A Purely Topological Form of Non-Aristotelian Logic Carl G. Hempel The Journal of Symbolic Logic, Vol. 2, No. 3. (Sep., 1937), pp. 97-112. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28193709%292%3A3%3C97%3AAPTFON%3E2.0.CO%3B2-J The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.
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TEE J O ~ N A LOF SYMBOLICLOGIC Volume 2. Number 3, September 1937
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC'
CARL G. HEMPEL
1. The problem. The aim of the following considerations is to introduce a new type of non-Aristotelian logic by generalizing the truth-table methods so far employed for establishing non-Aristotelian sentential calculi. We shall expound the intended generalization by applying it to the particular set of pluriOne will remark that the points valued systems introduced by J. Euka~iewicz.~ of view illustrated by this example may serve to generalize quite analogously any other plurivalued systems, such as those originated by E. L. P ~ s t by , ~ H. Reichenbach,' and by others. 2. J. Lukasiewicz's plurivalued systems of sentential logic. First of all, we consider briefly the structure of the Lukasiewicz systems themselves. As to the symbolic notation in which to represent those systems, we make the following agreements: For representing the expressions of the (two- or plurivalued) calculus of sentences, we make use of the Principia mathematics symbolism; however, we employ brackets instead of dots. We call the small italic letters ( ( ~ ~ 1(dQ",, wr", . . . sentential variables or elementary sentences, and employ the term "sentence" as a general designation of both elementary sentences and the composites made up of elementary sentences and connective 1 , < < 21 > 7, << = 1 , SymbolS( ( ( - 1 1 1 1 . 9 9 - ). Now, the different possible sentences (or, properly speaking, the different possible shapes of sentences, such as "pH, "p v q", "-p.(q v r)", etc.) are the objects to which truth-values are ascribed; and just as in every other case one wants a designation for an object in order to be able to speak of it, we want now a system of designations for the sentences with which we are going to deal in our truth-table considerations. We might choose as designation of a sentence the <(
<(
Received October 8, 1936. The basic ideas of this paper have been set forth by the author at the Second International Congress for the Unity of Science, held in June 1936in Copenhagen. A short synopsis of that communication appeared in the Proceedings of the Congress, published in Erkenntnis, vol. 6. See (a) J. Lukasiewiu and A. Tarski, Unkrsuckungen &a dcn A~~csagcnkalkJI, Comptes rendus des sdances de la Soc. des Sciences el des Lettres de Vcnsovie,d.iii, vol. 23 (1930), pp. 1-21 ;(b) J. Lukasiewiu, PkilosopkischcBwmerkungcn w mckrzuertigcn Systemen dcs Aussagenkalkds, ibid. pp. 51-77. A very dear and illustrative account of these systems and of the matrix-method in general is given in C. I. Lewis, Alicrndive syskmc oflogic, The monist, 1932, pp. 481-507, and in C. I. Lewis and C. H. Laneford, Symbdic Logic, ch. vii, Trulh-due s y s k m and the matrix &hod. a E. L. Post, Introduction to a general theory of Jcmmlary proposilim. American journal of mathematics, vol. 43 (1921), pp. 163-185. H. Reichenbach (a) Wakrscheinlickk&logik, Sitmrigsberichte der Preussischen A M - e der Wissenschajfen,phys.-math.Klasse, 1932,pp. 476-488;(b) Wahrscheinlichkeitslehte, A. W. Sijthoff, Leiden 1935.
