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A purely topological form of non-Aristotelian logic1

Published online by Cambridge University Press:  12 March 2014

Carl G. Hempel
Carl G. Hempel*
Affiliation:
Bruxelles-Forest, Belgium
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Extract

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 pluri-valued systems introduced by J. Łukasiewicz. One will remark that the points of view illustrated by this example may serve to generalize quite analogously any other plurivalued systems, such as those originated by E. L. Post, by H. Reichenbach, and by others.

2. J. Łukasiewicz's plurivalued systems of sentential logic. First of all, we consider briefly the structure of the Łukasiewicz 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 mathematica symbolism; however, we employ brackets instead of dots. We call the small italic letters “p”, “q”, “r”, … 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 symbols (“~”, “ν” “.”, “⊃” “≡”).

Now, the different possible sentences (or, properly speaking, the different possible shapes of sentences, such as “p”, “p∨q”, “~p.(q∨ 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 3 , September 1937 , pp. 97 - 112
Copyright
Copyright © Association for Symbolic Logic 1937

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Footnotes

1

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 1936 in Copenhagen. A short synopsis of that communication appeared in the Proceedings of the Congress, published in Erkenntnis, vol. 6.

References

2 See (a) Łukasiewicz, J. and Tarski, A., Untersuchungen über den Aussagenkalkül, Comptes rendus des séances de la Soc. des Sciences et des Lettres de Varsovie, cl. iii, vol. 23 (1930), pp. 1–21Google Scholar; (b) J. Łukasiewicz, Philosophische Bermerkungen zu mehrwertigen Systemen des Aussagenkalküls, ibid. pp. 51–77. A very clear and illustrative account of these systems and of the matrix-method in general is given in Lewis, C. I., Alternative systems of logic, The monist, 1932, pp. 481–507Google Scholar, and in C. I. Lewis and C. H. Langford, Symbolic Logic, ch. vii, Truth-value systems and the matrix method.

3 Post, E. L., Introduction to a general theory of elementary propositions. American journal of mathematics, vol. 43 (1921), pp. 163–185CrossRefGoogle Scholar.

4 Reichenbach, H. (a) Wahrscheinlichkeitslogik, Sitzungsberichte der Preussischen Akademie der Wissenschaften, phys.-math.Klasse, 1932, pp. 476–488Google Scholar; (b) Wahrscheinlichkeitslehre, A.W. Sijthoff, Leiden 1935Google Scholar.

5 Łukasiewicz and Tarski, in their papers (a) and (b), 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 reasons for adhering to that distinction, see R. Carnap, Logische Syntax der Sprache, section 42: Notwendigkeit der Unterscheidung zwischen einem Ausdruck und seiner Bezeichnung.

6 Loa cit., footnote 2, (b), p. 63.

7 See the proof given by Post in his article cited in footnote 3.

8 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 Lichte der neuen Logik, Sijthoff, Leiden 1936.Google Scholar) 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 considerations, in which that concept of truth which is employed in connection with the matrix method is supposed to be topologically graduable.

9 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 it does not fulfill this condition.

10 See Camap, Logische Syntax der Sprache.

11 Cf. Carnap, loc. cit., pp. 1 ff., pp. 120 ff.

12 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 make use of the general syntactical concepts and methods developed by Tarski, A. (Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I. Monatshefte f. Math. u. Physik, vol. 37, pp. 361–404CrossRefGoogle Scholar and by R. Carnap (see above, footnote S). It 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 “cs” as fundamental concept.

14 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). It 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 Tarski and Carnap in their general syntactical researches (see above, footnote 14).

15 See Carnap, , Logische Syntax der Sprache, p. 126Google Scholar.

16 See Carnap, loc. cit., pp. 123–4.

17 These definitions are taken from Carnap, Logische Syntax der 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).