Topological and multitopological frames in the context of. Indeed, there is not even any general agreement on what the intuitionistic analogue of the basic modal logic, k, is. Intuitionistic modal logic treats possibility and necessity as not perfectly symmetric. Systematic verification of the intuitionistic modal logic cube in. In this paper we consider an intuitionistic variant of the modal logic s4 which we call is4. Preface this book is an introduction to logic for students of contemporary philosophy. In the authors recent paper 7, the kripke semantics of propositional modal logic is extended to intuitionistic modal logic by considering a topology on the set of worlds, and interpreting propositions as open sets in the topology. Natural deduction systems for various intuitionistic modal logics are presented. This is usually accepted in classical modal logic, but intuitionistically it is not always the case. Intuitionistic frames and formulas 40 intuitionistic calculus 45 embeddings of cl into int 46 basic properties of int 49 realizability logic and medvedevs logic 52 exercises \ 54 notes 56 3 modal logics 61 3. Pdf in this paper we consider an intuitionistic variant of the modal logic s4 which we call is4.
Just as intuitionistic propositional logic can be embedded into the modal logic. Priest 2001 intuitionistic and paraconsistent logic. On intuitionistic modal and tense logics and their. Kripkestyle models with two accessibility relations, one intuitio nistio and the other modal, are given for analogues of the modal system k based on heytings propositional logic. In this section we compare the logics in this paper to related logics in the literature. Our course will mostly concentrate on propositional calculus. Justification logics are explicit modal logics in the sense that they unfold the 2 modality in families of socalled justification terms. Toward intuitionistic nonnormal modal logic and its calculi.
The proof theory and semantics of intuitionistic modal logic. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. For further details, motivation and examples, we refer the reader to the earlier paper. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by hilbertstyle axiomatizations. For logical aspects, adding axiom schemata is a simple and popular way to. This understanding of mathematics is captured in paul erd. It is shown that these two relations can combine with. The subsystem of intuitionistic logic with the false axiom removed is known as minimal logic. Relational sheaves and predicate intuitionistic modal logic. The novelty of this paper is that we place particular importance on the natural deduction formulation of is4 our formulation has several important metatheoretic properties. Lewitzka, \\textita denotational semantics for a lewisstyle modal.
The pdf presentation of this paper is automatically generated from the isabellehol. On intuitionistic modal and tense logics and their classical. Intuitionistic logic an overview sciencedirect topics. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multitopological. The novelty of this paper is that we place particular importance on the natural deduction formulation of is4our formulation has several important. A modal logic amalgam of classical and intuitionistic. In the stream of studies on intuitionistic modal logic, we can find mainly three kinds of natural deduction systems. A little earlier again is the 1933 g odel translation of intuitionistic logic into classical s4. Intuitionistic logic is intended to provide a constructive subset of classical logic. A bidirectional decision procedure for intuitionistic modal logic is5. In mathematical logic, proceedings of the conference on mathematical logic dedicated to a.
In chapter 3, we introduce an intuitionistic version of ltl with the next temporal operator. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. Intuitionistic logic and modality via topology leo esakia departmentoflogic,georgianacademyofsciences,shroshast. Stirling 38 uses an intuitionistic modal logic to capture a notion of bisimilarity of divergent processes. In section 4, the applicability of the logic to mathematical morphology on graphs is considered. A linearlogical reconstruction of intuitionistic modal logic s4. We present three examples of topological semantics for intuitionistic modal logic with one modal operator. Models for normal intuitionistic modal logics abstract. We present a modal extension of classical propositional logic in which intuitionistic propositional logic is mirrored by means of the modal operator. Both intuitionistic modal logic and nonnormal modal logic have a strong tradition. Bishop and his followers, intuitionistic logic may be considered the. A little earlier again is the 1933 g odel translation of intuitionistic logic. This logic, denoted by g, can be regarded as the axiomatic extension of intuitionistic modal logic intk 11 by the prelinearity axiom.
Categorical and kripke semantics for constructive s4 modal logic. A view of its evolution 5 was a variable neither always true nor always false. Pdf intuitionistic modal logic michael zakharyaschev. A modal frame is a frame or locale a together with two monotone. Intuitionistic logic can be understood as a weakening of classical logic, meaning that it is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Cut elimination in nested sequents for intuitionistic. Simpson doctor of philosophy university of edinburgh 1994 graduation date november 1994 abstract possible world semantics underlies many of the applications of modal logic in computer science and philosophy. In particular, there is no single semantic framework rivalling that of possible world semantics for classical modal logic.
Intuitionistic modal logics have been extensively investigated by simpson 11, whose main. Anyways intuitionistic modal logic and applications imla is a loose association of researchers, meetings and a certain amount of mathematical common ground. Intuitionistic modal logics originate from different sources and have different areas of application. The present paper attempts to extend the results of l, in the domain of the propositional calculus, to a class of modal systems called normal. Kripke semantics also known as relational semantics or frame semantics, and often confused with possible world semantics is a formal semantics for nonclassical logic systems created in the late 1950s and early 1960s by saul kripke and andre joyal. In this sense, the modal extension combines classical and intuitionistic propositional logic avoiding the \\textitcollapsing problem. A brief introduction to the intuitionistic propositional calculus.
