An Invitation to Formal Reasoning

' be the state of affairs in which p (i.e., the state signified by 'p') and let the term 'p' denote any domain or world characterized by

(i.e., any p-world). By convention, lower case letters are statement letters. But in our new way of construing these letters they are a special type of term that apply to the members of a domain of worlds, 'p' applying top-worlds, 'q' to q-worlds and so forth. But in this case, the domain of worlds, which is the domain of the claim of any propositional statement, contains only one world as its member (by hypothesis the actual world). In effect, all propositional statements are now construed as singular statements about the one actual world. Since in any context of discourse only one domain is under consideration, all propositional statements are semantically singular. It is as if all propositional (lower case) letters were starred as uniquely denoting terms. We are saying that all propositional statements are about one and only one thing: the world itself. We may read 'p' categorically as the statement: 'the world is a p-world'; moreover 'p' is true if and only ifthe world is a pworld. We now read our formulas '+p+q', '- p+q', etc., as 'some p-world is a q-world. 'every p-world is a q-world', etc. Since the categorical version of a propositional statement is a statement of the form 'some/every xis a y', the bare assertion 'p' may be understood as the categorical assertion 'some world is a p-world'. However, since all statements are about the one world, 'some p-world is a q-world' entails 'every p-world is a q-world'. We rewrite propositional statements in categorical form in accordance with the following rules:

Statement Logic 1. 2. 3. 4.

p&q p ::> q

p -p

some pis a q every pis a q some thing is a p (some pis a p) no thing is a p (no p is a p)

207 +p+q -p+q +p+p -(+p+p)

Because of their singularity, propositional assertions are subject to special laws that distinguish them from ordinary categorical statements. The latter are about many things. But propositional assertions are all about one and the same thing: the world itself. In particular, the following two propositional laws are especially important: 1. some world is p-world = every world is a p-world 2. some pis q =every pis q. According to the first law any statement 'p' claims that 'some world is a pworld'. But since only one world is under consideration, this claim is tantamount to the claim that every world is a p-world. The truth of the second law is also obvious; there being only one world, if that world is both a p-world and a q-world, then every world is a q-world and in particular every p-world is a q-world. By construing every propositional statement as a categorical statement about worlds, we carry out Leibniz's terminist program. Leibniz correctly believed that statement logic could be understood as a special branch of the logic of terms. This terminist doctrine is encouraged when we note (a) that compound statements have the same formal structure as categorical statements whose elements are terms and (b) that many of the laws of statement logic (e.g., the law that denying a conjunction is equivalent to affirming the disjunction of denials) are the same as the laws of term logic (e.g., that denying that some A is B is equivalent to affirming that every A is not B since both laws have the structure: -(+x+y) = +(-x-y)). We noted, however, that terminism seemed wrong when we consider that the categorical versions of propositional statements led us to a branch of term logic that differs in fundamental ways from the rest ofterm logic. For example, we know that in the ordinary logic of terms covalence and equality are necessary conditions for logical equivalence. But covalence and equality are not necessary conditions for the equivalence of propositional statements. Other discrepancies are equally serious. For example, we noted that 'some A is B /every A is B' is not valid in ordinary term logic, but its analogue, 'p and

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q /p::::~q', is valid in statement logic. We note also that the pair of statements 'some A is A; some nonA is nonA' can both be true in ordinary term logic but their propositional analogue in statement logic is the pair 'p and p' and 'not p and not p', which can't be jointly true. But we now see that all differences between term and statement logic can be explained once we recognize that propositional statements are unique in having a singleton domain as their universe of discourse so that all propositional terms denote one and the same world. A logic of terms for a singleton domain has just the properties that distinguish statement logic from ordinary term logic. In a context where all statements apply to only one thing, 'every A is B' does follow from 'some A is B' and 'some A is A' is incompatible with 'some nonA is nonA'. Interpreting the propositional statements of statement logic by treating the statement letters in the formulas '+p+q' and '-p+q' as propositional term letters does not violate any of the laws of a term logic whose statements have a universe of discourse that has exactly one member. Thus, while the move from 'some A is B' to 'every A is B' is prohibited, this isn't the case where the terms are UDTs. Thus from 'some pis q' to 'every p is q' is permitted. For if it is true that something (viz., The World) is both p and q then it is true that everything is both p and q. And if every thing is both p and q then everything is either not p or q. This explains why '+p+q' entails '-p+q'. For, where there is only one world to consider, 'some p-word is a q-world' does entail 'every p-world is a q world'. The singularity of propositional statements also accounts for their divergence from the law of equivalence governing general statements. According to that law two statements are logically equivalent if and only if they are covalent and equal. But in propositional logic, when two statements are both covalent and equal, that suffices for their equivalence. However, neither of these conditions is necessary. One of the divergent cases is the law equating 'p&p' to 'p v p ', which in categorical form is the equivalence of 'some pis p' to 'every non-p isn't non-p' . For ordinary terms this equivalence between divalent statements is invalid: 'some P is P' is not equivalent to 'every nonP isn't nonP'. The right hand side is equivalent to 'no nonP is nonP'. (Consider 'some ape is an ape', which is true and 'no non-ape is a non-ape' or 'every non-ape is an ape', which are false.) But in propositional logic the divalent equivalence holds. Here again the validity of the propositional equivalence is due to the circumstance that only one thing is being talked about. In a singleton domain of worlds 'some p-world is a p-world' = 'no p-world is a non-p-world', since

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209

where something (the actual world ) is a p-world it follows that nothing fails to be a p-world. Thus the crucial difference between ordinary term logic and propositional 'term' logic is that the latter makes claims about a singular domain and in a singular domain 'something is X' is equivalent to 'everything is X'. This crucial difference gives statement logic its special character. And, once this is understood, we are free to understand statement logic as a special branch of the logic of terms that we have all along been studying. We have shown that the logic of propositional statements can be construed as a special branch of the logic of terms. But having shown this, it still remains true, as a practical matter, that the techniques we have learned for dealing with arguments in statement logic are especially suited to it. In any case, even though '+p+q' has the same form as a statement '+P+Q', which we read 'some P is Q', it would be pointless to change our way of expressing ourselves, by reading, say, 'roses are red and violets are blue' as 'some world in which roses are red is a world in which violets are blue'. We therefore continue to read '+p+q' as 'p and q' and '-p+q' as 'if p then q'. For all intents and purposes, we continue to treat statement logic as an independent logic even as we bear in mind the terminist interpretation that Leibniz adumbrated.

17.

Venn Diagrams for the Singleton Universe of Propositional Logic

Consider the Venn diagram for 'someS is P' where 'S' and 'P' are ordinary terms: Figure 23

s

p

The cross mark inside the SP segment indicates the presence of an SP thing. But what about the other segments? Since no information is given we leave

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this empty~ nothing may or may not be in any one of them. Now consider the Venn diagram for 'some p-world is a q-world' where it is understood that no more than one world is under consideration.

Figure 24

Note that we have shaded everything except the pq area. Note now that an unshaded area represents the presence of a pq-thing (in this case a pq-world). The difference between a regular Venn diagram and a Venn diagram for a singleton universe is this: on a regular diagram we have three possibilities for any given area:

1. it has nothing in it (marked by a cross) 2. it is definitely empty (shaded) 3. it may be empty or not (unmarked and unshaded)

In a Venn diagram for a singleton universe we have only two possibilities: 1. it has something in it (unshaded) 2. it is empty (shaded) There is no third possibility. Consider the Venn diagram for 'p pis aq':

Figure 25

:;:~

q' in its categorical form 'every

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Looking at Figure 24 we note that Figure 25 is already depicted: the p(not q) area is shaded. This graphically shows that 'some pis q' entails 'every pis q' in the singleton universe of discourse of propositional logic.

***************************************************************** Exercises: Construct Venn diagrams for each of the following, showing that the conclusion follows from the premises. l.p=>q

-s q=>s I -p&-s 2. p&q I P => q 3. not p I ifp then q 4. not (p and not q) not q lnotp 5. ifp then q ifq then r I ifp then r

*******************************************************************

8 Modem Predicate Logic

1. Syntax The logical system that we have studied in this book is known as Term Logic or Term/Functor Logic (TFL). TFL parses sentences as 'dyads' made up of two terms and a functor that connects them. Not only sentences, but relational terms, compound terms and compound sentences are given dyadic parsings. Term Logic goes back to Aristotle. In this chapter we shall acquaint ourselves with another logical grammar and another system oflogic known as Predicate Logic or Modern Predicate Logic (MPL). For discussion of the differences between TFL and MPL, the reader may wish to look again at Chapter 2 section 11, where we distinguished two approaches to logical syntax: 'the Term Way' and 'the Predicate Way'. We have so far followed the Term Way. In this chapter we follow the Predicate Way. The grammar of a logical language is called logical syntax. Looking at a sentence, the logician distinguishes between its material elements, which carry its meaningful content, and its formative elements, which determine its form. For example, looking at 'some apes are omnivores' the logician who follows the term way distinguishes between the two material elements 'apes' and 'omnivores', on the one hand, and the functor expression 'some are' that joins these terms to give us a sentence of the form 'some X are Y', on the other. The material elements ofTFL are nouns or noun phrases. By contrast, the material elements of Predicate Logic are verbs or verb phrases that serve as predicates. Let us recapitulate the parsing style of the Term Way. Consider the fact that 'some bird sings' entails 'some singer (singing thing) is a bird'. The terminist represents this inference as '+B+S I +S+B'. In so representing it one first paraphrases 'some birds sing' as 'some bird is a singer'. In so 'regimenting' the sentence, we treat the verb 'sing' in the premise as if it were the predicate verb phrase 'is a singer'. We then use 'singer' as a term that could take either subject or predicate position. In effect, TFL splits the verb

213

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'sing' into a copula, 'is (a)', and a term, the noun or noun phrase 'singer' or 'one who sings'; this term then reappears in the subject position of the conclusion. We similarly regiment 'all birds sing' as 'all birds are singers', again analyzing it as a two term sentence, '- B+S'. In TFL each sentence has two terms and a functor that connects them. The same two term parsing is given to a singular sentence like 'Caruso sings'. Here the (singular) term 'Caruso' and the general term 'singer' are connected by the commutative functor 'some is a' and the sentence transcribes as '+C*+S'. In TFL a singular sentence such as 'Caruso sings' has 'wild' quantity; it is particular, but because its subject term, 'Caruso', denotes uniquely, 'some Caruso sings' entails its corresponding universal, 'every Caruso sings'.

