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Modalities and Quantification

  • Rudolf Carnap (a1)

The purpose of this article is to give a survey of some results I have found in investigations concerning logical modalities. The results refer: (1) to semantical systems, i.e., symbolic language systems for which semantical rules of interpretation are laid down; (2) to corresponding calculi, i.e., syntactical systems with primitive sentences and a rule of inference; (3) to relations between a semantical system and the corresponding calculus.

The semantical systems to be dealt with are the following: pro positional logic (PL), functional logic (FL), and the corresponding modal systems, viz. modal propositional logic (MPL) and modal functional logic (MFL).

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1 Lewis C. I. and Langford C. H., Symbolic logic, 1932; the systems are developed from those in Lewis's earlier book (1918).

2 I shall hereafter refer to the following publications of mine by the signs in square brackets:

[Syntax] The logical syntax of language, (1934) 1937.

[I] Introduction to semantics, 1942.

[II] Formalization of logic, 1943.

3 I have indicated the parallelism between the modal concept of the necessity of a proposition and the meta-concept of the analyticity of a sentence first in [Syntax] §69 (where, however, ‘analytic’ was still regarded as a syntactical term), and, more clearly, in [I] pp. 91 ff.

4 For this concept, the term ‘tautologous’ is sometimes used; see Quine W. V., Mathematical logic, 1940, p. 50, and [I], p. 240. In [II] D11–30, I have used the term ‘L-true by NTT.’

5 Hilbert D. and Ackermann W., Grundzüge der theoretischen Logik, (1928) 1938.Bernays P., Mathematische Zeitschrift, vol. 25 (1926).

6 See, e.g., Hilbert and Ackermann, op.cit., Kap. I, §§3 and 4, or Hilbert and Bernays , Grundlagen der Mathematik, vol. 1, 1934, pp. 53f.

7 The completeness of PC was first proved by Post E. L., American journal of mathematics, vol. 43 (1921). See W. V. Quine, this Journal, vol. 3 (1938), pp. 37ff.

8 An earlier system MPC, which I constructed in 1940, was slightly different from the one here given; I constructed a proof for its completeness with the help of the reduction procedure explained in the next section. I found later that my system was equivalent to, but simpler than, Lewis's system S5. While writing this article, I found that M. Wajsberg had given a still simpler form (Ein erweiterter Klassenkalkül, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 113-126); therefore I now adopt (in D4-2d) his axioms, with the following two inessential changes. (1) I take (d1), where Wajsberg takes the four axioms of the propositional calculus of Hilbert and Ackermann with the symbol of necessity added to each. (2) In (d3) I have ‘⊃’, while Wajsberg has a symbol corresponding to inline-graphic this change is due to the fact that I do not use, like Wajsberg, a rule of strict implication but a rule of material implication in order to have MPC and MFC contain PC. Wajsberg's calculus is primarily intended as a class calculus with a symbol for class universality added to it, but he remarks that it can also be interpreted as an extended propositional calculus corresponding to Lewis's systems of strict implication. In this interpretation, Wajsberg's ‘|Χ|’ corresponds to ‘Np’, and therefore his ‘Χ<Υ’ to my inline-graphic and to Lewis's ‘pq’ In the same paper, Wajsberg gave a proof for the completeness of his calculus (see below, T6–2f).

9 See Lewis and Langford (op. cit.), especially Appendix II; for S5, see p. 501. Compare also Parry W. T., Modalities in the Survey system of strict implication, this Journal, vol. 4 (1939), pp. 137154; concerning S5, see pp. 151 ff.

10 See Parry, op. cit., pp. 152 ff.

11 This use of the term ‘modal function’ is narrower than that of Parry, op. cit., p. 144, who takes it to include the truth-functions.

12 See J. Dugundji, this Journal, vol. 5 (1940), pp. 150 f., with references to Gödel and McKinsey.

13 More detailed explanations and discussions of the L-concepts are given in [I] §§14 ff. For the definitions of these concepts with the help of ‘range,’ based on conceptions of Wittgenstein, see [I] §§18 and 19; the method used in the present paper is similar to procedure E in §19, but it can take a simpler form here because FL contains atomic sentences for all atomic propositions.

14 Quine, op. cit., p. 88, *100, 102, 103, 104. G. Berry, has shown (this Journal, vol. 6 (1941), pp. 23–27) that Quine'e schema *101 may be omitted if the closure of a matrix is denned by referring to the inverse alphabetical order of the quantifiers. This simplification has here been adopted for FC (and MFC).

15 Quine, op. cit., §§17-21.

16 Hilbert and Bernaye, op. cit., pp. 179 ff.

17 Meaning and necessity: A study in semantics and modal logic. (To appear soon.)

18 Quine W. V., Notes on existence and necessity. The journal of philosophy, vol. 40 (1943), pp. 113127.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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