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CARL G. HEMPEL
sentence itself, put between quotation marks, as we have done so far. But this would be too cumbersome for our purposes. Therefore we agree t o designate (1) (a) small italic letters, representing sentential variables, by the corresponding capital letters, (b) the connective symbols of negation, disjunction, conjunction, implication, and equivalence by the following small italic letters: "n", "a", l
((p.q) v ( p . - q ) f l 1 will be "pi ( (PkQ)a(PknQ) ". The symbols introduced by the stipulations (1) do not belong to the sentential calculus, but to its syntax langzrage, with respect to which the sententid calculzcs is the object Zang~age.~ Thus, whilst the first of the two symbol-series just considered is a sentence of the sentential calculus, the second one does not belong to that calculus, but to its syntax language, and it is not a sentence itself, but the designation of a sentence, i.e. of one of the objects to which truth-values are ascribed. Now we come to consider the general truth-tables by which Eukasiewicz determines his plurivalued systems. According to traditional Aristotelian logic, every proposition is either true or false, and correspondingly the truth-tables of the usual sentential calculus (e.g. that of Principia mathematics) admit only of two truth-values, 0 and 1, for every elementary sentence; Eukasiewicz generalizes this principle, which he calls the Zweiwertigkeitssatz16by introducing n different truth-values: 0, l / n - 1, 2/n- 1, . . . , 12-2/n- 1, 1. I n terms of these values, he first erects truth-tables for negation and implication. The general n-valued matrix of negation may be characterized by the following formula, which belongs to the syntax language of the sentential calculus:
I
Here, "X" is a (free) variable, the constant values of which are sentence designations (such as "P" or "PeQ"), and "Tr( . - . ) " is a syntactical functor which indicates the truth-value of the expression designated by its argument. The general implication matrix is determined by the following stipulation:
Disjunction, conjunction, and equivalence are defined by negation and implication; the definitions imply the following, matrix-stipulations for "a", "k", "e": 5 Lukasiewicz and Tarski, in their papers (a) and (h), cited in footnote 2, differentiate very strictly between the expressions of the sentential calculus and their syntactical (metalogical) designations; see also the explicit remark in paper (a), page 2. As to the reasonsfor adhering to that distinction, see R. Camap, Logische Syntax der Sprache, section 42: ~Votii~endigkeit der Unterscheidzhng zx~ischeneinem Ausdruck zind seiner Bezeichnung. 6 Loc. cit., footnote 2, (b), p. 63.
A P U R E L Y TOPOLOGICAL FORM O F NOK-ARISTOTELIAX LOGIC
99
I n the case n = 2 , the Lukasiewicz matrices are identical with those of the ordinary sentential calculus; this shows that the stipulations (2) furnish a genuine generalization of two-valued logic. By means of the truth-tables thus established, the valid formulae, or, as we shall say, the tautologies of n-valued lcgic, are defined as the sentences to which the n-valued tables ascribe the truth-value 1 for all possible truth-values of the elementary sentences of which they are composed. The class of all the tautologies determined by the n-valued tables is called L,. Thus, Lt is the well known class of the tautologies of two-valued sentential logic. It is identical with the class of those sentences which follow from the axiom system erected in Principia mathemati~a.~ Every tautology of L,(n? 2) obviously is also a tautology of L;but the converse does not hold. Thus, for example, the formulae representing the classical "Aristotelian" principles of exbelong cluded middle-PanP-and of excluded contradiction--n(PknP)-both to L,but not to any L, with n > 2 . Moreover, the systems L, are all different from one another; hence Lukasiewicz's generalized matrix method gives rise to an infinite set of non-Aristotelian systems of logic. 3. T h e matrices of a purely topological logic of sentences. The construction of the Lukasiewicz systems is based, as we have seen, on assuming a finite or infinite scale of different numerical truth-values. Here the question arises whether it is not possible to erect a sentential logic on the weaker basis of a purely topological serial order of sentences, in which the falser of two sentences precedes the truer one and equally true sentences stand a t the same place-without the introduction of any numerical truth-value.s We shall try to answer this question by developing what might be called a purely topological logic of sentence^.^ This logic represents a generalization of classical sentential logic which lies still beyond
' See the proof given by Post in his article cited in footnote 3. This way of putting the problem was suggested by certain formally similar questions which the author is investigating in collaboration with Dr. P. Oppenheim, and which concern the logical significance of purely topological order for empirical science, in particular for the introduction of "graduable" concepts possessing no numerical degrees. (See Hempel and Oppenheim, Der Typusbegriff im Lichfz der neuen Logik, Sijthoff, Leiden 1936.) In this context, Dr. Oppenheim raised the question whether the concept of truth could not also be considered as such a graduable concept; this induced the author to develop the present consideratisns, in which that concept of truth which is employed in connection with the matrix method is supposed to be topologically graduable. The expression "topological logic" has already been used by Reichenbach, but in quite a different sense. Reichenbach (see note 4, (a) pp. 10-11, (b) p. 383) calls a plurivalued logic metrical if all its truth-values can immediately be interpreted as probabilities in the sense of relative frequencies; and he designates a certain three-valued logic topological in order to express the fact that ~t does not fulfill this condition.