2.

MPL: The Predicate Way

An alternative to the two term way of construing sentences follows the linguist's familiar parsing of sentences into Noun Phrase and Verb Phrase. Consider 'Caruso sings'. The linguist parses this as 'Caruso/sings' with 'Caruso as subject-name and 'sings' as predicate-verb. This style of parsing is adopted by MPL for all singular sentences. Note that in dividing the sentence into the two material parts consisting of 'Caruso' and 'sings', MPL does not further analyze the verb 'sings' as 'is a singer'. Thus in 'Caruso sings' there is no 'connective' expression that ties the verb to the noun. Indeed, that is one very important difference between TFL and MPL; in TFL we always have a formative element, a 'connective' that ties the two material elements. But in MPL the verb 'sings' follows the noun 'Caruso' without benefit of a connective joining one to the other. In the relational sentence 'Paris loves Helen', MPL recognizes the verb 'loves' as a two place predicate. Both 'Paris' and 'Helen' are subjectnames. By contrast, TFL parses 'Paris loves Helen' as consisting of two 'subsentences': 'Paris loves' [+P* 1+L 12] and 'Helen is loved' [+H*2+L 12], both implicit in 'P 1+L 12+H* 2'. Note that 'P* 1+(L 12+H* 2)' transforms into '+P* 1+(+H2+L 12+)' [read 'Paris is what Helen is loved by'], a dyad whose second term is itself a dyad: the sub-sentence '+H* 2+L 12 '.

Modern Predicate Logic

3.

215

General Sentences in MPL

Predicate Logic treats a general sentence like 'some birds sing' as a two verb sentence. The contrast between TFL and MPL can be better appreciated when we look at the different way they approach a simple inference like some birds sing I some singers are bird

In representing this little inference the logician must identify two 'material' (nonformal, 'extra-logical') expressions that appear in converse order in the two sentences. As we said, TFL chooses nouns; MPL chooses verbs. In line with its terminist parsing policy, TFL regiments 'some birds sing' as 'some birds are singers'; the two expressions that convert are the nouns 'singers' and 'bird. MPL takes verbs (predicates) as the interchangeable expressions. A verb may be simple like 'ran' or 'sings' or it may be formed by taking a noun and putting a copula in front of it.' For example, 'is a singer' is a verb formed in this manner. Thus any predicate of the form 'is an X' is a verb. Using verbs as the basic material elements, MPL regiments 'some bird sings' as a sentence containing the two verbs or predicates, 'is a bird' and 'sings'. So analyzed, the inference looks like this: something is such that it is a bird and it sings I so something is such that it sings and it is a bird. Note that the sentence, as parsed in MPL, is now understood to contain pronouns as well as predicates; the word 'something' behaves as the antecedent to the pronoun 'it', which appears as the subject in the two component (singular) sentences 'it is a bird' and 'it sings'. Because the MPL construal of 'some bird sings' is a bit cumbersome, practitioners ofMPL have introduced certain symbols to make it tidier. (MPL is often popularly called 'symbolic logic'.) Let '(Ex)' abbreviate the phrase 'something, x, is such that', let 'Bx' stand for 'xis a bird' and let 'Sx' stand for 'x sings'. Then MPL represents the inference thus: (Ex)(Bx&Sx)

I (Ex)(Sx&Bx) In words:

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An Invitation to Formal Reasoning something x is such that x is a bird and x sings I something x is such that x sings and x is a bird

The expression '(Ex)' is sometimes read as 'there exists an x such that'. Grammatically, this expression serves as the antecedent to the pronoun, 'x', that comes after it in 'x is a bird' and 'x sings'. Logicians call '(Ex)' 'the existential quantifier'. It serves to introduce the pronoun variable 'x', which is said to be 'bound' by the quantifier antecedent. Thus, in MPL pronouns are 'bound variables' and antecedents are 'quantifiers;' in the logical language of MPL, the relation of binding is the relation that an antecedent expression (e.g., the quantifier 'something') has to its pronouns (the variables it binds). That '(Ex)' is related to the variables it binds in the way that an antecedent is related to its pronouns is evident if we reads '(Ex)(Bx&Sx)' thus: there exists a thing [antecedent] such that it (that thing) [pronoun] is a bird and it [pronoun] sings Note (again) that the letter 'B' in 'Bx' does not stand for the noun 'bird'; it is not a term letter but a predicate letter representing the verb 'is a bird'. By contrast, in the terminist formula '+B+S' the letter 'B' is a term letter representing the noun 'bird'.

4.

The Logical Language of MPL

Compare '+S+P' as a way of representing 'some Spaniard is a painter' to '(Ex)(Sx&Px)'. The TFL formula is a straightforward transcription ofthe English sentence. The MPL formula is more complicated. And indeed, the language of MPL is not easy to master. To help us in learning it we shall make use ofthe more natural and familiar (to us) formulas ofTFL. Thus in 'translating' a sentence like 'some boy loves a girl' into MPL we shall first transcribe it as 'B 1+L 12+G2 ' and then use the transcribed formula as a bridge to get to '(Ex)(Bx&(Ey)(Gy&Lxy))', treating the TFL formulas as a bridge to the corresponding formulas of MPL. Before learning how to reckon arguments in MPL, we must learn the logical language ofMPL. The symbolic formulas ofMPL are not algebraic: they contain no plus or minus signs. Thus 'p and q' is not represented as '+p+q' and 'ifp then q is not represented as '-p+q'. Instead, the symbolic ways of representing compound sentences is exclusively used:

Modern Predicate Logic

English form pandq ifp then q p orq p if and only if q notp

Algebraic Form

Symbolic Form

+p+q -p+q +[- p+q]+[- q+p]

p&q p=>q pvq p-q

-p

-p

-(-p)-(-q)

217

The following symbolic expressions are used in forming elementary sentences: English Form X is P SomeS is P Every Sis P

Algebraic Form +X*+P +S+P -S+P

Symbolic Form Px (Ex)(Sx&Px) (x)(Sx => Px)

We use the TFL algebraic formulas as bridges to MPL formulas. The rules for translating any sentence of TFL into a sentence of MPL are called 'TR' (for 'translation rules'). The first of these rules, TRI, tells how to translate particular TFL sentences into MPL. The TR rule for particular sentences is TRI: TRJ:

English TFL MPL some A is B ==> +A 1 +B1 ==> (Ex)(Ax&Bx)

For example, applying TRI to 'some American is a banker' we use the TFL bridge, '+A1+B 1', to get to '(Ex)(Ax&Bx)' as the MPL formula. Note the use of numerical pairing indices in the TFL formula. These are usually omitted in transcribing non-relational sentences since it is obvious in such simple sentences that the two terms of the sentence are the only terms being paired. Nevertheless, in using a TFL sentence as a bridge for translating into MPL, we shall give each term pair (arbitrarily chosen) numerals; these indices are then replaced by bound variables in the corresponding MPL formulas. The expression that follows a quantifier is called the 'matrix'. For example the matrix of the formula '(Ex)(Ax&Bx)' is '(Ax&Bx)'. In translating '+A 1+B 1' as '(Ex)(Ax&Bx)' we treat the numeral as the bound variable pronoun 'x' and we treat '+A 1+B 1' as a conjunction of form '+p+q'. This gives us the matrix formula 'Ax&Bx' or 'it is A and it is B', which needs

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a quantifier antecedent to the pronoun, 'x'. MPL prefixes the conjunctive formula (the matrix) with the quantifier antecedent '(Ex)' or 'something, x, is such that'. There is a corresponding TR for universal sentences in MPL. Consider the universal form 'every A is B', which the terminist transcribes as '- A+B'. MPL gives ita conditional reading, treating it as 'Ax-::JBx'. But now the quantifier antecedent that 'binds' the pronouns is universal. The MPL way of expressing 'every A is a B' is every thing is such that if it is an A then it is a B Using the bound variable 'x' in place of 'it' we have the formula: every thing x is such that: if x is an A then x is a B As we will see quite shortly, a translation rule applied to the 'matrix' gives us (Ax-::JBx). The symbolic abbreviation for 'every thing xis such that' is '(x)'. Thus the MPL formula corresponding to '- A+B' is (x)(Ax -::J Bx) The expression 'every thing xis such that', which is symbolically abbreviated as '(x)' is called 'the universal quantifier'. The universal quantifier in '(x)(Ax -::J Bx)' binds the variables of the component sentences 'Ax' and 'Bx' in the matrix, 'Ax -::J Bx'. The TR for translating 'every A is B' into MPL is:

TR2:

every A is B

==>

-A 1+B1 ==> (x)(Ax

~Ex)

For example, to translate 'every American is a banker' we use the bridge '-A 1+B 1'. Applying TR2, yields '(x)(Ax->Bx)' as the MPL formula. There are three TR rules for translating compound forms. Wherever 'p' and 'q' are sentences, the formula '+p+q' translates as 'p&q'. The rule that takes us from '+p+q' to 'p&q' is 'TR/and', the And rule:

TR!and:

p and q ==> +p+q ==> p&q

A second rule, called, 'TR/if, or the If rule, is used to translate conditional forms. Thus, given 'ifr then s', we first transcribe it as '-r+s' and then, by TR/if, translate it as 'r -::J s' :

Modern Predicate Logic TR!if

219

ifp then q ==> -p+q ==> p:::xJ

A third rule, called 'TRior', tells us how to render 'p or q' into MPL: TRior:

p or q ==>

--p--q

==>

pvq

In applying TRI to a compound form like 'some roses are pink and some are white' we first transcribe it in the usual way +[+R+P]+[+R+W] which, byTR/and, becomes '[+R+P]&[+R+W]'. WenextapplyTRl to each conjunct. However, in applying TRI we must treat each conjunct separately giving each conjunct its own quantifier; we may not bind all the pronoun variables by a single antecedent quantifier. The correct MPL translation is (Ex)(Rx&Px)&(Ex)(Rx&Wx) Although 'x' is used throughout, it has different antecedent quantifiers in the two conjuncts. Another (easier to read) MPL version would use 'y' for the second conjunct: (Ex)(Rx&Px)&(Ey)(Ry&Wy) These two versions are equivalent. But the following formula which binds 'x' by only a single quantifier is not a correct translation: (Ex)((Rx&Px)&(Rx&Wx)) TR/and and TRior apply to compound terms as well as to compound sentences. In translating a sentence like 'some A is either B or C', we first apply TRI to give us '(Ex)(Ax&(B or C)x)' and then apply TRior to the disjunctive expression. The result is '(Ex)(Ax&(B v C)x)'. A later rule {TR4) allows us to represent '(B v C)x' as 'Bx v Cx'.

