (86): Law of the uniqueness of the conjunctive propositions (87): Law of the existence of the conjunctive unary relations (88): Law of the uniqueness of the conjunctive unary relations (89): Law of the composition of the conjunctive propositions (90): Law of the composition of the conjunctive unary relations (91): Law of the existence of the implicative propositions (92): Law of the uniqueness of the implicative propositions (93): Law of the existence of the implicative unary relations (94): Law of the uniqueness of the implicative unary relations (95): Law of the composition of the implicative propositions (96): Law of the composition of the implicative unary relations (97): Law of the existence of the prohibitive propositions (98): Law of the uniqueness of the prohibitive propositions (99): Law of the existence of the prohibitive unary relations

- 675 -

(100): Law of the uniqueness of the prohibitive unary relations (101): Law of the composition of the prohibitive propositions (102): Law of the composition of the prohibitive unary relations (103): Law of the existence of the bijective propositions (104): Law of the uniqueness of the bijective propositions (105): Law of the existence of the bijective unary relations (106): Law of the uniqueness of the bijective unary relations (107): Law of the composition of the bijective propositions (108): Law of the composition of the bijective unary relations (109): Law of the existence of the exclusively disjunctive propositions (110): Law of the uniqueness of the exclusively disjunctive propositions (111): Law of the existence of the exclusively disjunctive unary relations (112): Law of the uniqueness of the exclusively disjunctive unary relations (113): Law of the composition of the exclusively disjunctive propositions (114): Law of the composition of the exclusively disjunctive unary relations (115): Law of the existence of the disjoint propositions (116): Law of the uniqueness of the disjoint propositions

- 676 -

(117): Law of the existence of the joint propositions (118): Law of the uniqueness of the joint propositions (119): Law of the existence of the disjoint unary relations (120): Law of the uniqueness of the disjoint unary relations (121): Law of the existence of the joint unary relations (122): Law of the uniqueness of the joint unary relations (123): Law of the complementarity of the disjoint and the joint propositions (124): Law of the complementarity of the disjoint and the joint unary relations (125): third Law of the composition of the disjoint propositions (126): second Law of the composition of the disjoint propositions (127): first Law of the composition of the disjoint propositions (128): third Law of the composition of the disjoint unary relations (129): second Law of the composition of the disjoint unary relations (130): first Law of the composition of the disjoint unary relations (131): Law of the associativity of the disjoint propositions (132): Law of the joint distributivity of the disjoint propositions (133): Law of the associativity of the disjoint unary relations

- 677 -

(134): Law of the joint distributivity of the disjoint unary relations (135): Law of the resolution of the disjoint propositions (136): Law of the resolution of the joint propositions (137): Law of the resolution of the disjoint unary relations (138): Law of the primary disjoint propositions (139): Law of the secondary disjoint propositions (140): Law of the primary disjoint unary relations (141): Law of the secondary disjoint unary relations (142): Law of the existence of the disjunctively conditional propositions (143): Law of the uniqueness of the disjunctively conditional propositions (144): Law of the existence of the conjunctively conditional propositions (145): Law of the uniqueness of the conjunctively conditional propositions (146): Law of the existence of the disjunctively conditional unary relations (147): Law of the uniqueness of the disjunctively conditional unary relations (148): Law of the existence of the conjunctively conditional unary relations (149): Law of the uniqueness of the conjunctively conditional unary relations (150): Law of the double conditional disjunctively conditional propositions

- 678 -

(151): Law of the double conditional disjunctively conditional unary relations (152): Law of the composition of the conjunctively conditional propositions (153): Law of the composition of the conjunctively conditional unary relations (154): Law of the disjunctively conditional propositions (155): Law of the disjunctively conditional unary relations (156): Law of the negatively conjunctive disjunctivly conditional propositions (157): Law of the negatively conjunctive disjunctivly conditional unary relations (158): Law of the disjoint disjunctively conditional propositions (159): Law of the disjoint disjunctively conditional unary relations (160): Law of the disjunctively conditioned disjunctively conditional propositions (161): Law of the disjunctively conditioned disjunctively conditional unary relations (162): Law of the self conditional disjunctively conditional propositions (163): Law of the self conditional disjunctively conditional unary relations (164): Law of the basic disjunctively forbidden disjunctively conditional propositions (165): Law of the basic disjunctively forbidden disjunctively conditional unary relations (166): Law of the negatively conditional disjunctively conditional propositions (167): Law of the negatively conditional disjunctively conditional unary relations

