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We investigate a first-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specifically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.
The system obtained by adding full propositional quantification to S5 is known to be decidable, while that obtained by doing so for T is known to be recursively intertranslatable with full second-order logic. Recently it was shown that the system with two S5 operators and full propositional quantification is also recursively intertranslatable with second-order logic. This note establishes that the map assigning p to \squarep provides a validity and satisfaction preserving translation between the T system and the double S5 system, thus providing an easier proof of the recent result.
This paper presents a logical characterization of coalgebraic behavioral equivalence. The characterization is given in terms of coalgebraic modal logic, an abstract framework for reasoning about, and specifying properties of, coalgebras, for an endofunctor on the category of sets. Its main feature is the use of predicate liftings which give rise to the interpretation of modal operators on coalgebras. We show that coalgebraic modal logic is adequate for reasoning about coalgebras, that is, behaviorally equivalent states cannot be distinguished by formulas of the logic. Subsequently, we isolate properties which also ensure expressiveness of the logic, that is, logical and behavioral equivalence coincide.
This paper defines reduction on derivations (cut-elimination) in the Strict Intersection Type Assignment System of an earlier paper and shows a strong normalization result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterization of normalizability of terms using intersection types.