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An Invitation to Formal Reasoning

****************************************************************** Exercises: Translate into MPL: Example: if every A is B then some C is D 1. - [- A+B)+[+C+D] Algebraic Transcription 2. [- A1+BJ1 => [+C2+D2] TR/if 3. (x)(Ax=>Bx) => (Ey)(Cy&Dy) TR2, TRl i) if any A is B then every C is D ii) no C is D but some C is K iii) only A is B iv) every J is K unless some J is M (hint: If 'ifu't' were a word, 'unless' would be its synonym.) v) some rose is pink and it is white ********************************************************************

5. Singular Sentences in MPL

In TFL singular and general sentences are syntactically alike. In MPL singular sentences are treated as being radically different in form from general sentences. The TFL manner of transcribing sentences whose subjects are proper names or other singular subjects does not differ from the manner in which general sentences are transcribed. Thus 'Caruso is a tenor' transcribes as '+C* 1+T 1' and 'Caruso admired Puccini' as '+C* 1+A 12+P* 2 '. By contrast, where the MPL form of the general sentence 'some Italian is a tenor' is '(Ex)(lx&Tx)', the MPL form for 'Caruso is a tenor' is 'Tc', in which the lower case letter 'c' represents the subject 'Caruso' and the upper case letter represents the predicate 'is a tenor'. Singular sentences that have two singular subjects are given similar treatment. The MPL form for the relational singular sentence 'Caruso admired Puccini' is 'Acp'. This formula has two lower case letters in subject position, respectively representing the singular nouns 'Caruso' and 'Puccini'. The predicate letter 'A' represents the transitive verb 'admires', or the relational

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221

predicate expression 'is an admirer of. Predicates like these take two subjects; we call them 'two place predicates'. Translating sentences containing singular terms into the notation of MPL is straightforward and we do not need to use a TFL formula as a bridge. The translation rule is

TR3 S*isP S* is R toP* xis P xis R toy

==> ==> ==> ==>

Ps Rsp Px Rxy

Applying TR3 to 'Tom loves Ella' [T* 1+L 12+E*21 we get 'Lte'. (Note that we simply replace the numerals of the TFL formula, 'L 12 ' by the names of the lover and the beloved. Sentences with three subjects have a three place predicate. For example, the MPL formula for 'Italy ceded Fiume to Yugoslavia' is 'Cify', in which 'C' is a three place predicate relating Italy as the country that ceded, Fiume as the city that was ceded and Yugoslavia as the country to whom Fiume was ceded by Italy. A fmal translation rule takes compound terms into MPL formulas. The MPL formula for 'some gentleman and scholar is a farmer' is '(Ex)((Gx&Sx)&Fx)'. Its TFL formula is '+<+G+S> 1+F 1'. In getting from the TFL formula to the MPL translation, we first apply TR1 and then apply TR/and to '<+G+S>' to get '(G&S)'. This gives us '(Ex)(x&Fx)'. A new rule, TR4, allows us to distribute a variable inward, treating a form like 'x' as '(Gx&Sx)'.

TR4:

<+A+B>x ==> x ==> Ax&Bx <--A--B>x ==> x ==> AxvBx We get the MPL translation of 'some gentleman and scholar is a farmer' by the following steps: 1. 2. 3. 4.

+<+G+S>I+FI (Ex)(<+G+S>x+Fx) (Ex)(x&Fx) (Ex)((Gx&Sx)&Fx)

TFL 1, TR1 2, TR/and 3, TR4

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6. How the Logical Syntax of MPL is 'Ontologically Explicit' The logical grammar ofMPL is the standard logical grammar taught today. One reason for this is that formulas written in the language of MPL are explicit about what makes them true; each MPL formula tells us exactly what sort of things have to be present in the world or absent from it for the formula to be true. Consider again the TFL and MPL renditions of a sentence like 'some Spaniards are painters'. According to TFL this is to be rendered as '+S+P'. According to MPL this is to be rendered as '(Ex)(Sx&Px)'. The TFL formula is simpler and closer to the original English. However, MPL has a special virtue that has recommended it to most contemporary logicians. In the words of one the foremost exponents of MPL in this century: The grammar that we logicians tendentiously call standard is. a grammar designed with no other thought than to facilitate the tracing of truth conditions. And a very good thought this is. 0'-/.V.O. Quine, Philosophy ofLogic, Prentice Hall, 1970, pp.35- 36) That '(Ex)(Sx&Px)' has very explicit truth conditions is evident when we consider that for 'some Spaniard is a painter' to be true the following conditions must hold: a Spaniard must exist, a painter must exist, the Spaniard in question must be the painter in question. All these conditions for the truth of 'some Spaniard is a painter' are expressly present in the MPL formula, which says 'there exists a thing such that it is a Spaniard and it is {also) a painter'. The condition that a Spaniard exists is there. The condition that a painter exists is there. And the condition that the Spaniard in question is the painter in question is explicit in the use of a single pronominal subject for the two predicates 'is a Spaniard' and 'is a painter', both of these predicates being said to hold of that subject. More generally, an MPL formula is always explicit about what things in the world must exist or fail to exist in order for the statement to be true. This is even more clearly seen in more complicated sentences like 'some boy loves no girl', whose MPL form is '(Ex)(Bx& -(Ey)(Gy&Lxy))' which says 'there exists an individual, x, such that x is a boy and there exists no individual, y, such that y is a girl and x loves y'. Here the existence of a boy and the nonexistence of any girl that is loved by that boy are the truth conditions of the sentence, and both ofthese truth conditions are explicitly expressed in the MPL formula. Quine is right in suggesting that one of the main reasons for the deserved popularity ofMPL lies in the 'ontological' explicitness of its

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logical grammar: any MPL statement as much as says what in the world there must be or fail to be in order for the statement to be true. Nevertheless, the virtue of explicitness is more than offset by the artificial complexity of the MPL formulas and by their 'distance' from the sentences of an ordinary language. The complexity is evident in the way it treats even the simplest sentences. Consider 'every horse is an animal', which TFL transcribes as '-H+A' but which MPL translates as '(x)(Hx =>Ax)'. The TFL formula is akin to a stenographic transcription (letters for terms, '- ' for 'every', '+'for 'is'). But the MPL formula paraphrases 'every A is B' as something like 'any individual is such that if it is a horse then it is an animal'. Such a paraphrase is aptly termed a 'translation' because it introduces novel syntactical elements not found in the original English sentence 'every horse is an animal'. In the first place, it treats 'every horse is an animal' as a pronominalization in which the quantifier expression 'any individual' serves as antecedent to the pronoun 'it'. The original sentence, of course, has no pronouns. In the second place, MPL introduces the sentential connective form 'if ... then' which connects two component 'sub-sentences', 'it is a horse' and 'it is an animal'. Again, the original sentence looks to be simple and not compound. The distance between the vernacular English sentence and the MPL formula is even greater for a relational sentence like 'some sailor is giving every child a toy'. TFL transcribes this in a stenographic way as '+S 1+G 123 -C2+T3 '. By contrast, MPL creatively paraphrases it, or 'translates' it, as 'there exists an individual, x, such that x is sailor and for any individual, y, ify is a child then there exists an individual, z, such that z is a toy and xis giving y to z'. The symbolic formula for this is (Ex)(Sx&((y)(Cy => (Ez)(Tz&Gxyz))) Another example of an MPL rendering of a relational sentence is every owner of a cow owns a barn which MPL understands to say 'for every x if something, y, is such that y is a cow and x owns y then something, z, is such that z is a barn and x owns z'. Symbolically: (x)((Ey)(Cy&Oxy) => ((Ez)(Bz&Oxz)))

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Formulas like these are examples of 'multiple quantification'; which is to say, they are sentences with more than one quantifier expression, each of which binds a different pronominal variable. It is often not easy to see how to give MPL renderings of English relational sentences that involve multiple quantification. How, for example, is one to translate 'some boy envies every owner of a dog'? Or 'some one who owns a barn does not own any animal'? The language of MPL is the standard language of modern logic. But it is an artificial language, so it is not easy to learn. However, anyone who has mastered the language of TFL can easily learn how to translate any TFL sentence into MPL. Translation from English into MPL may be made simple by using TFL transcriptions as bridging formulas. With the aid of the TFL bridging formulas, and using the TR rules we have just learned, the translation of multiply quantified relational sentences can be done in a fairly straightforward mechanical way. The TR rules that get us from 'some/every A is B' to the corresponding MPL formulas apply to dyads ofthe form '+X+Y'. But any sentence of TFL can be written as a sentence that is either itself a dyad or analyzable into dyads.

