Expressive power and semantic completeness: Boolean connectives in modal logic

Studia Logica 49 (2):197 - 214 (1990)
We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective).
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DOI 10.1007/BF00935599
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References found in this work BETA
Semantic Analysis of Tense Logics.S. K. Thomason - 1972 - Journal of Symbolic Logic 37 (1):150-158.
On Engendering an Illusion of Understanding.Dana Scott - 1971 - Journal of Philosophy 68 (21):787-807.
Decidability of S4.Krister Segerberg - 1968 - Theoria 34 (1):7-20.

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For Want of an 'And': A Puzzle About Non-Conservative Extension.Lloyd Humberstone - 2005 - History and Philosophy of Logic 26 (3):229-266.

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