Let bikt be the biintuitionistic tense logic obtained by extending biint with two pairs of adjoint modalities, and, with no explicit relationship between the modalities of the same colour, namely, and. Semantical study of intuitionistic modal logics computer software. In fact, this similarity of intuitionistic logic and classical modal logic is not a mere coincidence, but can be made formally precise by a translation of intuitionistic logic into classical modal logic where, obviously, the new concept of necessity plays an important role. Sotirov, modal theories with intuitionistic logic, in. In particular, duality theory came into existence see 11, xi, 26 and m ihe main novelty here is that we add a topology on any frame and extend the functors to the category of all frames. Priest 2001 intuitionistic and paraconsistent logic. Just as intuitionistic propositional logic can be embedded into the modal logic. It is shown that these two relations can combine with. Finally, we study the combined biintuitionistic tense logic, which contains the wellknown intuitionistic modal logic as a sublogic. The work in 9 shows how certain classes of axioms in modaltense logics can be transformed into logical rules in the display calculus and in nested sequents. Finite model property of intuitionistic modal logic was studied in, 15, 11.
S4, biskt can be embedded into a multimodal logic with. Topological and multitopological frames in the context of. This fact allows various possibilities to define intuitionistic modal logics. Article pdf available in annals of pure and applied logic 161. Many intuitionistic modal logics have been investigated in the literature from semantic, algebraic and prooftheoretic perspectives cf. Anotherimportantsource, particularlyforchapters6and8, wasedgettiers 1988 modal logic class at the university of massachusetts. After giving canonical topological kripke models for the hilbertstyle axiomatizations of the fischer servi logic and its classical companion logic, we use the canonical model in a second main result which char. Kurtz may 5, 2003 1 introduction for a classical mathematician, mathematics consists of the discovery of preexisting mathematical truth. On intuitionistic modal and tense logics and their classical companion logics.
Semantical study of intuitionistic modal logics department of intelligence science and technology graduate school of informatics kyoto university kensuke kojima january 16, 2012. Labelfree modular systems for classical and intuitionistic. Strong completeness and the finite model property for bi. Proof theory and proof search of biintuitionistic and tense. A brief introduction to the intuitionistic propositional calculus stuart a. This understanding of mathematics is captured in paul erd. Intuitionistic modal logics originate from different sources and have different. However, they are not supposed to be duals in intuitionistic modal logic. Modal logic is an extension of classic propositional and predicate logic that allows the use of modal operators. An introduction to modal logic and a companion to modal logic. Davoren and others published on intuitionistic modal and tense logics and their classical companion logics. We shall not here discuss the philosophical issues raised by our logics.
In this framework, the intuitionistic propositional logic ipcipc is embedded in the modal logic s4s4, s4s4 is realized in the logic of proofs lplp, and lplp has a provability interpretation in. 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. Axiom system intuitionistic logic kripke model tense operator classical propositional calculus these keywords were added by machine and not by the authors. It is possible to present natur al deduction rules in a sequentstyle, where. On intuitionistic modal and tense logics and their classical. A little earlier again is the 1933 g odel translation of intuitionistic logic into classical s4. A new introduction to modal logic is an entirely new work, completely rewritten by the authors. Clifford, tense and tense logic kuhn, steven, journal of symbolic logic, 1978.
We take the wellknown intuitionistic modal logic of fischer servi with semantics in bi relational kripke frames, and give the natural extension. Download pdf a new introduction to modal logic free. It is sometimes also used to refer to tense logic, a modal logic based system of temporal logic introduced by arthur prior in the late 1950s, with important contributions by hans kamp. Abstract we investigate several intuitionistic modal logics imls, mainly from semantical viewpoint. We give a nested sequent calculus for biintuitionistic tense logic that has cutelimination, and a derived deep inference nested sequent calculus that is complete with respect to the. Sequent and natural deduction calculi for imls are rarer,1,16,3,5,11,7. From syllogism to common sense modal and intuitionistic. Intuitionistic propositional logic ipc in an attempt to provide a. Gentzen sequent calculi for some intuitionistic modal logics. Intuitionistic logic, modal logic, intuitionistic modal logic, deep inference. Readers interested in the intuitionistic view of time and. Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality.