7. Dyadic Normal Forms

The formulas of TFL that are especially useful for translating into MPL are what we call 'Dyadic Normal Forms' (DNF). The DNF of a sentence is a normal form in which compound and relational terms are dyads. We here review the method of transforming any standard TFL sentence into DNF first introduced in section 11 of Chapter 4. A dyad is an expression of the form '±X±Y'. For example, a compound term such as 'gentleman and scholar', which transcribes as an expression ofthe form '<+X+Y>', is a dyad. Any sentence is a dyad in which the first part '±X' is the subject and the second part '±Y' is the predicate. A sentence is in DNF when its structure is fully dyadic, so that every expression in it other than its terms is a dyad ofthe form '±X±Y'. As often as not, a given sentence is not initially given to us in dyadic form. Thus in 'some boy is teaching some girl'[==> '+B 1+(T 12+G2) 1' ], the expression 'T12+G2 ' has the subject 'a girl' on the right. To rewrite it as a proper dyad we commute the phrase 'T 12+G2 ', putting the subject term, 'G2 ', to the left. In effect, we rewrite the sentence as '+B 1+(+G2+T 12) 1'. In this formula each subject phrase has its predicate to the right and the whole formula is said to be in dyadic normal form. The expression '+G2+T 12 ' is a 'sub-sentence' that may be

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understood to say 'some girl is taught (by)'. Generally, when we put a relational sentence into DNF, we place each subject phrase to the left of its own predicate. Let us take the sailor sentence, 'some sailor is giving every child a toy', as another example. Its transcription is '+S 1+(G 123 -C2+T3)'. This has three subject expressions: 'some sailor', 'every child' and 'some toy'. The first of these subjects, '+S 1', is already to the left, but the second and third subjects need to be relocated to the left of their own predicates. The resulting DNF formula is '+S 1+(-C 2+(+T3+G 123))'. Here the expression that follows the subject '+S 1' is a predicate that says of some sailor that he gives every child a toy. '-C2 ' has its predicate; this says of every child that it gets a toy. And '+T3' too has its own predicate, which says of a toy that it is given. Each subject is now in its own dyad and the whole sentence is in 'dyadic normal form'. The subject-predicate pairings may be made more explicit by giving each term pair its own numerical co- indexing:

The rules of translation that take us from TFL forms to MPL forms apply to dyads. In applying these rules to relational statements, we first reformulate the TFL transcriptions in DNF and then apply each rule in a stepwise fashion to the dyadic forms of the TFL formula. Consider how we should use the DNF of the sailor sentence to get to the MPL formula: l.+St+( -C2+(+T3+G 123)) 2.(Ex)(Sx& (-C 2+(+T3+Gx23 ))

DNF TRl

Note that we began with the outermost dyad that includes all the other dyads. In it we replaced the numerical index that pairs the terms 'sailor' and 'giver' by the pronoun 'x', which now serv~s as common subject to the predicates 'is a sailor' and 'gives' in the formula 'he is a sailor and he gives'. Moving inward, we proceed by applying rule TR2 to the next dyad, '- C2+( +T 3+Gx23 ) ', this time replacing '2' by the bound variable pronoun 'y': 3. (Ex)(Sx&((y)(Cy => (+T3+Gxy3))

TR2

Finally, we again apply TRl to the remaining (innermost) dyad, '+T3+Gxy3'.

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An Invitation to Formal Reasoning 4. (Ex)((y)(Cy => (Ez)(Tz&Gxyz)))

TRI

The MPL translation is now complete. Let us see how the rules can be applied to give us the MPL translation of 'every owner of cow is an owner of a bam'. This first transcribes as '-(0 12+C 2)+(0 13+B3)', but we need it in DNF so we put each subject to the left of its predicate:

1. -(+C2+0 12)+(+B3+0 13) 2. (x)((+C 2+0x2) => (+B3+0x3)) TR2 3. (x)((Ey)((Cy&Oxy) => ((Ez)(Bz&Oxz))

DNF TRI (twice)

Sometimes one relational phrase is embedded in another. An example is 'some adolescent is a cousin of a murderer of a painter' which transcribes as '+A 1+(C1 2+(M23+P3))'. DNF TRI (thrice)

1. +AI+(+(+P3+M23)+C12)) 2. (Ex)(Ax&(Ey)((Ez)Pz&Myz&Cxy)

Some sentences have one singular subject and one general subject. An example is 'Tom is memorizing every poem', whose TFL transcription is '+T*1+(M12- P2)'. DNF TR3

1. +T*1+(-P2+M12)

2.

(-P 2+M~)

(Note that in applying TR3 we eliminate 'T* 1' and elsewhere replace the codenoting numerical index '1' by the name 't' .) We next apply TR2 to the formula that remains:

3. (y)(Py => Mty)

TR2

The MPL formula, 3, may be read thus: anything, y, is such that if y is a poem then Tom is memorizing y or, alternatively as for anything, y, if y is a poem then Tom is memorizing y

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and, more colloquially, as if anything is a poem, Tom is memorizing it Consider 'some boy envies everyone who owns a dog', whose TFL transcription is '+B 1+(E 12- (0 23+03))'. 1. +Bt+(-(+D3+023)+Et2)) 2. (Ex)(Bx&(y)((Ez)(Dz&Oyz)

=>

Exy)))

DNF l,TRl, TR2, TRl

(After a while, one gets to collapsing several steps into one.)

8. Translating Pronominalizations The general form of a pronominalization is ' ... some s ...the s'. In the natural languages, words such as 'it', 'him(self)', 'she', stand in for a pronominal subject 'the S (in question)'. Thus we do not normally say 'some barber shaves the barber in question' but 'some barber shaves himself. Most pronominalizations are paraphrases of sentences that contain no pronouns. A pronominalization such as 'some A is a B; it is also C' is merely a loquacious way of saying 'some A and B is a C'. Similarly, 'if any A is a B then it is a C' is reducible to 'every A and B (thing, person) is a C', which again is not a pronominalization. (See the rules for pronominal expansions in section 8 of Chapter 6 above.) The style ofMPL is to pronominalize. A sentence like 'some ape is hungry' is not naturally construed as a pronominalization. But the logical syntax of MPL construes any general sentence, i.e., any sentence containing 'some' or 'every', as a pronominalization. Thus 'some ape is hungry' becomes 'something is such that it is an ape and it is hungry'. And while 'every boy loves some girl' is not a pronominalization in natural language, its MPL rendering has two pronouns: 'everything is such that if it is a boy then there exists someone such that it is a girl and the boy (he) loves the girl (her)'. By contrast, TFL transcribes a natural language sentence of the form 'some A is B' or 'every A is R to some B' without treating it as pronominalizations. For example, '- B1+L 12+G2' is a pronoun-free formula. The bridging rules that take us from TFL to MPL are thus designed to take us from a nonpronominal formula like 'some A and B is C' [ > '+<+A+B>+C'] to the

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corresponding quantified pronominalization formula '(Ex)(Ax&Bx&Cx)' of MPL. Some sentences are natural pronominalizations and TFL transcribes them with proterms. Oddly enough, when a TFL formula is itself a pronominalization, it is not really suitable as a bridge to MPL. For example, the transcription of'some A is a Band it is a C' is '+[+A' 1+BJJ+[+A\ +C 1]'. But this formula is not as good a bridge to the MPL formula as '+<+A+B>+C' which is pronoun- free. For that reason, in using the formulas of TFL as bridges to MPL, it is best to confine oneself to pronoun-free formulas wherever possible. We noted earlier that a pronominalizaion like 'some A is B; it is C' can be understood as a paraphrase of a pronoun-free formula 'some A and B is a C'. But certain English pronominalizations are not merely stylistic paraphrases ofnonpronominal sentences. Examples are 'some barber shaves himself' and 'some student eloped with his aunt'. These sentences contain 'reflexive pronouns' and they cannot be naturally rephrased in a way that eliminates the pronouns. In transcribing such pronominalizations, TFL uses super-scripted proterms. (See Chapter 4, section 8.) Thus, '+B' 1+S 12 +B' 2' transcribes 'some barber shaves himself' and '+S' 1+E 12+(A23 + S ' 3)' transcribes 'some student eloped with his aunt'. Note that in these formulas the common superscript signifies the common reference to the same barber again, the same student again. But the numerical indices of the proterms differ and we cannot automatically apply TRl and TR2 to the formulas by replacing the numbers by bound variables. The MPL version of'some barber shaves himself' is 'someone is such that it is a barber and it shaves it(self)'. Symbolically this is '(Ex)(Bx&Sxx)'. Suppose we tried to get this MPL formula by applying TRl to '+B' 1+S 12+B' 2', the transcription of 'some barber shaves himself. 1. +B' I+{+B' 2+S12) 2. (Ex)(B'x & (+B' 2 +Sx2)) 3. (Ex)(B'x&((Ey)(B'y & Sxy)

DNF TRl TRl

Arriving at 3 we can go no further. The reason is clear: the TFL formula has two numerical indices, which renders it unsuitable as a bridge to the MPL formula which has only one. If we are to apply our bridging rules to formulas that contain proterms, we shall need to modify the formulas in a way that renders them suitable as bridges to the MPL translation. As it stands, a formula like '+B' 1+{S 12+B' 2)' cannot serve as a bridge formula for purpose of

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translation to MPL. So we must modify it in a systematic way to serve that purpose.

9. Preparing the TFL Bridge We modify a TFL pronominalization such as +B' 1+(+B' 2+S 12) (the DNF form of 'some barber shaves himself) by applying to it our rule for internal pronoun elimination (IPE) [see section 17, chapter 4, and section 8, chapter 6]. Recall that this rule allows us to remove subsequent proterms and, in so doing, eliminate superscripts. Thus, our DNF formula for 'some barber shaves himself, viz., '+B' 1+(+B' 2+S 12)' becomes, by IPE, '+B 1+S 11 '. Applying TR1 to this formula we get '(Ex)Bx&Sxx)'. To translate 'some student eloped with his aunt' we begin with its transcription '+S' 1+(E 12+(A23 +S' 3))', and put that formula into DNF. 1. +S' 1+(+(+S\+A23 )+E 12) 2. +SI+(+A21+E12) 3. (Ex)(Sx&(Ey)(Ayx&Exy))

DNF IPE TR1 (twice)

which we read as 'there is an x such that x is a student and there is a y such that y is an aunt ofx and x eloped withy'

******************************************************************** Exercises: Translate the following sentences into MPL. 1. every A is Band C 2. everyone who kidnaps a son of a millionaire is a criminal 3. no mother hates her baby 4. some A or B is D but not E 5. every father of a bride mistrusts her groom

********************************************************************

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10. Identity in MPL Consider how MPL treats 'some author is a Missourian' as opposed to a singular sentence like 'Twain is a Missourian'. In MPL 'Twain is a Missourian' is translated as 'Mt'. By contrast, 'some author is Missourian' is a general sentence that translates as '(Ex)(Ax&Mx)'. TFL marks the proper name, but gives both the same treatment: Twain is a Missourian -> +T*+M some author is a Missourian => +A+M