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CARL G . HEMPEL
the plurivalued systems, and its construction leads to some considerations which may be of interest for the theory of deductive systems in general and for the theory of logic in particular. We will now first of all formulate precisely our initial assumptions. We consider a finite or infinite set of sentential constants "PI", "pz", . . . , "ql", tsQ2", . . . "rl", g'r22),. . . . (In case the set of the sentential constants under consideration has a greater cardinal number than go, one must introduce still further subscripts. For the considerations of this paper, however, the cardinal number of that set is irrelevant.) Each of these small italic letters with subscript may be looked upon as an abbreviation of a certain constant sentence such as "2X2=4" or "2)<2=5." Let B be the class of all the sentential constants under consideration and of the expressions which can be derived from them by negation, implication, disjunction, conjunction, and equivalence. As designation for a sentential constant occurring in B, we choose the corresponding capital italic letter with the same subscript; thus, for instance, "P7" will designate "p,". For designating the compound sentences contained in B, we adopt the conventions indicated above for the usual calculus of sentences. Now we suppose that the elements of B are put into a serial order, determined by two two-termed relations: being less true or preceding in truth-order, which we designate by " <", and being just as true or standing a t the same place in truth-order, designated by " = "; according to the serial structure of the order, these relations are supposed to satisfy the following conditions: Standing a t the same place is (3.1) symmetrical and (3.2) transitive; preceding is (3.3) irreflexive and (3.4) transitive, and the order determined by the two relations is connected in the following sense: for any two B-elements X, Y (3.5) if X < Y , then not X = Y ; and (3.6) if not X = Y , then X Y. We shall also characterize these three cases by saying that the topological truthvalue of the ordered pair (X, Y) is <, =, >, respectively. Starting from these basic assumptions, we will now erect our topological logic of sentences by following the procedure of Eukasiewicz as closely as possible in formal and in material respects; viz., we shall establish matrices and determine them in such a manner that they become, so to speak, topological abstracts of Lukasiewicz's general numerical matrices; that is, more precisely: if one interprets the two fundamental relations (<, =) of our topological system as having smaller numerical truth-value and having the same numerical truth-value, respectively, then the content of our topological matrices is to be contained in (is to be a logical consequence of) the five general numerical matrix-determinations (2a)-(2e).
,
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC
101
We begin with the negation matrix. What can be stated of its content in our topological terms is: the truer X, the falser nX; or, more precisely, if X stands before, a t the same place as, or after Y in the truth-order, then n X will stand after, a t the same place as, or before nY in the truth-order, respectively. We sum this up in a "topological matrix of negation." This matrix
shows the essential difference between the numerical and the topological truthtables. We will now express this difference with respect t o the tables of twomembered combinations. Numerical tables answer questions of the following type: Given the exact places of X and Y in the numerical truth-scale, what is the exact place of, say, XaY in the same scale? I n our topological terms, we cannot express the position of one, but only the relative position of two sentences in the truth-order. So, in analogy to considering single sentences as one does in the numerical case, we now consider ordered pairs of B-elements, and the topological truth-tables of two-termed combinations answer the following question: Given the relative position of two pairs (XI, Yl), (X2, Y2) of B-expressions, what is the relative position of XlaYl with respect to XtaY2? And analogously for k, i, and e. The number of lines in each of these truth-tables will therefore be equal to the number of the possible topologically different positions of (XI, YI) with respect t o (Xz, Y2); and this number is equal to that of the different possible arrangements of the four expressions XI, Y1, X2, Yz in truth-order. Each of these arrangements can be characterized by the topological truth-values which are taken by the following six pairs of expressions: (XI, YI), (X2, Y2), (XI, XI), (YI, Yt), (XI, YI), (X2, Yl). Take, say, the arrangement in which Y1 and Yt stand a t the same place, XI before and Xz after them; then the six topological truth-values in question are <, > , < , = , <, > , respectively. In the same way, every other arrangement of four B-sentences determines a certain distribution of the topological truth-values over those six places, but not vice versa: a distribution which has for instance "<" a t the first and fourth and "=" or ">" a t the fifth place is impossible, since preceding is transitive. A combinatorial consideration which takes into account the serial structure of the truth-order shows that only 75 of the 3O formally possible distributions of topological truthvalues over those six places determine a possible arrangement of XI, YI, XS, YI in truth-order. Thus, in general, the topological matrix of every two-membered sentential combination will have 75 lines (corresponding to the nZ lines in an n-valued system), 6 argument-columns (corresponding to 2 in a plurivalued system) and one value-column for every two-termed sentential connection (just as in the plurivalued systems). The first line of the matrices contains the valu* "< " in each of the 6 argumentplaces. What is the corresponding topological truth-value of (XlaYl, XtaYt)? From (2c), one sees that in our case XlaYl= YI and XtaYt= Yt; and as YI< Ys, one has XlaYl<X2aY2. Quite analogously, one finds that XlkYl<XtkY2. For
