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On the syntactical construction of systems of modal logic

Published online by Cambridge University Press:  12 March 2014

J. C. C. Mckinsey
J. C. C. Mckinsey*
Affiliation:
University of Nevada
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Extract

When C. I. Lewis developed his theory of strict implication, he left open the question which of his various systems should be regarded as being closest to our intuitions—though he was inclined to favor the system S2. There are to be found in the literature numerous discussions of this question; most of these have condemned S2 as being too strong, and have proposed ways of weakening it.

In the present paper I shall attempt to throw some light on this question by setting up a syntactical definition of “possibility.” I shall show that every system of modal logic constructed on the basis of this definition is at least as strong as the Lewis system S4.

As the intuitive basis for the syntactical definition of possibility, I take the position that to say a sentence is possible means that there exists a true sentence of the same form. Thus, for example, it would be said that the sentence, “Lions are indigenous to Alaska,” is possible, because of the fact that the sentence, “Lions are indigenous to Africa,” has the same form and is true.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 10 , Issue 3 , September 1945 , pp. 83 - 94
Copyright
Copyright © Association for Symbolic Logic 1945

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References

1 See Lewis, and Langford, , Symbolic logic, pp. 122–198Google Scholar, and pp. 492-502.

1 In the following papers, for example: Nelson, E. J., Intensional relations, Mind, n.s. vol. 39 (1930), pp. 440–453CrossRefGoogle Scholar; Duncan-Jones, A. E., Is strict implication the same as entailment?, Analysis, vol. 2 (1934–1935), pp. 70–78CrossRefGoogle Scholar; Vredenduin, P. G. J., A system of strict implication, this Journal, vol. 4 (1939), pp. 73–76.Google Scholar

3 This syntactical definition of possibility is based on certain ideas of Carnap. See The logical syntax of language, p. 181 and pp. 250-255. The inadequacy of the definition, however, has been more recently pointed out by Tarski, , in Über den Begriff der logischen Folgerung, Actes du Congrès International de Philosophie Scientifique, VII Logique, Actualités scientifiques et industrielles 394, pp. 1–11.Google Scholar For some remarks in this connection, see the third section of the present paper.

4 The formal description of our system, as well as the proof of theorems about it, becomes simpler if we suppose the sentential variables are simply those mentioned, instead of supposing, as Lewis does with regard to his systems, that we have different letters, sometimes affected with subscripts:

Since the choice of which letters are to be used as variables makes no difference in the formal properties of a system, we shall therefore hereafter suppose that the various Lewis calculi involve only the variables:

5 Since we are not concerned here with what variables a actually involves, but are interested only in the fact that it involves no variables except a certain finite class of variables, no loss of generality is involved in representing the variables by

instead by

for if α involves no variables except , and if γ is the maximum of the numbers ii, …, in, then α involves no variables except p1, …, pr. This observation will enable us considerably to simplify the notation to be used in later proofs.

In a similar way, if α involves no variables except ,and β involves no variables except , and if γ is the maximum of the numbers , then we can represent α as α, (pl, …, pr) and β as β(p1, …, pr)—where it is of course not assumed that all the variables p1, …, pr necessarily occur in α and β.

6 Parry, William Tuthill, Modalities in the Survey system of strict implication, this Journal, vol. 4 (1939), pp. 137–154Google Scholar. See especially page 149.

7 The method used in this proof is essentially the same as that used in Gödel's, Kurt paper, Zum intuitionistischen Aussagenkalkül, Akademie der Wissenschaften in Wien, Mathematisch-naturwissenschaftliche Klasse, Anzeiger, vol. 69 (1932), pp. 65–66.Google Scholar

8 Tarski, Alfred, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1936), pp. 261–405.Google Scholar