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Register a new userTowards a Proof Theory of Godel Modal Logics
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Added by
George Metcalfe
Publication date
2011
fields
Informatics Engineering
and research's language is
English
Authors
George Metcalfe
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Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of Godel logic. The calculi are used to establish completeness and complexity results for these fragments.
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In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $forall x_0 exists x_1 dots exists x_n bigwedge x_i R_lambda x_j$. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
Given a class $mathcal C$ of models, a binary relation ${mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $mathcal C$ in $L$ where the modal operator is interpreted via $mathcal R$. We discuss how modal theories of $mathcal C$ and ${mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Lowenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.
A new scheme for proving pseudoidentities from a given set {Sigma} of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when {Sigma} defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when {Sigma} defines a pseudovariety V which is {sigma}-reducible for the equation x=y, provided {Sigma} is enough to prove a basis of identities for the variety of {sigma}-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.
We give a sufficient condition for Kripke completeness of modal logics enriched with the transitive closure modality. More precisely, we show that if a logic admits what we call definable filtration (ADF), then such an expansion of the logic is complete; in addition, has the finite model property, and again ADF. This argument can be iterated, and as an application we obtain the finite model property for PDL-like expansions of logics that ADF.
A logic satisfies the interpolation property provided that whenever a formula {Delta} is a consequence of another formula {Gamma}, then this is witnessed by a formula {Theta} which only refers to the language common to {Gamma} and {Delta}. That is, the relational (and functional) symbols occurring in {Theta} occur in both {Gamma} and {Delta}, {Gamma} has {Theta} as a consequence, and {Theta} has {Delta} as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate Godel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate Godel logic. While this has no impact on settling the question for predicate Godel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.
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