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CARL G. HEMPEL
implication, one finds from (2b) that in the numerical case, under the conditions of our first line, Tr(XliYl)=Tr(XZiY2)= 1; therefore in our table, XliYl = XziYz. As to equivalence, the numerical stipulations (2e) do not allow us to infer from the argument-indications of our first line the relative position of XleY1 with respect to X2eY2;they rather admit each of the three possible positions. This may be seen from the following numerical esample. Let Tr(Xl), Tr(Yl), Tr(X2), Tr(Y2) be 1/10, 8/10, 7/10, 9/10, respectively; then Tr(XleY1) = 3/10 is smaller than Tr(XzeY2) =8/10; if the four truth-values are 1/10, 8/10, 2/10, 9/10, then Tr(XleY1)=3/10 is equal to Tr(X2eY-J; and for the values 1/10, 5/10, 2/10, 9/10, Tr(XleY1) =6/10 is greater than Tr(X2eYS = 3/10. We put an interrogation mark a t the first place of the value-column of equivalence in order to indicate that the case is left indeterminate in the sense that each of the three topological truth-values is possible. These examples will serve to show how the complete matrices can be established. The result of the construction is represented by the big table, p. 112. I t shows that indeterminate cases occur only for implication (6 cases) and for equivalence (24 cases). How are we to evaluate these topological matrices? In the case of numerical systems, one of the truth-values, as a rule the value 1, is selected, and by means of it the tautologies of n-valued logic are defined as the sentences the matrix of which contains the selected truth-value in every place of its value-column. In our case, one might try to establish an analogous criterion by selecting one of the topological truth-values, say <, and calling a B-sentence-say Pli(Q5anQs)-a tautology of topological logic if the value-column of the topological matrix for (Xli(YlanYl), X2i(YZan Y2)) contains exclusively "<"-symbols. But this criterion would not be fulfilled by any B-expression, since for XI = X2, Y1= Y2 (see the big table) the topological matrix of every formula contains " = " in the value-column. The same reasoning holds in the case of >. .4nd finally, if one selects = instead, the class of tautologies thus defined is no longer empty; it contains, amongst others, Pip, which has the form of a tautology of two- and plurivalued logic; but a t the same time, it contains n(PiP), which has the form of a contradiction in two- and plurivalued logic; the same holds for certain other tautologies of and their negations; therefore it i ~ o u l dnot be suitable, either, to define the tautologies of topological logic by the "="-criterion. Here is the point where the analogy with the construction of plurivalued systems comes to an end. We have been able to establish matrices just as in the numerical case, and to make their content conform to that of the general Lukasiewicz matrices; but for evaluating the tables thus constructed, we have to find another method. 4. Construction of language T. A suitable method of evaluation is suggested by the following consideration. As recent researches have made clear,1° lo See
Carnap, Logische Syntax der Sprache.
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC
103
the logic of a given language cannot be completely characterized by a system of tautologies or logically valid formulae of the language under consideration; it rather is determined by the rules of logical inference which govern the language, and which belong to its syntax language. By those rules, all the logically valid formulae of the language are determined, but not vice versa. Therefore instead of attempting to establish tautologies of topological logic, we will now interpret our topological matrices as a system of inferential principles by constructing a language T in which logical inference is governed by purely topological rules corresponding to the stipulations contained in the topological tables. We determine the language T by formation and transformation rules.ll The formation rules stipulate which kinds of expressions are to be called sentences of T or T-sentences. Now, we are going to give the T-sentences such a form that-putting it first in an intuitive manner-the language T does not enable us to ascertain a single logical or empirical fact, but only to say that a certain fact is less sure, just as sure, or more sure than another one. More exactly: T is to have the character of a syntax language with respect to the sentences of B, but its means of expression are to be so restricted that they only allow the formulation of propositions which affirm either that a certain B-sentence is less true or that it is just as true as another one. This idea is a t the root of the following formation rules: (4.1) A symbol or a symbol-series is called a sentence-member with respect to T, or briefly a T-member, if it has one of the following shapes: a. "P19',"PZ9',lip3,,, . . .
