Realization using the model existence theorem

8 years 13 days ago

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Justiﬁcation logics reﬁne modal logics by replacing the usual necessity operator with a family of justiﬁcation terms that embody reasons for the necessity of a formula, rather than simply recording the fact of necessity. Many common modal logics have justiﬁcation counterparts. The connection between a modal logic and its justiﬁcation counterpart is through a Realization Theorem, which says that modal operators can be replaced in a precise way with justiﬁcation terms so that modal theorems turn into justiﬁcation logic theorems. In this paper we present a new proof of Realization. We use the familiar machinery of consistency properties to prove a weak version, we call it Quasi-Realization. Then we show how to convert Quasi-Realizations into Realizations proper. Unlike most other treatments in the literature, the work here is not propositional, but ﬁrstorder. Only one modal/justiﬁcation logic is discussed, but the methods easily extend to other standard systems.

Melvin Fitting

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