An identity sentence in English is like any other singular sentence except for having singular terms in both subject and predicate positions (see section 23 of Chapter 3). And again, sentences affirming identity are not given special treatment in TFL. Consider 'Twain is Clemens', which TFL transcribes as '+T*+C*', in which 'T*' is the subject term and 'C*' is the predicate term. Apart from the fact that the proper names are starred to indicate that they are uniquely denoting terms, there is nothing special about this transcription. The form of 'Twain is Clemens' [=> '+T*+C*'] is the same as the form of 'some humorist is a Missourian' [=> '+H+M']. By contrast, MPL treats all proper names as subject terms and so it represents 'Twain' and 'Clemens' by lower case letters. But if both 't' and 'c' are subjects, they must both be attached to a two place predicate. What is the predicate? The answer given by MPL is that 'Twain is Clemens' should really read 'Twain is identical with Clemens'. More generally, in a statement of identity one would read 'is' as 'is identical with'. In translating 'Twain is Clemens' we first read it as 'Twain is identical with Clemens'. Letting 'I' represent the two place predicate 'is identical with', our translation is 'Itc'. It is also customary to use '=' for the identity predicate, in which case the translation is 't=c'. Thus, given an English sentence like 'Twain is Clemens', MPL treats it as a dyadic sentence ofthe form 'X*= Y*'. By contrast, TFL does not introduce a relation of identity. For TFL an identity sentence is simply one that has uniquely denoting terms in both subject and predicate positions. Thus 'Twain is Clemens' is 'monadic'. But since it is singular it has wild quantity. To put it bluntly; TFL does not recognize identity as any kind of relation. Even a sentence like '2 3 = 8' is read monadically as a particular sentence '(some) 23 is 8'. MPL, however, sees a sign of relation in 'Twain is Clemens'.

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Suppose A affirms that Twain is Clemens and B denies it. MPL represents their respective statements thus: A: Itc B: -ltc The denial of identity comes up when I say 'Twain is funnier than everyone else'. Anyone who asserts this is saying in effect that Twain is funnier than everyone who is not (identical with) Twain. MPL translates this as (x)(- Ixt => Ftx) i.e., 'for anyone x, ifx is not identical with Twain, then Twain is funnier than ' X.

To translate a sentence affirming identity or difference into MPL we need first to rephrase it in a way that explicitly brings in a two place predicate 'is identical with', 'is the same as' or 'is not identical with' or 'is other than'. Only then can we proceed to apply the rules of translation. Thus, given the sentence 'Twain is funnier than anyone else', we should first render it as 'Twain is funnier than everyone not identical with Twain'. 1. +T* 1+F 12 -((-I23)+T*3)

2. +T* 1+( -(+T*3+(- I23))+F12)

DNF

Eliminating 'T*' and replacing all subscripts pairing with 'T*' by 't' we get: 3. -(-12t)+F2t 4. (x)(- Ixt => Fxt)

TR2

This example also illustrates the general point that the bridge between TFL and MPL is more useful for sentences containing general terms than it is for sentences containing proper names or other uniquely denoting terms. The syntactical distance between the formulas of TFL and MPL is greatest when the sentences involved have proper names or other uniquely denoting subjects expressions (e.g., 'the King of France', Smith's eldest daughter'). In many cases it simply does not pay to use TFL as a bridge for translating singular statements (including identities) into MPL. Thus, given a simple singular sentence like 'Twain is an author', we should directly move to 'At'. (Why bother to get to it via '+T*+A'?)

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Sometimes the identity predicate comes between two pronouns or between a pronoun and a proper name. An example is there is only one divine being We understand this to say that some being is divine and no being other than that being is divine. TFL renders this as +[+B'+D]+[- (+(- B')+D)] There is no natural way to eliminate the pronouns. So we don't bother to use the formula as a bridge. MPL renders the sentence as (Ex)(Dx&(y)(- lyx

:J

-

Dy))

i.e., 'some being is divine and every being not identical with it isn't divine'.

******************************************************************** Exercises: 1. Translate 'nothing differs from itself into the language ofMPL. 2. Translate 'only Crick understands' into the language of MPL. (Hint: construe the sentence as 'Crick understands and no one else understands'.) 3. Translate into MPL: none except Teddy is loyal to Sally (Hint: no one not identical to Teddy)

******************************************************************** 11. Logical Reckoning in MPL Having learned how to translate into MPL language we will now learn some MPL techniques for evaluating arguments and drawing conclusions from premises. Consider the valid syllogistic argument 'every ape is hairy; some denizens of Madagascar are apes; so some denizens of Madagascar are hairy'. To show that it is valid in TFL we could transcribe it, deny its conclusion and derive a contradiction:

Modern Predicate Logic

1. -A+H 2. +D+A 3. -(+D+H) 4. -D+(-H) 5. +A-H 6. +H-H

233

premise premise negation of conclusion 3,PEQ 4,2, DDO 5,1, DDO

This evaluates the argument as valid. For we have shown that denying the conclusion leads to contradiction. (Another method could be used: show that 1, 2, and 3 give you a P/Z conjunction (see section 5 of Chapter 5).) A similar technique may be used in MPL. First we should represent the premises and the denied conclusion in MPL thus: 1. (x)(Ax :::> Hx) 2. (Ex)(Dx&Ax) 3. - (Ex)(Dx&Hx)

premise premise negation of conclusion

But now we stop: we lack MPL rules analogous to the TFL rules such as PEQ and DDO that would enable us to proceed to derive a contradiction. In the next two sections we present the MPL rules that are used in evaluating arguments like the above.

12. Transformation Rules Rules of one important type are called Rules of Transformation (also called Rules of Substitution). A rule of transformation permits us to transform a formula into an equivalent formula thereby permitting us to replace the former by the latter. One such rule allows us to change a formula that has an initial sign of negation into one that has no such sign. For example, given the TFL formula '-(+T+A)' [read: nothing is an A], we clear the negation sign by driving it inward algebraically by 'obversion' to give us '-T+(-A)' [everything is a nonA]. The analogous move for MPL is - (Ex)(Ax) = (x)- (Ax) Note that this transformation involves three changes: 1. we change the outermost sign

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An Invitation to Formal Reasoning 2. we change the quantifier 3. we change the sign of the Matrix

In MPL, the law that corresponds to TFL obversion is called 'The Law of Quantifier Interchange' (QI). The following transformations are justified by the law:

Law of Quantifier Interchange (QI): -(x)( .. x...) = (Ex) -( .. x .. .) -(Ex)( .. x... ) = (x) -( .. x.. .) (x)( .. x.. .) = -(Ex) -( .. x ... ) (Ex)( .. x.. .) = -(x) -( .. x.. .) Applying QI we may make the following simple inferences: not everything is created I something is un-created -(x)(Cx) I (Ex)-Cx nothing is created I everything is not created - (Ex)Cx I (x)- Cx everything is created I not: something is not created (x)Cx I -(Ex)-Cx something is created I not: everything is not created. (Ex)Cx I - (x)- Cx A number of transformation laws of Statement Logic that deal with negation and the connectives ' & ', 'v', and '::J' figure prominently in reckoning with the formulas ofMPL. The following rules of transformation are known as DeMorgan's Laws:

DeMorgan 'sLaws (DML): Symbolic form -(p&q) = ( -p) v ( -q) -(p v q) = (-p)&(-q) p&q = -((-p) v (-q)) p v q = -((-p)&(-q))

Algebraic Form -(+p+q) = -p-q -(--p--q) = +(-p)+(-q) +p+q = -(-p-q) --p--q = -(+(-p)+(-q))

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Any conditional formula may be replaced by a disjunction. We call this the Conditional/Disjunction equivalence. Conditional/Disjunction (C/D): p ~ q = -p v q -p+q

=

- -( -p)- -q

Another important rule oftransformation is the Law of Double Negation: Double Negation (DN): p = --p

The following transformation laws are called Iteration Iteration (IT): p= p&p p= pvp

13. Rules of Inference Rules oflnference license the steps in a derivation leading to a conclusion from given premises. The following rules of inference are much used in justifying steps in a derivation. Modus Ponens (MP):

p p

~q

lq Example: if there is smoke, there is fire there is smoke so, there is fire Modus To/lens (MT): p ~ q

-q I -p

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Example: if there is smoke, there is fire there's no fire hence there's no smoke

Simplification (SIMPL): p&q lp Example: roses are red and violets are blue hence roses are red

Conjunction (Conj): p q lp&q Example: roses are red violets are blue hence roses are red and violets are blue

Addition (Add): p lpvq Example: Tony is home hence Tony is home or roses are red Some times an inference may be justified by simply appealing to a transformation rule. The transformation laws are then treated as rules of inference. For example, suppose we wished to show that 'not every A is B' entails 'some A is not B'. The MPL translation of 'not every A is B' is the premise of the following example. 1. - (x)(Ax => Bx)

prenuse

Modern Predicate Logic 2. 3. 4. 5.

(Ex)- (Ax ::l Bx) (Ex)-(-Ax v Bx) (Ex)((- (-Ax))&(- Bx)) (Ex)(Ax&(-Bx))

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1, QI 2, C/D 3,DML 4,DN

The last line is MPL for 'some A is not B'.

14. Literal Formulas Literal formulas are simple sentences consisting ofa predicate and one or more singular subjects. The following sentences are examples ofliteral formulas: Caruso sings Plato admired Socrates x loves y xisnnullng Sally gave Tommy a cookie

Sc Aps Lxy

Rx Gstc

If a sentence is literal, so is its negation. For example, since 'Gstc' is a literal sentences, so is '- Gstc'. (Literal sentences that are positive are sometimes called 'atomic' sentences. Negative literals are not atomic.) We shall need four more rules before we get down to reckoning with the formulas ofMPL. The rules in question are called Rules ofInstantiation. A rule of instantiation permits us to replace a pronoun by an arbitrary name. For example, given the sentence 'someone is such that he is a Spani.ard and he is proud' we may decide to call that 'someone' 'Abe' and then say: 'Abe is a Spaniard and Abe is proud'. The formal rule justifying this in MPL is called 'existential instantiation' and looks like this:

Existential Instantiation (El): (Ex)( .. x.. .)