- 679 -

(168): Law of the positively conditional disjunctively conditional propositions (169): Law of the positively conditional disjunctively conditional unary relations (170): Law of the resolution of the disjunctively conditional unary relations (171): Law of the primary disjunctively conditional propositions (172): Law of the secondary disjunctively conditional propositions (173): Law of the primary disjunctively conditional unary relations (174): Law of the secondary disjunctively conditional unary relations (175): Law of the elemental existential propositions (176): Law of the elemental universal propositions (177): Law of the elemental propositions (178): Law of the joint elemental propositions (179): Law of the basic existential propositions (180): Law of the basic universal propositions (181): Law of the basic propositions (182): Law of the joint basic propositions (183): Law of the Dirac propositions’ conjunctive content completeness (184): Law of the Dirac propositions’ conjunctive content consistency

- 680 -

(185): Law of the Dirac propositions’ joint basic composition (186): Law of the existence of the particular Dirac propositions (187): Law of the Dirac unary relations’ conjunctive content completeness (188): Law of the Dirac unary relations’ conjunctive content consistency (189): Law of the existence of the particular Dirac unary relations (190): Law of the elemental conjunctive content completeness of the reduced Dirac propositions (191): Law of the elemental conjunctive content consistency of the reduced Dirac propositions (192): Law of the joint elemental composition of the reduced Dirac propositions (193): Law of the existence of the particular reduced Dirac propositions (194): Law of the primary content completeness of the particular element identifying unary relations (195): Law of the primary content consistency of the particular element identifying unary relations (196): Law of the conjunctive content non recursion of the particular element identifying unary relations (197): Law of the existence of the particular element identifying unary relations (198): Law of the basic composition of the particular element identifying unary relations (199): Law of the primary composition of the particular element identifying unary relations

- 681 -

(200): Law of the particular non empty world identifying unary relations (201): Law of the existence of the particular empty world identifying unary relations (202): Law of the uniqueness of the particular empty world identifying unary relations (203): Law of the joint basic composition of the particular non empty world identifying unary relations (204): Law of the existence of the deterministic worlds (205): Law of the existence of the candidate world identifying unary relations (206): Law of the basic composed propositions (207): Law of the extended composed propositions (208): Law of the basic composed unary relations (209): Law of the extended composed unary relations (210): Law of the basic reduced propositions (211): Law of the extended reduced propositions (212): Law of the basic composed, primary unary relations (213): Law of the extended composed, primary unary relations (214): Law of the existence of the disjunctively deterministic generalised element existential world attributes (215): Law of the uniqueness of the disjunctively deterministic generalised element existential world attributes

- 682 -

(216): Law of the existence of the disjunctively deterministic generalised element universal world attributes (217): Law of the uniqueness of the disjunctively deterministic generalised element universal world attributes (218): Law of the existence of the conjunctively deterministic generalised element existential world attributes (219): Law of the uniqueness of the conjunctively deterministic generalised element existential world attributes (220): Law of the existence of the conjunctively deterministic generalised element universal world attributes (221): Law of the uniqueness of the conjunctively deterministic generalised element universal world attributes (222): Law of the composition of the conjunctively deterministic generalised element existential world attributes (223): Law of the composition of the conjunctively deterministic generalised element universal world attributes (224): first Law of the distributivity of the deterministic world attributes (225): second Law of the distributivity of the deterministic world attributes (226): Law of the deterministic world attributes’ conjunctive attribute transfer (227): Law of the deterministic wold attributes’ disjunctive attribute transfer (228): Law of the deterministic world attributes’ disjunctive separation (229): Law of the deterministic world attributes’ conjunctive separation (230): first Law of the conditional deterministic world attributes

- 683 -

(231): second Law of the conditional deterministic world attributes (232): first Law of the deterministic world attributes’ primary composition (233): second Law of the deterministic world attributes’ primary composition (234): first Law of the basic composed deterministic world attributes (235): second Law of the basic composed deterministic world attributes (236): first Law of the resolution of the deterministic world attributes (237): second Law of the resolution of the deterministic world attributes (238): Law of the existence of the generalised element disjunctively deterministically empty element world attributes (239): Law of the uniqueness of the generalised element disjunctively deterministically empty element world attributes (240): Law of the existence of the generalised element conjunctively deterministically empty element world attributes (241): Law of the uniqueness of the generalised element conjunctively deterministically empty element world attributes (242): Law of the existence of the general element disjunctively deterministically empty element world attributes (243): Law of the uniqueness of the general element disjunctively deterministically empty element world attributes (244): Law of the existence of the general element conjunctively deterministically empty element world attributes