CCQ177, <1Q27?, 11Qa92,
...
l
f
(in other words: if it is one of the capital italic letters "P", "Q", "R", with an integer-sign as subscript. If one supposes the cardinal number of the set B of all sentential constants with which T deals to be greater than go, one has to modify this stipulation by admitting further subscripts (see also above p. 100). But as will be seen, the cardinal number of that set-even its being finite or infinit-is of no importance for the following considerations.) The symbols characterized under a. will also be called elementary T-members. b. "-(U,)"; c. "(Ul)i(U,)"; d. "(Ul)a(Ut)"; e. "(U1)k(U2)"; f. "(Ul)e(Uz)",
where "U1" and "UZ" are T-members.
Let M be the class of all T-members, determined by this recursive definition. (4.2) A symbol-series is called a T-sentence if it has one of the following forms: b. "U1= U2",
a. "U1< Us"; where "U1" and "Uz" are T-members.
A T-member may be considered as a syntactical designation of a B-sentence, Cf. Carnap, loc. cit., pp. 1 ff., pp. 120 ff.
JI
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CARL G. REMPEL
and a T-sentence as a sentence saying that a certain B-sentence U1 is less true than a certain B-sentence U2 (or that U1 is just as true as UZ,as the case may be). However, the definitions (4.1) and (4.2) do not refer to the class B, and therefore the following considerations are independent of that special interpretation. In conformity to the initial considerations of section 4, we express the logic of T in a system of general syntactical propositions (stipulations and consequences of such) on T-members and T-sentences.12 For this purpose, we make use of member-variables "XI", "xz", " ~ a " , . . , llyl", 16y2", llya", . . , the valuedomain of which consists of the designations of T-members. When formulating general propositions by means of these syntactical variables, we designate the following symbols of the language T by themselves: "n", "i", "a", "k", "e", 11 9 9 ( L 9 , ( , ) , '(<", "=". On the basis of these conventions, we can, for example, formulate the following genera1 statement, which is a consequence of (4.1): For any two T-members xl, a, the expression (nxl)a(zlix2) is again a T-member. But the chief purpose of the notation just introduced is to express the transformation rules for T. These rules will be formulated as postulates concerning the class cs of the closed systems of T-sentences. Intuitively put, we call a class a a closed system (or briefly a cs), if for every two T-members zl, 22, the class a contains a t least one of the sentences xl < x 2 , X I = 2 2 , %<XI, and if further the consequences of every set of elements of a are likewise elements of a. In the systematic formal procedure, we shall expressly stipulate by the single transformation rules, what further sentences are to be elements of a cs, if sentences of such and such a form are contained in it; thus every transformation rule will be a postulate concerning cs; and we shall then define the concept "consequence (in T)" by means of "cs", saying that a T-sentence is a consequence of a class a of T-sentences if it is an element of every cs which contains a as a subclass. Designating the consequence relation by "Co", this definition can be put as follows:
-
-
Here, "a" and "8" are variables the value-domain (range) of which consists of the designations of the various possible classes of T-sentences. The consequence relation is transitive in the following particular sense: if a sentence is a consequence of a class a of sentences each of which is a consequence of 8, then it is also a consequence of 8. This follows immediately from the above definition of "Co". For all transformation rules except one, it is more convenient to formulate them, not in terms of the fundamental concept "cs", but by means of "Co".la
* If one considers every T-member as a syntactical designation of a B-element, those general propositions belong to a syntax language of the second order with respect to B, for they are expressed in the syntax language of T. 13 This has also the great advantage that for the formulation of many of the transformation rules and for drawing conclusions from them (which will be done in section 5) we shall be able to
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC
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The transformation rules for T will be taken from the above topological matrices and from the postulates (3.1)-(3.6) concerning the truth-order. For the latter also can serve as a basis of inferences in T; thus, for example, "P1
This formula postulates, in correspondence to (3.1)) that for every two T-members zl and a,the T-sentence 22 = xl is a consequence of the class which contains the T-sentence X I = % as its only element. At first glance, it might seem possible to express (3.1) in T itself by an axiom of the following kind: ( X = Y) (Y=X). However, our formation rules exclude the occurrence of a T-formula of such a form: T contains neither the implication-symbol (only its designation "i" can occur) nor does it contain any variable, and this excludes the possibility of formulating any (general) axiom in T. (3.2) can be put as follows: I n correspondence to (3.3), we introduce two stipulations:
This means: a class containing a T-sentence of the form xl<xl has every T-sentence as its direct consequence ("a" and "x~)'are free variables, and by substituting suitable constant values for them in "22 <xS" and " % = ~ 3 ) ) , one can arrive a t any T-sentence whatsoever). We shall also say: a class containing a T-sentence of the form xl <xl is contravalid.14 The following formula corresponds to (3.4) :