/ ... a... EI tells us that if we are given any step in a derivation that is existentially quantified we may remove its quantifier and substitute a name for the variables that it binds. Thus we may be given the premise 'some student is Pakistani', whose MPL form is '(Ex)(Sx&Px)'. If this sentence is true there must exist something (someone)--call it (he or she) 'a' --that is S and P. If so, 'Sa&Pa'

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is true. Again, this move eliminates the quantifier '(Ex)' and replaces the bound variable by an arbitrary name, giving us a formula that is free of quantifiers and variables. EI may be performed only once. For we have no right to assume that more than one student is Pakistani. Thus we cannot arbitrarily choose a second name, say 'Bob' and also infer 'Bob is a student and Bob is Pakistani'. Moreover, once 'a' has been introduced as a name via EI, it cannot be used in an another existential instantiation. Thus suppose we are now told that 'some student is a billionaire'. This translates as '(Ex)(Sx&Bx)'. Applying EI, we can derive 'Sb&Bb' by deciding to call the student in question 'b'. But we cannot instantiate '(Ex)(Sx&Bx)' by again using 'a' as the name for the billionaire student since that name has already been used for the student who is Pakistani. We have no grounds for assuming that they are the same person. A second rule, called 'universal instantiation', enables us to eliminate the universal quantifier by instantiation.

Universal Instantiation (UI): (x)(. .. x ...)

/ ... a... UI says that, given a universally quantified step in a derivation, we may remove its quantifier and replace the variables it binds by any name we choose. Thus, given the formula '(x)(Sx : : > Px)' as the translation of 'every Spaniard is proud', we may instantiate to 'Sa::::> Pa', since ifthe premise is true for any x, it is true for any arbitrary person, 'a'. For example, if it is true that every Spaniard is proud, then it is true that if Abe is a Spaniard, then Abe is proud. Universal instantiation is unrestricted; we can instantiate as many times as we please. For if it is true that every Spaniard is proud then it is true that if Bob is a Spaniard, then Bob is proud, and also true that if Zelda is a Spaniard then Zelda is proud, and so on for any name one cares to put in for the universally bound variable. The reverse of EI also holds. Given the premise 'Abe is a Spaniard and Abe is proud' we may deduce the conclusion 'someone is such that he is a Spaniard and he is proud', thereby replacing the name 'Abe' by pronouns. Because we move from a singular to a general sentence, the rule permitting this is called 'existential generalization':

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Existential Generalization (EG):

... a... I (Ex)( .. x .. .)

Applying EG to the premise' Adam was created' we may conclude 'something was created'. Thus: Ca I (Ex)(Cx) Finally, if we have a formula ' ... a .. .' in which 'a' appears as an arbitrary name so that ' ... b ... ' would also be true and ' ... c .. .' would also be true, etc., then we may 'generalize universally' to '(x)( ... x ... )': Universal Generalization (UG):

... a... I (x)(. .. x .. .) For example, suppose we chose 'Jupiter' as an arbitrary name to fill the gap in' ... was created'. Assume we could just as well have chosen any other name to fill that gap and that the resulting sentence would be taken as true. In that case we could have series of sentences of the form ' ... was created', all of which are assumed to be true, and we have the right to generalize to 'everything was created'. Thus, by applying UG, we have the inference 'Cj I (x)(Cx)'. As an example ofhow instantiation and generalization may figure in logical reckoning in MPL, consider how we may derive a conclusion from 'every A is B' and 'something is an A': 1. 2. 3. 4. 5. 6.

(x)(Ax => Bx) (Ex)(Ax) Aa Aa => Ba Ba (Ex)(Bx)

prenuse premise

2, EI 1, UI 3,4,MP 5,EG

We have derived the conclusion 'Something is aB'. Using 'T' for 'thing', the same conclusion may be quickly derived in TFL:

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1. -A+B 2. +T+A 3.+T+B

prenuse prenuse 1+2, DDO

We offer now a summary of important rules for reckoning in MPL: 1. Rules dealing with negation: DN: p= --p DML: -(p&q) = -p v -q -(p v q) = (-p)&(-q) C/D: -(p&-q)=p=>q=-pvq 2. Rules of iteration: CI: p =p&p DI: p = p vp 3. Rules dealing with quantifiers:

QI: -(Ex) = (x)-(x) =(Ex)(Ex) = -(x)(x) =-(Ex)4. Rules of instantiation:

EI: (Ex)( ... x ... ) / ... a .. . UI: (x)( ... x ... ) / ... a .. . 5. Rules of generalization:

EG: ... a ... I (Ex)( ... x ... ) UG: ... a ... I (x) ( ... x ... )

15. Reckoning in MPL Consider the valid argument 'every Ape is furry, some denizens of Madagascar are apes, so, some denizens of Madagascar are furry'. To prove it is valid in TFL we might use a reductio proof, denying the conclusion and showing that conjoining the denial with the premises is inconsistent by P/Z. But a direct proof could also be given:

Modern Predicate Logic 1. -A+F 2. +D+A 3. +D+F

241

premise premise 1+2, DDO

We now show how to do a direct proof in MPL: 1. (x)(Ax => Fx) 2. (Ex)(Dx&Ax) 3. Da&Aa 4. Aa=> Fa 5.Aa 6.Fa 7.Da 8. Da&Fa 9. (Ex)(Dx&Fx)

premise premise 2, EI 1, UI 3, SIMPL

4,5,MP 3, SIMPL 6, 7, Conj 8,EG

Step 9 is the conclusion 'some denizen ofMadagascar is furry'.

16. Canonical Normal Forms (CNF)

It is often useful to transform the sentences we are reckoning with into a 'standard' or 'normal' form. Any sentence ofMPL can, by transformation, be reduced to an equivalent sentence that has the following two characteristics: 1. Except for quantifiers, predicate letters and bound variables, the formula contains no signs other than '- ',' & ' and 'v'. 2. '-'appears only in literal sentences. Any sentence that has these two characteristics is said to be in 'Canonical Normal Form' (CNF). If a sentence is not in CNF it is always possible to transform it into one that is. We want to learn how to transform sentences into CNF because CNF sentences are especially suited for logical reckoning in MPL. As an example of a CNF transformation we use QI, C/D, DML, and DN to tum '- (x)(Ax => Bx)' into the CNF formula '(Ex)(Ax&(- Bx)' by the following series of steps. 1. - (x)(Ax => Bx) 2. (Ex)- (Ax => Bx)

Given 1, Ql

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An Invitation to Formal Reasoning 3. (Ex)-(-Ax v Bx) 4. (Ex)(--Ax&-Bx) 5. (Ex)(Ax&- Bx)

2,CD 3,DML 4,DN

Line 5 is a CNF formula. One technique for getting a CNF formula for an English sentence is to transcribe into a TFL 'bridge' formula and then to clear the minus signs from the bridge formula before translating it into MPL. Consider, for example, 'no boy hates every girl', whose TFL formula is '-(+B 1+H 12 -G2)'. If we applied TRI and TR2 to the DNF of this formula the result would be '-(Ex)(Bx&(y)(Gy => Hxy))'. To get this into CNF we should have to apply QI and DML. But suppose we had first cleared the external minus sign from the bridging formula. We should then have'- B 1-(H12 -G2)', from which we would again clear an external minus sign, giving us '- B 1+((- H 12)+G2)'. The DNF ofthis formula is '-B 1+(+G2+(-H 12))', which we use as a bridge and which translates into '(x)(Bx => (Ey)(Gy&- Hxy))'. Now this formula contains only literal components, but it just falls short of being in CNF since it contains the conditional sign'=>'. However, we may apply C/D to give us the CNF: (x)(- Bx v (Ey)(Gy&- Hxy)) The moral of this example is that the more we can simplify in TFL before we translate into MPL, the better off we are when it comes to doing logic in MPL.

17. Indirect Proofs in MPL Indirect proofs tell us whether a given argument is valid or not. But for that we need to be given a complete argument. In using an indirect method to evaluate an argument for validity, we (1) deny its conclusion, conjoining the denial to the premises, (2) get all conjuncts into CNF, (3) instantiate each conjunct, using EI or UI or both and (4) tree the instantiations and test the tree for inconsistency. An inconsistent result shows that the original argument is valid. Thus, the general method consists of denying the conclusion and then instantiating the counterclaim. This gives us a conjunction that has no quantifiers. We are then able to examine this conjunction in the usual way: by treeing it to see whether we have got a contradiction. Consider again the argument 'every A is F, some Dis A I some Dis F'. Denying the conclusion gives us the following counterclaim of three conjuncts:

Modern Predicate Logic 1. (x)(Ax ::1 Fx) 2. (Ex)(Dx&Ax) 3. -(Ex)(Dx&Fx)

243

premise prenuse negation of conclusion

We now proceed by applying the rules of reckoning: 4. (x)-(Dx&Fx) 5. (x)((-Dx) v (-Fx)) 6. (x)(- Ax v Fx)

3, QI 4,DML 1, C/D

Lines 2, 5 and 6 are all in CNF. We now instantiate them: 7. Da&Aa 8. -Da v -Fa 9.-AavFa

2, EI 5, UI 6, UI

The conjunction of 7, 8 and 9 can be treed: Da Aa

I -Da

\ -Fa

I -Aa

\ Fa

The tree has all of its paths 'closed', revealing that 7, 8 and 9 are jointly contradictory. Since 7, 8 and 9 follow from the counterclaim, this shows that the counterclaim of the original argument is inconsistent. The argument itself is therefore valid. We could save ourselves several steps if we clear the minus from our TFL formulas before using them as the bridge to MPL. 1. -A+F 2. +D+A 3. -(+D+F) (== -0+(-F))

=> => =>

(x)(Ax ::1 Fx) (Ex)(Dx&Ax) (x)(Dx ::1 - Fx)

Applying C/D to 1 and 3 gives us three CNF formulas ready for instantiation and treeing:

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An Invitation to Formal Reasoning 1.* (x)(-Ax v Fx) 2. (Ex)(Dx&Ax) 3.* (x)(-Dx v -Fx)

A word of advice is in order. It is better (and is often necessary) to use EI before using UI. This is due to the restriction that applies to EI but not to UI. Instantiating in the order 2, 1*, 3* gives us the same closed tree as before. The general technique for indirectly proving the validity of an argument in MPL is this: 1. Form the counterclaim of the argument in TFL. Clear all external minus signs from the formulas and translate into MPL. 2. Transform all sentences to CNF. 3. Instantiate--beginning with EI and then going on to UI. 4. Tree the resulting sentences. 5. Now check to see whether the tree closes. If it does close, the argument is valid. An invalid argument will not give a closed tree.

