Abstract
The central task of this chapter is to show the Soundness and Completeness of our axiomatizations of the various tense logic systems. This amounts to showing that a statement A is provable (in a given system) from a set S of statements if and only if S entails A (in that system). The upshot of this result is the exact correspondence of the syntactical-deductive and the semantic accounts given for the system.
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Notes
The general outlines of our Soundness and Completeness proofs are due to Makinson, 1966 and Leblanc, 1976.
It is a routine matter, familiar from standard treatments of classical logic, to arrange the statements of a system in a definite order, So we feel free to speak here of the alphabetical place of a statement in such an ordering, and presume the statements of Kt to have been so ordered. We shall also assume that this ordering is based on the complexity of a statement, so, e.g., A preceeds —A, A and B preceed A D B, and A preceeds FB and PB.
S certain to be of cardinality alepho, hence the appropriateness of the superscript.
This ordering of the members of Sls -- and assigning them indices based upon it — is necessitated by the fact that two such sets may have identical memberships. The ordering, found in McArthur, 1972, Appendix I, assigns to each set a positive integer, An upshot of this procedure is the denumerability of 12s no matter what the initial set S. Hence S2 in the constructed historical moment (n, R, P) is sure to be denumerable (at most), and thus a Lowenheim-Skolem Theorem is available for K t insofar as this shows it to have a denumerable model.
Since we presume the statements of Kt to have an alphabetical order based on complexity (see note 2) we can run this induction on the complexity of A.
See McArthur and Leblanc, 1975.
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© 1976 Springer Science+Business Media Dordrecht
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McArthur, R.P. (1976). Soundness and Completeness Theorems for Tense Logic. In: Tense Logic. Synthese Library, vol 111. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3219-2_5
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DOI: https://doi.org/10.1007/978-94-017-3219-2_5
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