(3.5) can be expressed by the following pair of postulates: (6.5.1) Co(x3 < XP, ( ~ < 1 22, Xi = (6.5.2) CO(S = x4, { X I < a , XI = (Interpretation similar to that of (6.3.1, 6.3.2).) make use of the general syntactical concepts and methods developed by A. Tanki (Fundamentale BegriJe der Methodologie der deduktioen Wissenschjten. I. Momtshefte f. Math. u. Physik, vol. 37, pp. 361-404 and by R. Carnap (see above, footnote 5). I t may be of interest to notice here that the idea of these authors to take "consequence" as the fundamental concept in establishing the syntax of a language leads to certain difficulties in its application to the syntax of T. This has been the reason for our choosing "u" as fundamental concept. l4 The idea of defining an inconsistent class of sentences as a class which has every sentence (of the language under consideration) as its consequence is due to Post (see footnote 3). I t does not refer to the concept of negation and therefore is much more general than the usual definition of inconsistency, which would not be applicable in our case, as the language T does not contain any negation-symbol: the formation rules do not provide the possibility of symbolically negating a T-sentence. Post's idea has been adopted and developed by Tanki and Carnap in their general syntactical researches (see above, footnote 14).
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CARL G. HEMPEL
Finally, we introduce the following postulate as a correlate to (3.6) : (a s CS)3 { [(XI < a ) e a ] v [(XI = xz) s a ] v [ ( a
< XI)s a ] J
This means, that every closed system a contains, for every two T-members x1 and xz, a t least one of the sentences XI <s,xl = xz, xz <XI. This postulate represents a generalization of the Zweiwertigkeitssatz, which stipulates that the truth-value of every sentence is either 0 or 1-tertium non datur-and which, therefore, may also be called the syntactical principle of the excluded middle. (It must be distinguished from the formula "p v -p" of the sentential calculus, which is also often called the principle of .the excluded middle.) The general Eukasiewicz systems are based on replacing this principle by what might be called the n-Wertigkeitssatz or the syntactical principle of the excluded (I$+ l)st (see above, p. 98); intuitively put, this principle affirms that the possible truth values of any sentence are limited to n well determined places in the scale of real numbers. Now (6.6) can be considered as a topological generalization of this principle, because it affirms that of any two sentences the first stands before or after or a t the same place as the second, and consequently every sentence has its place in the truth-order. Thus (6.6) guarantees that no two sentences arc incomparable with respect to their relative truth, and we may therefore call it the principle of excluded incomparability. In a similar way, one can establish a syntactical version of the principle of generalize it to the n-valued case, and excluded contradiction ("-(p.-p)"), then formulate a topological correlate for it which affirms that the three cases al <x2, x1= x2, x2
We come now to the evaluation of the topological matrices. I n principle, we interpret every line of the five matrices as a transformation rule for T ; but it will turn out that the determinations thus resulting can be reduced to a relatively small number of postulates, from which they are all deducible. The first two lines of the negation table furnish the following formulae:
The formula corresponding to the third line follows from (7.1) by interchanging "1" and "2". The matrix of disjunction may be translated line by line as follows:
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC
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But the 75 formulae thus resulting can be replaced by the following three, which correspond to the stipulations (2c):
From these formulae, together with those established under (6), one easily deduces the 75 formulae just mentioned. Quite similarly, the transformation rules resulting from the conjunction matrix can be summed up by the following three formulae, which correspond to (2d) :
As to the implication matrix, its content can be summed up in 12 postulates: C o ( x ~ i y=~ xziyz,
Co(xliy1 = xziyz,
Co(xliy1 = xziyz, XI = yl, xz = ~ 2 ) )
Co(xziyz < xliyl, XI < yl, yz < X Z ) )
Co(xziy2 < xliyl, xl = yl, y2 < xz))
Co(xliy1 = xziyz, yl < XI,y2 < 22, xl = xz, YI = y2))
Co(xliy1 < xziyz, yl < XI,y2 < xz, XI = x2, yl < y2))
Co(xziy2 < xliyl, (yl < XI, y2 < xz, XI = xz, YZ < yl))
Co(xziy2 < xliy~,{ yl < xl, y2 < xz, xl < XZ,yl = yz)
Co(xziyz < x~iyl, yl < XI,y2 < xz, XI < xt, y2 < yl))
Co(xliy1 < xziy2, yl < XI,yz < xz, xz < XI,yl = Y Z ) )
Co(xliy1 < xziyz, YI < XI, y2 '2 xz, xz < XI, yl < YZ]
From these formulae one can deduce 69 theorems, corresponding to the single lines of the implication matrix, the six lines which contain an interrogation mark and hence do not determine any transformation principle excepted. And finally, we can sum up the stipulations of the topological equivalence matrix in the following seven postulates: (11.1) (11.2) (11.3) (11.4) (11.5) (11.6) (11.7)
Co(xley1 < qeyz, ( X I < y1, xz Co(xley1 < xzeyz, $1 < y l , Co(sey2 < qeyl, xl < yl, x2 Co(xley1 = xley2, XI < yl, XI Co(xley1 < xzeyz, 8 < y l , Co(xleyl = xzeyz, XI < yl, zi Co(xley1 = sey2, 21 = yl,%
< y2, XI < xz, yl = < ~ ~ 2 ~< x 1a ,YZ < YI ) < y2, xl = xz, yl < ~ 2 ) )
y2i'
= xz, yl = y2j) = ~ 3)) = y2, xz = YI ) ) = ~2)).