18. Relational MPL Arguments We now apply the technique to some more complicated arguments. Consider how we could show that the following argument is valid. every colt is a horse I every owner of a colt is an owner of a horse Denying the conclusion we have

1. -C2+H2 2. - (- (0 12+C2)+(012+H2))

prenuse negation of conclusion

1 and 2 jointly form the counterclaim of the argument. Before proceeding to translate into MPL clear the external minus sign of 2: 2, peq

Modern Predicate Logic

245

A further simplification of 3 is available:

3,PEQ We now put 4 into DNF: 4,DNF We now translate 5 and 1. 6. (Ex)(+C 2+0x2)&(- H 2+0x2) 7. (Ex)((Ey)(Cy&Oxy)&(y)(Hy => (-Oxy))) 8. (y)(Cy => Hy)

5, TR1 6, TR1, TR2 1, TR2

Using C/D we get 8 and 7 into CNF: 9. (y)(-Cy v Hy) 10. (Ex)((Ey)(Cy&Oxy)&(y)(-Hyv -Oxy))

8, C/D 7,C/D

Instantiating 10 gives us

11. Cb&Oab&(-Hb v -Oab)

10, EI,Ul

Instantiating 9 gives us 9, UI

12. -Cb vHb Treeing 11 and 12 we get: Cb Oab

I -Hb

I -Cb

\ -Oab

\ Hb

The tree for the instantiations closes; we have a contradiction. To show that 'every girl is loved by some boy' follows from 'some boy loves every girl' we counterclaim and get:

246

An Invitation to Formal Reasoning 1. +B 1+(L 12 -G2) 2. - (- G 2+(L 12+B 1)

pretruse negation of conclusion

We then simplify 2 by driving in the minus signs:

2,PEQ

3. +G2+((- L 12 - B 1)) 4. +B 1+(-G2+Ln)) 5. +G2+(- B 1+(- L 12))

1,DNF 3,DNF

4 and 5 are our bridging formulas. We now translate into MPL: 6. 7. 8. 9.

4, TR1, TR2

(Ex)(Bx&(y)(Gy;:, Lxy) (Ey)(Gy&(x)(Bx;:, - Lxy) (Ex)(Bx&(y)(-Gy v Lxy)) (Ey)(Gy)&(x)(- Bx v - Lxy))

5, TR1, TR2 6, C/D 7, C/D

8 and 9 are now in CNF so we instantiate them: 8, EI, UI 9, EI, UI

10. Ba&Gb&(-Gb v Lab) 11. Gb&(- Ba v -Lab) We now tree 10 and 11: Ba Gb

I

\

-Gb

Lab

I -Ba

\ -Lab

All paths close so we have derived a contradiction from the counterclaim. Here is a more complicated argument: anyone who ~nvies anyone who Qwns a §.lave is immoral every (captured) Akkadian was a slave I no Akkadian is owned by anyone that any moral person envies We will use a direct TFL proof to derive the conclusion and then follow this by an indirect MPL proof of validity.

Modern Predicate Logic 1. - (E 12+(0 23 +S 3))+(- M 1) 2. -A3+S3 I -(+A3+(0 23+(+M 1+E12))) 3. - M~-(E 12+(023+S3)) 4. - M 1+(- E 12)- (0 23 +S3) 5. -(023 +S 3)+(-E 12)-M 1 6. -(-((-E 12)-M 1)))+(-023 )-S3 7. -S 3+(-0 23 )-(EI2+MI) 8. - A3+(- (0 23 )- (E 12+M1) 9. -(+A3+0 23 +(E 12+M 1)

247

prenuse premise conclusion 1, peq 3, peq 4, LLC (twice), LLA 5, peq 6, LLC (twice), LLA 2+7, DDO 8, peq

Line 9 is the conclusion we sought to derive. In the indirect MPL proof we shall use steps 1 and 2 and the denial of the conclusion as bridges to the MPL formulas. 1* -(+(+S 3+0 23 )+E 12))+(-M 1) 1, DNF 1** (x)((Ey)((Ez)(Mx&Exy)&Oyz))) 1*, TR2, TR1 2, TR2 2* (z)(Az -> Sz) 3* +A3+(0 23 +(+M 1+Ed negation of conclusion 3*, DNF 3** +A3+(+(+M 1+Ed+023 )) 3*** (Ez)(Az&(Ex)((Ey)(Mx&Exy)&(Oyz))) 3**, TR1 (thrice) We now tree instantiations of3***, 1**and 2*.

4*

Ac Ma Eab Obc

I - Sc

I -Ac

I \ \ - Obc - Eab - Ma

3***, EI

1**, UI

\ Sc

2*, UI

The tree closes, which shows that denying the conclusion entails a contradiction. To show that 'some Greek shaves himself follows from 'every barber shaves himself and 'some barber is a Greek' we deny the conclusion and

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An Invitation to Formal Reasoning

conjoin it with the premises:

1. no Greek shaves himself

negation of conclusion premise prenuse

2. every barber shaves himself 3. some barber is a Greek

To show that these three statements are jointly inconsistent we transcribe them, simplify, translate them, get them into CNF and then instantiate. Having simplified the transcriptions we arrive at:

1. -G'1+(-S12)-G'2 2. - B' I+SI2+B' 2 3. +BI+GI

negation of conclusion premise premise

As a first step we apply IPE to 1 and 2: 1, IPE 2, IPE

4. -GI+(-SII) 5. - BI+SII

After transformation, instantiation and treeing, we have: Ga Ba

I

\

-Ba

Saa

I -Ga

\ -saa

It is instructive to compare how one could do this example in TFL:

1. 2. 3. 4.

- B\+S 12 - B' 2 +BI+GI - BI+SII +GI+S11

prenuse prenuse 1, IPE 2+3, DDO

[Note that an English language sentence like 'some boy envies every owner of a dog' has no pronouns in it. MPL 'translates' it with three pronouns [(Ex)(Bx&(y)((Ez)Dz&Oxy => Exy))], but TFL transcribes it as a proterm free formula [- B+(E-(O+D)]. More often than not, even when a sentence

Modern Predicate Logic

249

explicitly contains pronouns, it can be rephrased and transcribed in a pronoun-free way. For example, 'if any A is B, then it is C' can be rephrased as 'any A that is B is C' and transcribed without proterms as '-<+A+B>+C'. Indeed, most logical reckoning can be done without the pronouns (bound variables) that are an indispensable feature of MPL. This gives TFL a decided advantage over MPL in naturalness and ease of reckoning. Having said this, it should be pointed out that reflexive pronouns are special; in transcribing a sentence like 'some barber shaves himself TFL needs proterms (prior to application of IPE). Though we prefer TFL to MPL, it must be conceded that in the area oflogical reckoning that involves reflexive pronouns, TFL has no special advantage over MPL.]

******************************************************************** Exercises: I. Show that 'Aristotle is not wiser than himself follows from 'no one is wiser than Aristotle'. (hint: use UI). 2. Using MPL, derive 'every senator admires a fool' from 'every senator admires himself and 'if any(one) is a senator, then he is a fool'.

******************************************************************* 19. Identity Arguments in MPL

As we have seen above (section 10 of this Chapter), statements such as 'Mark Twain is Sam Clemens' and 'the square root of twenty-five is five' are often called identities. An identity statement in TFL is simply a statement both of whose terms are uniquely denoting terms. Again, an identity is a statement of the form 'some X* is Y*'. Arguments that contain an identity premise or conclusion require special treatment in MPL. The reason for this is that in MPL a sentence like 'Twain is Clemens' is construed as a dyadic statement whose 'is' is understood to mean 'is identical with'- -a two place relational term that ties 'Twain' and 'Clemens' as two subjects in the way 'loves' ties 'Paris' and 'Helen' in 'Paris loves Helen'. Having introduced identity as a relation, MPL must also introduce special laws governing it. The following 'Laws of Identity' are often used when dealing with identity arguments.

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An Invitation to Formal Reasoning

Laws ofldentity: LI.l ifx=y, theny=x (I'he law ofsymmetry) LI.2 ifx=y and y=z, then x=z (I'he law oftransitivity) Ll.3 x=x (I'he law ofrejlexivity) Ll. 4 ifPx and x=y, then Py (Leibniz's Law)

LI.l asserts that identity is a symmetrical relation. According to LI.l, if Twain is Clemens, then Clemens is Twain. LI.2 asserts that identity is a transitive relation, so that if Clemens is Twain and Twain is the author ofLife on the Mississippi, then Clemens is the author of Life on the Mississippi. The third law asserts that identity is reflexive. According to LI.3, we can always assert that 'Twain is Twain', 'Socrates is Socrates' and so on. LI.4 allows us to replace 'x' by 'y' once we know that x=y. Thus, given that Twain is a humorist, and given that Twain is Clemens, we can conclude that Clemens· is a humorist. This law is often called Leibniz's Law. Thus, in MPL one or more special laws governing the identity relation must be added as 'axioms of identity'. In TFL identity statements are not relational and the analogous laws can be proved by ordinary means. For example, according to Leibniz's Law, 'Y* is P' follows from 'X* is P' and 'X* is Y*'. Since identities are singular statements, they have wild quantity. By assigning universal quantity to 'X* is P' and particular quantity to 'X* is Y' we can prove the fourth law syllogistically. Thus, an argument like the Twain example comes out as a valid syllogism: +T*+C* -T*+H /+C*+H

Twain is Clemens Twain is a humorist I Clemens is a humorist

We now offer a pair of identity arguments to illustrate the way that MPL deals with them by applying the laws of identity. (i) Show that 'Mark is an author' follows from the premises 'Clemens is Twain', 'Twain is an author' and 'Clemens is Mark'. 1. c=t 2. At 3. c=m

4. m=c

premise premtse premise 3, symmetry

Modern Predicate Logic

5. m=t 6. t=m 7. Am

251

4, 1, transitivity 5, symmetry 2, 6, L.L. (Leibniz's Law)

(ii) Show that 'Twain is Clemens' follows from 'Twain is funnier than anyone else' and 'Twain isn't funnier than Clemens'. One way is to deny the conclusion and show that this gives us a contradiction:

1. (x)(-(x=t) 2. -Ftc

:::>

Ftx)

3. -(t=c) 4. (x)(x=t) v Ftx)

premise prenuse negation of conclusion 1, C/D,DN

4, U.I.