- 684 -

(245): Law of the uniqueness of the general element conjunctively deterministically empty element world attributes (246): Law of the composition of the generalised element disjunctively deterministically empty element world attributes (247): Law of the composition of the generalised element conjunctively deterministically empty element world attributes (248): Law of the composition of the general element disjunctively deterministically empty element world attributes (249): Law of the composition of the general element conjunctively deterministically empty element world attributes (250): Law of (251): Law of (252): Law of (253): Law of (254): Law of (255): Law of

the basic deterministic existential world attributes the basic deterministic universal world attributes the basic deterministic world attributes the determinism of the basic deterministic world attributes the basic world attributes the joint basic world attributes

(256): Law of the existence of the particular world existential, particular element existential basic propositions (257): Law of the uniqueness of the particular world existential, particular element existential basic propositions (258): Law of the existence of the particular world existential, particular element universal basic propositions

- 685 -

(259): Law of the uniqueness of the particular world existential, particular element universal basic propositions (260): Law of the existence of the particular world universal, particular element existential basic propositions (261): Law of the uniqueness of the particular world universal, particular element existential basic propositions (262): Law of the existence of the particular world universal, particular element universal basic propositions (263): Law of the uniqueness of the particular world universal, particular element universal basic propositions (264): propositional Law of the world of candidate elements (265): Law of the basic propositions’ conjunctive parcelling (266): Law of the basic propositions’ disjunctive parcelling (267): Law of the basic propositions’ conjunctive attribute transfer (268): Law of the basic composed basic propositions (269): Law of the secondary basic propositions (270): Law of the primary basic propositions (271): Law of the normal composition of the primary elemental propositions (272): Law of the elemental propositions’ basic composition (273): Law of the basic propositions’ conjunctive normalisation

- 686 -

(274): Law of the complements of the generalised world existential, generalised element existential basic propositions (275): Law of the complements of the generalised world existential, generalised element universal basic propositions (276): Law of the composition of the particular world universal, generalised element existential basic propositions (277): second Law of the distributivity of the basic propositions (278): third Law of the distributivity of the basic propositions (279): first Law of the distributivity of the basic propositions (280): final Law basic unary natural languages

- 687 -

I.2. Index Laws Unary Languages’ Dependencies (d1): Law of the disjunctively content present propositions (d2): Law of the disjunctively content absent propositions (d3): Law of the conjunctively content present propositions (d4): Law of the conjunctively content absent propositions (d5): Law of the disjunctively content present unary relations (d6): Law of the disjunctively content absent unary relations (d7): Law of the conjunctively content present unary relations (d8): Law of the conjunctively content absent unary relations (d9): Law of the disjunctively content necessary propositions (d10): Law of the disjunctively content forbidden propositions (d11): Law of the conjunctively content necessary propositions (d12): Law of the conjunctively content forbidden propositions (d13): Law of the disjunctively content necessary unary relations (d14): Law of the disjunctively content forbidden unary relations (d15): Law of the conjunctively content necessary unary relations

- 688 -

(d16): Law of the conjunctively content forbidden unary relations (d17): Law of the element disjunctively content recursive propositions (d18): Law of the element conjunctively content recursive propositions (d19): Law of the element disjunctively content recursive unary relations (d20): Law of the element conjunctively content recursive unary relations (d21): Law of the set disjunctively content recursive propositions (d22): Law of the set conjunctively content recursive propositions (d23): Law of the set disjunctively content recursive unary relations (d24): Law of the set conjunctively content recursive unary relations (d25): Law of the set set disjunctively content recursive propositions (d26): Law of the set set conjunctively content recursive propositions (d27): Law of the set set disjunctively content recursive unary relations (d28): Law of the set set conjunctively content recursive unary relations (d29): Law of the disjunctively content recursive propositions (d30): Law of the conjunctively content recursive propositions (d31): Law of the disjunctively content recursive unary relations (d32): Law of the conjunctively content recursive unary relations

