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Denotational semantics for modal systems S3–S5 extended by axioms for propositional quantifiers and identity. (English) Zbl 1371.03026
The author focuses on a non-Fregean identity relation among propositions. That is, an identity relation for propositions that satisfies reflexivity and Leibniz’s law but not an identity of propositions on the ground of their being equivalent. In modal contexts, a non-Fregean approach will reject assuming an identity of propositions on the basis of their equivalence being necessary. The author will formulate certain second-order extensions of the modal systems S3, S4 and S5 that will follow the approach in question. These extensions are systems whose logical syntax includes as primitive symbols second-order quantifiers over propositions as well as an identity relation for propositions. In addition to the proper modal axioms of the S3–S5 modal systems, their axiomatic basis includes logical principles for the second-order quantifiers, non-Fregean principles for the identity sign, the Barcan formulas and their converse. Closures of all of these axioms under universal quantification as well as necessitation will also constitute axioms. Modus ponens is their only primitive rule. The modal systems in question are shown to be sound and complete with respect to an algebraic semantics formulated by the author. In the last section of the paper, the author characterizes another semantic system. Relative to this additional semantics, soundness and completeness theorems are proved for the second-order modal systems in question but without the Barcan formulas, their converse and certain extensional schema. The author also shows that his non-Fregean modal systems are conservative extensions of S3–S5.

##### MSC:
 03B45 Modal logic (including the logic of norms)
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