The formulae (6.1)-(11.7) exhaust the transformation rules of the language
T; they determine its principles of logical inference or, briefly, its logic; in this sense, the language T is governed by a purely topological logic. 5. Some theorems of the logic of T. In order to render more explicit the
108
CARL G. HEMPEL
nature of the topological logic thus established, we now consider some characteristic theorems which can be deduced from the above postulates. I n the formulation of some of these theorems, we shall make use of the concept of validity,* calling a T-sentence valid if it is a consequence of the null class of T-sentences. As a consequence of this definition and of (5.1), (5.2), a T-sentence is valid if and only if i t is an element of every closed system. Designating validity by "Vld" and the null class by "A" we state the definition and its consequence: Vld(xi < %) Vld(x1 = 23) V l d ( ~ i< ~ 2 ) Vld(x1 = 12)
~r C O ( X< ~ X*, A)
DI CO(xl = X2, A)
(a)(a c cs (a < x2) e a )
(a) ( a e cs > (XI = a)e a).
Introducing axioms into a formalized system can always be interpreted a s establishing certain transformation rules for that system, namely postulating that certain formulae (the axioms) are consequences of the null class.16 The converse of this remark does not hold; i t may be that the formation rules of a language do not allow one to formulate the intended axioms in the language itself, a s we saw in the case of T (see remark between (6.1) and (6.2)). This shows that the way in which T has been set up can be regarded a s a generalization of the axiomatic method, and that the T-sentences which we shall prove to be valid play a r61e corresponding to that of the consequences of the axioms in an axiomatic system, e.g. to the tautologies in the axiomatized sentential calculus. The first theorem expresses the reflexivity of standing a t the same place in truth-order :
The proof runs as follows. Because of the last of the four formulae just established, (12.1) affirms that every closed system contains, for every constant value of "xl", the T-sentence x l < r l as an element. Now, according to (6.6), every cs contains, for any value of "XI", either r l = x l or xl<xl. But also in the second case, i t contains xl=xl, as follows from (6.3). Similarly, the following theorems can be deduced from (6.1)-(6.6): (12.2) (12.3) (12.4.1) (12.4.2)
Co(x1 < 2.3, { X I < xz, x2 = 2-31) Co(x1 < x3, { X I = XZ,x2 < x3l) C O ( X< ~ ~ 4 {, xi < ~ 2 x2 , <XI]) Co(x.3 = x4, ( 3-1 < 2-2, X* < XI f ).
The two theorems (12.4) express the irreflexivity of preceding in truthorder. Some of these formulae are needed for proving the theorems mentioned above, which correspond to the single lines of the topological matrices. Let us now consider some theorems concerning negation: l6 l8
See Carnap, Logische Syntax der Sprache, p. 126.
See Carnap, loc. cit., pp. 1234.
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC
Co(nnx1 < nnxz, Co(nna = nnxr,
109
XI < xz )
1s = GI).