- 689 -

(d33): Law of the conjunctively necessary propositions (d34): Law of the conjunctively forbidden propositions (d35): Law of the disjunctively necessary unary relations (d36): Law of the disjunctively forbidden unary relations (d37): Law of the conjunctively necessary unary relations (d38): Law of the conjunctively forbidden unary relations (d39): Law of the disjunctively deterministic propositions (d40): Law of the disjunctively indeterministic propositions (d41): Law of the conjunctively deterministic propositions (d42): Law of the conjunctively indeterministic propositions (d43): Law of the disjunctively deterministic unary relations (d44): Law of the disjunctively indeterministic unary relations (d45): Law of the conjunctively deterministic unary relations (d46): Law of the conjunctively indeterministic unary relations (d47): Law of the disjunctively underdeterministic propositions (d48): Law of the disjunctively overdeterministic propositions (d49): Law of the conjunctively underdeterministic propositions

- 690 -

(d50): Law of the conjunctively overdeterministic propositions (d51): Law of the disjunctively underdeterministic unary relations (d52): Law of the disjunctively overdeterministic unary relations (d53): Law of the conjunctively underdeterministic unary relations (d54): Law of the conjunctively overdeterministic unary relations (d55): Law of the disjunctively necessary third propositions (d56): Law of the disjunctively forbidden third propositions (d57): Law of the conjunctively necessary third propositions (d58): Law of the conjunctively forbidden third propositions (d59): Law of the disjunctively necessary third unary relations (d60): Law of the disjunctively forbidden third unary relations (d61): Law of the conjunctively necessary third unary relations (d62): Law of the conjunctively forbidden third unary relations (d63): Law of the disjunctively implicitly deterministic third propositions (d64): Law of the disjunctively implicitly indeterministic third propositions (d65): Law of the conjunctively implicitly deterministic third propositions (d66): Law of the conjunctively implicitly indeterministic third propositions

- 691 -

(d67): Law of the disjunctively implicitly deterministic third unary relations (d68): Law of the disjunctively implicitly indeterministic third unary relations (d69): Law of the conjunctively implicitly deterministic third unary relations (d70): Law of the conjunctively implicitly indeterministic third unary relations (d71): Law of the disjunctively implicitly underdeterministic third propositions (d72): Law of the disjunctively implicitly overdeterministic third propositions (d73): Law of the conjunctively implicitly underdeterministic third propositions (d74): Law of the conjunctively implicitly overdeterministic third propositions (d75): Law of the disjunctively implicitly underdeterministic third unary relations (d76): Law of the disjunctively implicitly overdeterministic third unary relations (d77): Law of the conjunctively implicitly underdeterministic third unary relations (d78): Law of the conjunctively implicitly overdeterministic third unary relations (d79): Law of the element disjunctively explicitly complete second propositions (d80): Law of the element disjunctively explicitly incomplete second propositions (d81): Law of the element conjunctively explicitly complete second propositions (d82): Law of the element conjunctively explicitly incomplete second propositions (d83): Law of the element disjunctively explicitly complete second unary relations

- 692 -

(d84): Law of the element disjunctively explicitly incomplete second unary relations (d85): Law of the element conjunctively explicitly complete second unary relations (d86): Law of the element conjunctively explicitly incomplete second unary relations (d87): Law of the element disjunctively explicitly consistent second propositions (d88): Law of the element disjunctively explicitly inconsistent second propositions (d89): Law of the element conjunctively explicitly consistent second propositions (d90): Law of the element conjunctively explicitly inconsistent second propositions (d91): Law of the element disjunctively explicitly consistent second unary relations (d92): Law of the element disjunctively explicitly inconsistent second unary relations (d93): Law of the element conjunctively explicitly consistent second unary relations (d94): Law of the element conjunctively explicitly inconsistent second unary relations (d95): Law of the disjunctively implicitly G¨odel like third propositions (d96): Law of the conjunctively implicitly G¨odel like third propositions (d97): Law of the disjunctively implicitly G¨odel like third unary relations (d98): Law of the conjunctively implicitly G¨odel like third unary relations (d99): Law of the disjunctively G¨odel like second propositions (d100): Law of the conjunctively G¨odel like second propositions

