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ManyValued Modal Logics
 Fundamenta Informaticae
, 1992
"... . Two families of manyvalued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite manyvalued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds a ..."
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Cited by 272 (16 self)
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. Two families of manyvalued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite manyvalued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be manyvalued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established. 1 Introduction The logics that have appeared in artificial intelligence form a rich and varied collection. While classical (and maybe intuitionistic) logic su#ces for the formal development of mathematics, artificial intelligence has found uses for modal, temporal, relevant, and manyvalued logics, among others. Indeed, I take it as a basic principle that an application should find (or create) an appropriate logic, if it needs one, rather than reshape the application to fit some narrow class of `established' logics. In this paper I want to enlarge the variety of logics...
Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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Cited by 139 (25 self)
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
SSemantics Approach: Theory and Applications
, 1994
"... The paper is a general overview of an approach to the semantics of logic programs whose aim is finding notions of models which really capture the operational semantics, and are therefore useful for defining program equivalences and for semanticsbased program analysis. The approach leads to the intr ..."
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Cited by 126 (28 self)
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The paper is a general overview of an approach to the semantics of logic programs whose aim is finding notions of models which really capture the operational semantics, and are therefore useful for defining program equivalences and for semanticsbased program analysis. The approach leads to the introduction of extended interpretations which are more expressive than Herbrand interpretations. The semantics in terms of extended interpretations can be obtained as a result of both an operational (topdown) and a fixpoint (bottomup) construction. It can also be characterized from the modeltheoretic viewpoint, by defining a set of extended models which contains standard Herbrand models. We discuss the original construction modeling computed answer substitutions, its compositional version and various semantics modeling more concrete observables. We then show how the approach can be applied to several extensions of positive logic programs. We finally consider some applications, mainly in the area of semanticsbased program transformation and analysis.
ManyValued Modal Logics II
 Fundamenta Informaticae
, 1992
"... Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible  in other words each of the experts has their own Kripke model in ..."
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Cited by 34 (0 self)
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Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible  in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal language, and on what sentences will they agree? This problem can be reformulated as one about manyvalued Kripke models, allowing manyvalued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the manyvalued version and the multiple expert one will be formally established. Finally we will axiomatize manyvalued modal logics, and sketch a proof of completeness.
Arithmetic as a theory modulo
 Proceedings of RTA’05
, 2005
"... Abstract. We present constructive arithmetic in Deduction modulo with rewrite rules only. In natural deduction and in sequent calculus, the cut elimination theorem and the analysis of the structure of cut free proofs is the key to many results about predicate logic with no axioms: analyticity and no ..."
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Cited by 29 (3 self)
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Abstract. We present constructive arithmetic in Deduction modulo with rewrite rules only. In natural deduction and in sequent calculus, the cut elimination theorem and the analysis of the structure of cut free proofs is the key to many results about predicate logic with no axioms: analyticity and nonprovability results, completeness results for proof search algorithms, decidability results for fragments, constructivity results for the intuitionistic case... Unfortunately, the properties of cut free proofs do not extend in the presence of axioms and the cut elimination theorem is not as powerful in this case as it is in pure logic. This motivates the extension of the notion of cut for various axiomatic theories such as arithmetic, Church’s simple type theory, set theory and others. In general, we can say that a new axiom will necessitate a specific extension of the notion of cut: there still is no notion of cut general enough to be applied to any axiomatic theory. Deduction modulo [2, 3] is one attempt, among others, towards this aim.
Intuitionistic modal logics as fragments of classical bimodal logics
, 1998
"... Gödel's translation of intuitionistic formulas into modal ones provides the wellknown embedding of intermediate logics into extensions of Lewis' system S4, which reflects and sometimes preserves such properties as decidability, Kripke completeness, the finite model property. In this paper ..."
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Cited by 21 (5 self)
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Gödel's translation of intuitionistic formulas into modal ones provides the wellknown embedding of intermediate logics into extensions of Lewis' system S4, which reflects and sometimes preserves such properties as decidability, Kripke completeness, the finite model property. In this paper we establish a similar relationship between intuitionistic modal logics and classical bimodal logics. We also obtain some general results on the finite model property of intuitionistic modal logics first by proving them for bimodal logics and then using the preservation theorem.
On the relation between intuitionistic and classical modal logics. Algebra and Logic
, 1996
"... Intuitionistic propositional logic Int and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded just as fragments of classical modal logics containing S4. Atthe syntactical level, the Godel translation t embeds every intermediate logic L = Int+ into m ..."
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Cited by 19 (5 self)
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Intuitionistic propositional logic Int and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded just as fragments of classical modal logics containing S4. Atthe syntactical level, the Godel translation t embeds every intermediate logic L = Int+ into modal log1 ics in the interval L = [ L = S4 t (); L=Grz t ()]. Semantically this is re ected by the fact that Heyting algebras are precisely the algebras of open elements of topological Boolean algebras. From the latticetheoretic standpoint the map is a homomorphism of the lattice of logics containing S4 onto the lattice of intermediate logics, while, according to the Blok{Esakia theorem, is an isomorphism of the latter onto the lattice of extensions of the Grzegorczyk system Grz. Atthe philosophical level the Godel translation provides a classical interpretation of the intuitionistic connectives. And from the technical point of view this embedding is a powerful tool for transferring various kinds of results from intermediate logics to modal ones and back via preservation theorems.
Semantic cut elimination in the intuitionistic sequent calculus
 Typed Lambda Calculi and Applications, number 3461 in Lectures
, 2005
"... Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to ..."
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Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.
Bimodal Logics for Reasoning About Continuous Dynamics
 Advances in Modal Logic
, 2000
"... We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme #a## # #a##. In the intended semantics, the plain is given the McKinseyTarski interpretation as the interior operator of a topology, while the labelled [a] is given the ..."
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Cited by 10 (2 self)
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We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme #a## # #a##. In the intended semantics, the plain is given the McKinseyTarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation Ra . The interaction axiom expresses the property that the Ra relation is lower semicontinuous with respect to the topology. The class of topological Kripke frames axiomatised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a di#erential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although decidability is still an open question. 1 Introduction We study a propositional bimodal logic consisting of two S4 modalities # and [a], to...
Mathematical Knowledge Management in MIZAR
 Proc. of MKM 2001
, 2001
"... We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowled ..."
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We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.