In addition, we study models of is4 not in the framework of kirpke semantics, but in the more general framework of. Semantical study of intuitionistic modal logics computer software. We propose a modal linear logic to reformulate intuitionistic modal logic s4is4 in terms of linear logic, establishing an s4version of girard translation from is4 to it. Simpson doctor of philosophy university of edinburgh 1994 graduation date november 1994. Intuitionistic logic is related by duality to a paraconsistent logic known as brazilian, anti intuitionistic or dual intuitionistic logic. A linearlogical reconstruction of intuitionistic modal. Intuitionistic epistemic logic volume 9 issue 2 sergei artemov, tudor protopopescu. Kurtz may 5, 2003 1 introduction for a classical mathematician, mathematics consists of the discovery of preexisting mathematical truth. Thus, intuitionistic logic has a more finegrained view than just truefalse signified in the classical axiom. A brief introduction to the intuitionistic propositional calculus stuart a. Fairtlough and mendler 15 use an intuitionistic modal logic 4 to reason about the behaviours of hardware circuits. Information and computation intuitionistic modal logic and.
Intuitionistic logic encompasses the general principles of logical reasoning which have been abstracted by logicians from intuitionistic mathematics, as developed by l. Intuitionistic modal logic and set theory the journal of. A new introduction to modal logic is an entirely new work, completely rewritten by the authors. Topological semantics for intuitionistic modal logics, and.
While the girard translation from intuitionistic logic to linear logic is wellknown, its extension to modal logic is. Concerning the former we can very schematically identify two traditions. Labelfree modular systems for classical and intuitionistic modal logics sonia marin ens, paris, france lutz stra. Labelfree modular systems for classical and intuitionistic. A bidirectional decision procedure for intuitionistic.
This generalises the topological semantics of intuitionistic logic. S4, biskt can be embedded into a multi modal logic with. In these theories the underlying logic of properties and sets is intuitionistic, but there is a subset of formulae that are crisp, classical and twovalued, which represent the certain information. A widespread misconception has it that intuitionistic logic is the logic underlying brouwers intuitionism. An important example of the constructive aspect of intuitionistic logic is the brouwerheytingkolmogorov bhk constructive interpretation of logic. Certainly classical predicate logic is the basic tool of sequential program verification, but modal and temporal logics are increasingly being used for distributed and concurrent systems and intuitionistic logic provides a basis. Because these principles also hold for russian recursive mathematics and the constructive analysis of e. Intuitionistic modal logic made explicit michel marti.
The proof theory and semantics of intuitionistic modal logic alex k. Cut elimination in nested sequents for intuitionistic modal logics 3 in the following, we recall the birelational models 21,7 for iml, which are a combination of the kripke semantics for propositional intuitionistic logic and the one for classical modal logic. Justi cation logic is a re nement of modal logic which studies the concepts of knowledge, belief. Is5 is an intuitionistic variant of s5 modal logic, one of the normal modal logics, with. Intuitionistic modal logic based on neighborhood semantics. From one point of view, these systems are selfjustifying in that a possible world. In chapter 3, we introduce an intuitionistic version. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely.
It covers i basic approaches to logic, including proof theory and especially. Topological semantics for intuitionistic logic and for the classical modal logic s4 have a long history going back to tarski and coworkers in the 1930s and 40s, predating the relational kripke semantics for both 15, 18. Epistemic extensions of combined classical and intuitionistic propositional logic. The aim of this paper is to develop a natural logic and set theory that is a candidate for the formalisation of the theory of fuzzy sets. Information and computation intuitionistic modal logic. Artemovs justi cation logics including the original logic of proofs is a program unfolding since the early 90s. Terminating sequent calculi for two intuitionistic modal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other nonclassical systems.
Topological semantics for intuitionistic logic and for the classical modal logic s4 have a long history going back to tarski and coworkers in the 1930s and 40s, predating the relational kripke semantics for both 29, 36. In his thesis simpson formulates six requirements that an intuitionistic modal logic should obey. Intuitionistic logic stanford encyclopedia of philosophy. They have incorporated all the new developments that have taken place since 1968 in both modal propositional logic and modal predicate logic, without sacrificing tha clarity of exposition and approachability that were essential features of their. Download pdf a new introduction to modal logic free. Intuitionistic and modal logic homepages of uvafnwi staff. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in. Topological duality for intuitionistic modal algebras ii.
1289 1489 508 456 385 36 258 1074 834 1177 838 670 785 249 931 1392 1440 926 1407 703 384 1111 1444 1309 569 772 216 1082 1211 558 1165 72 356 933 357 1273 98 687 1248 801 1348 16 755 561 1109 144 718 1092