I n this context, it is interesting to notice that
does not follow from our stipulations (though in the syntax of each of the plurivalued Eukasiewicz systems "Tr(nnX)=Tr(X)"--which might be called the syntactical version of the principle of double negation-holds). The proof will be given in section 6 . We come now to some theorems concerning implication:
Theorems concerning disjunction:
Theorems concerning conjunction:
(15.2), (15.3), (16.2) and (16.3) show, that the commutativity and associativity of disjunction and conjunction are guaranteed also in our topological logic. The same is true for the associative laws:
The following theorems may be considered as topological correlates to the syntactical version of two of the De Morgan rules:
The correlates to the other two De Morgan rules do not follow from our stipulations; indeed, the following formulae can easily be demonstrated: Vld ( ~(nxlknx2) t = nn(xlax2))
Vld(n(nxlanx2) = nn(xlkx2)).
But since "Vld(nnxl = xl)" is not a consequence of our stipulations (as mentioned above), one can not deduce from (18.3), (18.4) the following formulae, which would be the correlates of the other two De Morgan rules:
CARL G. HEMPEL
I n the syntax of the plurivalued systems, the truth-values of negation, implication, disjunction, conjunction, and equivalence are one-many functions of the truth-values of the component sentences. Exactly correspondingly, the following uniqueness theorems can be proved for T:
6. Final remarks. The language T thus established is consistent but incomplete. Here we call a language consistent if not every one of its sentences is valid; and we call it incomplete if not every one of its sentences is either valid or contravalid (a sentence is said to be contravalid if the class which contains the sentence as its only element is contravalid in the sense defined above, p. 105); a sentence which is neither valid nor contravalid will be called indeterminate." Therefore an incomplete language is consistent; and for proving our assertion concerning T it will be sufficient to show that according to our stipulations for T, not every T-sentence is either valid or contravalid. For this purpose, consider the case in which T contains only two elementary members, say "PI" and ((P2".NOWwe define a subclass K of the class of all Tsentences such that K is a cs. We arbitrarily ascribe to "PI" the value V("P,") = O and to "P2"the value V("P2") = 1, and further we attribute to every T-member a value which is exactly the same as that defined by the Eukasiewicz stipulations for the sentence designated by that member. Thus, we have for example, V("PlaP2") = 1, V("PliP2") = 1, V(('P2iP1") = 0, etc. Now, we take as elements of K those and only those T-sentences which fulfill one of the following conditions: (a) the sentence has the form x l < q , and its members are such that V(xl) =0, V(x2) = 1; (b) the sentence has the form xl =a,and the members are such that either V(xl) = V(x2) = O or V(xl) = V(x2)= 1. The class K thus determined fulfills, as is easily verified, all our conditions (6.1)-(11.7) for a cs. But because of our stipulations, the T-sentence "P2". Then also K' is a cs; it is not contravalid, because it does not contain every T-sentence (e.g. not "P1
"
These definitions are taken from Carnap, @ische Syntax dm Sprache. The basic ideas of these general definitions of consistency and completeness are due to Post; they have undergone a further development in the researches of Tarski and of Carnap (see footnote 15).
A PURELY TOPOLOGICAL FORM OF NON-ARISTOTELIAN LOGIC
111
it can be proved generally that every T-sentence of the form xl<- (where xl and xz can be highly complex T-members) is indeterminate with respect to our transformation rules. (Hence every T-sentence that can be proved to be valid must have the form xl =&.) Finally, we give in this connection the proof that every T-sentence of the form nnxl=xl is indeterminate, a fact which was mentioned in connection with the theorems (13). We take the case in which T contains as many elementary members as there are rational numbers in the closed interval <0, 1 >; we will denote them by "P,", where r is rational and 0 S r S 1. Then we introduce the following correlation : V("Pr") = r for every r ;
V(nxl)
=
2
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2
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The V-values of the other member-forms may be defined, as above, in correspondence to (2b)-(2e). Then, as can easily be verified, V(nx1) is a rational number belonging to the interval <0, 1> if the same is true for V(xl); and V(nr1) is smaller, equal to, or greater than V(nx2) if V(xl) is greater, equal to, or smaller than V(XZ),respectively; but V(nnxl) =V(xl) only in the case that V(rl) = 0 or V(x1) = 1. Then we define a class L by stipulating that a T-sentence of the form xl
and maintains all the other definitions given above, then, instead of L, another closed system L' results, which contains every T-sentence of the form nnxl=xl, but not every T-sentence whatsoever. This proves, that the formula ( 1 Vld(nnxl=xl)" is not incompatible with the transformation rules of T , and that no T-sentence of the form nnxl = X I is contravalid, which, together with the result obtained before, shows that every sentence of that form is indeterminate, q.e.d.
CARL G. HEMPEL
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