5. c=t v Ftc 6. t=c v Ftc

5, symmetry

The tree for 2,3, and 6 is: -Ftc -(t=c) I \ t=c Ftc The tree closes. Denying the conclusion leads to contradiction so the argument is valid.

******************************************************************** Exercises: 1. Show that 'the Evening Star is Hesperus' follows from 'Hesperus is Phosphorus', 'Phosphorus is the Morning Star' and 'the Evening Star is the Morning Star'. (in MPL and then TFL) 2. Given the premises 'only Clemens knows' and 'Twain knows', prove that Twain is Clemens (in MPL and then TFL).

*******************************************************************

Rules, Laws and Principles

TFL Logical Law of Commutation (LLC): +(+X+Y)

= +(+Y+X)

Law ofObversion (Obv): -(+/-X+/- Y) = +(-/+X-/+Y) Principle of Equivalence (PEQ): Two statements are logically equivalent if and only if they are covalent and equal. Logical Law of Association (LLA): +X+<+Y+Z> = +<+X+Y>+Z Law of Simplification (LS): Any well-formed dyad may be detached from an expression whose terms are connected by a binary commutative and functor). associative functor (viz., the'+ ... +' Internal

Pronoun Elimination (IPE): If P' m... P' n is an internal pronominalization, remove P 'n and replace any remaining occurrence of n by m·

The REGAL Principle (REGAL): A syllogism is valid if and only if its mood is regular and it 'adds up'. Principle of Validity (PV): An argument, A, is valid if and only if its counterclaim, C(A), is inconsistent. Principle of Transitivity: If being B characterizes every A and being C characterizes every B, then being C characterizes every A. P/Z Criteria: A syllogistic conjunction is inconsistent if and only if it is itself canonically inconsistent or else is equivalent to a canonically inconsistent conjunction.

253

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An Invitation to Formal Reasoning

Laws of Identity: Law of Symmetry: A* is B* I B* is A* Law of Transitivity: A* is B* and B* is C* I A* is C* Law ofReflexivity: A* is A* Dictum de Omni (DDO): (Aristotelian version): Whatever characterizes every X characterizes any X. (Generalized version): What is true of every X is true of whatever is an X. (Host/Donor version): E(M), E*(- M) I E(E*) Reassignment Rule (R- A): The numerical index on a universally distributed occurrence of a term may be replaced by any other numeral. ... -Ti··· I ... -Ti··· (R- A2): The superscript of any distributed proterm may be changed to form another pronominalization . ... Ti ... I ... Ti ... Pronominalization (Pl): some X is Y I so it (the X in question) is Y +X'+Y I +X'+Y (Pla): +X+Y I +X'+Y' Rule ofWild Quantity (WQ), (P2): +X'+Y I- X'+Y Pronominal Expansions: some A is B = something is an A and it is a B +A+B = +[+T'+A]+[+T'+B] some A is B = an A exists (is a thing) and it is a B +A+B = +[+A'+T]+[+A'+B] every A is B = if any thing is an A, it is a B -A+B = -[+T'+A]+[+T'+B] every A is B = if an A exists, it is a B -A+B = -[+A'+T]+[+A'+B]

Rules, Laws and Principles

255

STATEMENT LOGIC principle of equivalence (peq): Two compound statements are equivalent if (but not 'only if) they are covalent and equal. Modus Ponens (MP): Given any premise ofthe form E(ifp), where p has negative occurrence, and another premise M(p ), where p has positive occurrence, the conclusion M(E) follows. Conjunction (Conj): From any two premises the conjunction ofthose premises follows. Simplification (Simpl): From any conjunction either conjunct follows. Conjunctive Iteration (CI): Any statement is equivalent to its conjunction with itself. Disjunctive Iteration (DI): Any statement is equivalent to its disjunction with itself. Disjunctive Addition (DA): From any premise the disjunction of that premise and any other statement follows. Law of 'And/Or' Distribution (AOD): (p or p) and (q orr)= (p and q) or (p and r) +[- -p- -p]+[- -q-- r] = -- [+p+q]-- [+p+r] Law of 'Or/And' Distribution (OAD): (p and p) or (q and r) = (p or q) and (p orr) --[+p+p]--[+q+r] = +[--p--q]+[--p--r]

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An Invitation to Formal Reasoning

MPL Translation Rules (TFL => MPL): TRI: some A is B -> +A1+B 1 => (Ex)(Ax&Bx) every A is B > - A1+B 1 => (x)(Ax ;:, Bx) TR2: TR/and: p and q => +p+q -> p&q TR/if: ifp then q > -p+q > p;:, q TRior: porq > --p--q > pvq TR3: S* is P > Ps S* is R toP* > Rsp xis P > Px x is R to y --> Rxy TR4: <+A+B>x > x => Ax&Bx <--A--B> >x >Ax vBx Law of Quantifier Interchange (QI): -(x)( ... x ... ) = (Ex)-( ... x ... ) -(Ex)(... x ... ) = (x)-( ... x ... ) (x)( ... x ... ) = -(Ex)-(... x ... ) (Ex)( ... x ... ) = -(x)-( ... x ... ) DeMorgan's Laws (DML): -(p&q) = -p v -q -(p v q) = -p&-q p&q = -(-p v -q) p v q = -(-p&-q) Conditional/Disjunction (C/D): p;:, q Double Negation (DN): p = - - p Modus Ponens (MP): p ;:, q, p I q Modus Tolens (MT): p;:, q, -q I -p Simplification (Simpl): p&q I p Conjunction (Conj): p, q I p&q

-(+p+q) = -p-q -(--p--q) = +(-p)+(-q) +p+q = -(-p-q) --p--q = -(+(-p)+(-q))

= -p v q

-p+q = --(-p)--q

Rules, Laws and Principles

257

Addition (Add): p I p v q Existential Instantiation (EI): (Ex)( ... x ... ) I ...a ... Universal Instantiation (UI): (x)( ... x ... ) I ... a ... Existential Generalization (EG): ... a ... I (Ex)( ... x ... ) Universal Generalization (UG): given that 'a' is an arbitrary name so that ' ... b .. .', ' ... c .. .', etc. would also be true: ... a ... I (x)( ... x ... ) Laws of !~entity: Law of Symmetry (LI.l ): if x=y then y=x Law ofTransitivity (L1.2): ifx=y and y=z, the x=z Law of Reflexivity (LI.3): x=x Leibniz's Law (L1.4): ifPx and x=y, then Py

ANote on Further Reading

Having mastered An Invitation to Formal Reasoning, the reader may wish to pursue one or more of the topics discussed or alluded to here. Term Functor Logic (TFL) was given its present formulation by Fred Sommers in a series of articles published in philosophical journals. You might want to look at 'The Calculus of Terms', reprinted in The New Syllogistic, George Englebretsen, editor, New York, Peter Lang Publ., 1987. Sommers provided an extensive introduction to and philosophical foundation for TFL in his The Logic of Natura/Language, Oxford, ClarendonPress, 1982. David Kelley's textbook, The Art ofReasoning, 2nd expanded edition, New York, W.W. Norton & Co., 1994, includes a chapter presenting the main elements ofTFL. Englebretsen' s Something to Reckon With, Ottawa, University of Ottawa Press, 1996, offers a summary account of term logic as well as an extensive survey of the historical antecedents of Sommers' logic. The history of all logic, not just TFL, begins of course with Aristotle. The bold reader may wish to consult one of the many editions and translations of his Prior Analytics, the book in which Aristotle invented syllogistic logic. A very comprehensive history of logic is William and Martha Kneale's The Development ofLogic, Oxford, Clarendon Press, 1962. The reader interested in exploring further Leibniz's contributions to logic might examine some ofhis essays in Leibniz: Logical Papers, G.H.R. Parkinson, editor and translator, Oxford, Clarendon Press, 1966. While Aristotle was the first logician, and Leibniz was perhaps the greatest traditional logician, Lewis Carroll (pen name of the mathematician Charles L. Dodgson) was almost the last- but certainly the most entertaining-traditional logician. His books, Symbolic Logic and The Game of Logic, first published a hundred years ago, have served as a source of delightful but challenging logic problems for generations of teachers and writers, including ourselves. Both books have been printed in a single volume by Dover Publ., Inc., New York, 1958. Modem Predicate Logic (MPL) has had a very brief (just over a century), but very rich and eventful, history. There are a very large number

259

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An Invitation to Formal Reasoning

of textbooks available now presenting the standard version of that logic. I. Copi's Introduction to Logic, 9th edition, New York, Macmillian, 1994, and S. Barker's Elements ofLogic, 5th edition, New York, McGraw-Hill, 1989, are two of the more popular of these. W.V.O. Quine's Philosophy ofLogic, Englewood Cliffs, N.J., Prentice Hall, 1970, is an excellent, brief account of the philosophy behind MPL. The system of diagrams used here was invented by the mathematician John Venn in the nineteenth century. An interesting survey of systems of logic diagrams is found in Martin Gardner's Logic Machines and Diagrams, Chicago, University of Chicago Press, 1982. A system of diagrams quite different from Venn's, but built with Sommers' version ofTFL in mind, has been developed by Englebretsen. A full presentation of it is found in his Line Diagrams for Logic: Drawing Conclusion, Lewiston, N.Y., Mellen, 1998.