- 693 -

(d101): Law of the disjunctively G¨odel like second unary relations (d102): Law of the conjunctively G¨odel like second unary relations (d103): Law of the disjunctive G¨odel second propositions (d104): Law of the conjunctive G¨odel second propositions (d105): Law of the disjunctive G¨odel second unary relations (d106): Law of the conjunctive G¨odel second unary relations (d107): Law of the element type A disjunctively scientific third propositions (d108): Law of the element type B disjunctively scientific third propositions (d109): Law of the element type A conjunctively scientific third propositions (d110): Law of the element type B conjunctively scientific third propositions (d111): Law of the element type A disjunctively scientific third unary relations (d112): Law of the element type B disjunctively scientific third unary relations (d113): Law of the element type A conjunctively scientific third unary relations (d114): Law of the element type B conjunctively scientific third unary relations (d115): Law of the element disjunctively scientific third propositions (d116): Law of the element conjunctively scientific third propositions (d117): Law of the element disjunctively scientific third unary relations

- 694 -

(d118): Law of the element conjunctively scientific third unary relations (d119): Law of the set type A disjunctively scientific third propositions (d120): Law of the set type B disjunctively scientific third propositions (d121): Law of the set type A conjunctively scientific third propositions (d122): Law of the set type B conjunctively scientific third propositions (d123): Law of the set type A disjunctively scientific third unary relations (d124): Law of the set type B disjunctively scientific third unary relations (d125): Law of the set type A conjunctively scientific third unary relations (d126): Law of the set type B conjunctively scientific third unary relations (d127): Law of the set disjunctively scientific third propositions (d128): Law of the set conjunctively scientific third propositions (d129): Law of the set disjunctively scientific third unary relations (d130): Law of the set conjunctively scientific third unary relations (d131): Law of the type A disjunctively scientific third propositions (d132): Law of the type B disjunctively scientific third propositions (d133): Law of the type A conjunctively scientific third propositions (d134): Law of the type B conjunctively scientific third propositions

- 695 -

(d135): Law of the disjunctively scientific third propositions (d136): Law of the conjunctively scientific third propositions (d137): Law of the disjunctively scientific third unary relations (d138): Law of the conjunctively scientific third unary relations

- 696 -

I.3. Index Laws Probabilistic Languages (p1): Law of the real number reference sets (p2): Law of the probability sets (p3): Law of the existence of the affirmative probabilities (p4): Law of the uniqueness of the affirmative probabilities (p5): Law of the existence of the dismissive probabilities (p6): Law of the uniqueness of the dismissive probabilities (p7): Law of the composition of the affirmative probabilities (p8): Law of the composition of the dismissive probabilities (p9): Law of the probabilities’ upper bound (p10): Law of the probabilities’ lower bound (p11): Law of the deterministic probabilities (p12): Law of the indeterministic probabilities (p13): Law of the existence of the complementary probabilities (p14): Law of the uniqueness of the complementary probabilities (p15): Law of the composition of the complementary probabilities

- 697 -

(p16): Law of the existence of the disjunctive probabilities (p17): Law of the uniqueness of the disjunctive probabilities (p18): Law of the existence of the conjunctive probabilities (p19): Law of the uniqueness of the conjunctive probabilities (p20): Law of the conjunctive probabilities (p21): Law of the disjunctively certain propositions (p22): Law of the conjunctively certain propositions (p23): first Law of the resolution of the disjunctively conditional propositions (p24): second Law of the resolution of the disjunctively conditional propositions (p25): Law of the existence of the disjunctively probabilistic generalised element existential world attributes (p26): Law of the uniqueness of the disjunctively probabilistic generalised element existential world attributes (p27): Law of the existence of the disjunctively probabilistic generalised element universal world attributes (p28): Law of the uniqueness of the disjunctively probabilistic generalised element universal world attributes (p29): Law of the existence of the conjunctively probabilistic generalised element existential world attributes (p30): Law of the uniqueness of the conjunctively probabilistic generalised element existential world attributes

- 698 -

(p31): Law of the existence of the conjunctively probabilistic generalised element universal world attributes (p32): Law of the uniqueness of the conjunctively probabilistic generalised element universal world attributes (p33): Law of the composition of the disjunctively probabilistic generalised element universal world attributes (p34): Law of the composition of the conjunctively probabilistic generalised element universal world attributes (p35): Law of the composition of the conjunctively probabilistic generalised element existential world attributes (p36): Law of the basic probabilistic world attributes’ distributivity (p37): first Law of the composition of the basic deterministic world attributes (p38): second Law of the composition of the basic deterministic world attributes (p39): Law of the basic probabilistic world attributes’ conjunctive attribute transfer (p40): Law of the resolution of the basic probabilistic world attributes (p41): Law of the primary composition of the basic world attributes (p42): Law of the basic composition of the indeterministic world attributes (p43): Law of the basic indeterministic world attributes

- 699 -

.

- 700 -