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, ((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.
100
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) > < < > > 11 12 < < > > = < > > a < < > > > < > > 13
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