The bi intuitionistic tense logic studied in 17 can be regarded as both a tense expansion of bi intuitionistic logic and an expansion of intuitionistic modal logic studied in, 18 with. It has been further developed by computer scientists, notably amir pnueli, and logicians. In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Neighborhood semantics has originally been considered in classical setting as a semantics of nonnormal modal logics modal logics which kripke semantics cannot handle 12, 29, and ours is its intuitionistic analogue.
In these models we have a collection of the worlds w and the relation of accessibility rwhile, in the case of intuitionistic logic, the elements of. Fairtlough and mendler 15 use an intuitionistic modal logic 4 to reason about the behaviours of hardware circuits. Intuitionistic logic stanford encyclopedia of philosophy. Temporal logic, knowledge operators, bdi models, denotational semantics, effects, security modelling and verification, natural language understanding and. We remark that this notion of a model for in tuitionistic modal logic was originally introduced in a tenselogical version by ewald in 8 and it has also been used by simpson in 17, p. The series of workshops on intuitionistic modal logic and applications imla owes its existence to the hope that philosophers, mathematical logicians and computer scientists would share information and tools when investigating intuitionistic modal logics and modal type theories, if they knew of each others work. 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. A brief introduction to the intuitionistic propositional calculus. The study of intuitionistic modal and tense logic has been done in the several literature, e.
Stirling 38 uses an intuitionistic modal logic to capture a notion of bisimilarity of divergent processes. Pdf in this paper we consider an intuitionistic variant of the modal logic s4 which we call is4. Models for normal intuitionistic modal logics abstract. Modality and possibility in some intuitionistic modal. Intuitionism and indeterminism tenselogical considerations. In these decades, in the area of type theory for programming language, various type systems corresponding via. Because these principles also hold for russian recursive mathematics and the constructive analysis of e. In chapter 3, we introduce an intuitionistic version of ltl with the next temporal operator. On an intuitionistic modal logic 9 the complete set of natural deduction rules for is4 is given in figur e 3. This longawaited book replaces hughes and cresswells two classic studies of modal logic.
This generalises the topological semantics of intuitionistic logic. Kripke semantics for an intuitionistic version of lineartime temporal logic ltl, which corresponds to a lambdacalculus for bindingtime. Pdf on intuitionistic modal and tense logics and their. Just as intuitionistic propositional logic can be embedded into the modal logic s4, biskt can. Mathematical logic, proceedings of the conference on mathematical logic, dedicated to the memory of a. Kripke models 3 for intuitionistic propositional logic are similar to kripke models for modal logic constructed over classical propositional logic. Topological semantics for intuitionistic modal logics, and. A modala word that expresses a modalityqualifies a statement. Proceedings of the 6th workshop on intuitionistic modal logic and applications imla 20, electronic notes in theoretical computer science, volume 300, 2014 n. On intuitionistic modal and tense logics and their. The first main result is that this notion of topological bisimulation yields semantic preservation w. Labelfree modular systems for classical and intuitionistic modal. Intuitionistic hybrid logic university of birmingham. This process is experimental and the keywords may be updated as the learning algorithm improves.
Pdf on some intuitionistic modal logics researchgate. Pacuit, neighborhood semantics for modal logic, springer international publishing ag 2017. Intuitionistic modal logics originate from different sources and have different areas of application. The literature on intuitionistic modal tense logics imtls is vast 6,23 and typically uses hilbert calculi with algebraic, topological or relational semantics. Hence we prefer the term tensed intuitionistic logic to intuitionistic tense logic when describing the subject of this paper. Kripke semantics for modal logic and correspondence theory.
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