In addition to fitting s multiexpert motivation, there are a number of applications of many valued modal logics, mostly in cases where fuzziness, uncertainty or truthlikeness play a role. Jan 01, 2011 dualities for algebras of fitting s many valued modal logics dualities for algebras of fitting s many valued modal logics maruyama, yoshihiro 20110101 00. In this family, the underlying propositional logics employ finite heyting algebras for the space of truth. Classical twovalued logic may be extended to nvalued logic for n greater than 2. A more detailed presentation can be found in the article on modal logic in this encyclopedia. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Finally we will axiomatize many valued modal logics, and sketch a proof of completeness. Technically, however, there is no reason why this has to be the case. This very extensive volume represents the current stateofa airs in modal logic.
An introduction to modal logic 2009 formosan summer school on logic, language, and computation 29 june10 july, 2009. Theorem 1 for any two modal theories t and e, e is a consistent stable expansion oft if and only i e thw for some nonempty autoepistemic model w t. This cited by count includes citations to the following articles in scholar. Elsevier fuzzy sets and systems 101 1999 115123 fuizy sets and systems an extended gentzentype formulation of a manyvalued modal propositional logic based on zadehs similarity relation osamu morikawa fukushima national college technology, taira, iwa, nagao 30, kamiarakawa, fukushimaken, japan received october 1995. The ones marked may be different from the article in the profile. Manyvalued reflexive autoepistemic logic manyvalued reflexive autoepistemic logic cd koutras, s zachos 20000101 00. Notions of bisimulation for heytingvalued modal languages. A many valued modal logic is introduced that combines the usual kripke frame semantics of the modal logic k with connectives interpreted locally at worlds by lattice and group operations over the real numbers.
Lecture 2 basics in this lecture we will introduce the basics of modal logic. Melvin fitting 199192 considers systems that define such modalities by. A third type of application to logic is the modeling of partial predicates and truth value gaps. Mathematics and computer science lehman college cuny, bronx, ny 10468 depts. The logic presented is a hybrid extension of fittings manyvalued modal logic given in \tableaus for manyvalued modal logic. A fuzzy modal logic is a combination of a fuzzy logic and a modal logic such that a formula at a given world may have a truth value other than true and false manyvalued modal logic 17. Intensional logic stanford encyclopedia of philosophy. Manyvalued modal logics ii 3 values in a many valued logic at possible worlds, but otherwise the general structure of a kripke frame was not altered. For example, the statement john is happy might be qualified by saying that john is usually happy, in which case the term usually is functioning as a modal. Pdf on the minimum manyvalued modal logic over a finite. Manyvalued modal logics model modal notions such as necessity, belief, and. This article deals with many valued modal logics, based only on the necessity operator, over a residuated lattice.
Melvin fitting 199192 considers systems that define such modalities by merging modal and many valued logic, with intended applications to problems of artificial intelligence. Melvin mel fitting born january 24, 1942 is a logician with special interests in philosophical logic and tableau proof systems. Now, many valued modal models where introduced by fitting in fit92b as an extension of heyting semantics for intuitionistic propositional logic to modal logic. This paper presents one simple approach to a major family of many valued modal logics, together with an illustration of why this family is philosophically interesting. Syntax and semanticslogicsfiltrations on many valued kripke modelsapplications. This is a problem about modal logic that cannot use tableaus because there arent any reasonable ones for the logic being considered.
Jan 01, 2000 many valued reflexive autoepistemic logic many valued reflexive autoepistemic logic cd koutras, s zachos 20000101 00. January 19, 1995 abstract we continue a series of papers on a family of many. However, the term modal logic is often used to include other extensions, for instance temporal logic, that allows for the expression of past or future truths. Many valued modal logics ii 3 values in a many valued logic at possible worlds, but otherwise the general structure of a kripke frame was not altered. Ian horrocks, ullrich hustadt, ulrike sattler, renate schmidt. A labelled tableau system is provided and a conexptimeupper bound obtained for checking validity in the logic. Dualities for algebras of fittings manyvalued modal logics. Finally we will axiomatize manyvalued modal logics, and sketch a proof of completeness. In logic, a manyvalued logic also multior multiplevalued logic is a propositional calculus in which there are more than two truth values. Semantically, one family is characterized using kripke models that allow formulas to take values in a finite many valued logic, at each possible world. Modal logic is the study of modal propositions and the logical relationships that they bear to one another. The introduction of systems of mvl by lukasiewicz 1920 was initially guided by the finally unsuccessful idea of understanding the notion of possibility, i. An advanced, but very accessible, textbook focusing on the main technical results in the area.
Computer science, philosophy, mathematics graduate center cuny, 33 west 42nd street, nyc, ny 10036. On the axiomatizability of modal many valued logics amanda vidal institute of computer science, czech academy of sciences. Manyvalued reflexive autoepistemic logic, logic journal. Dualities for algebras of fittings manyvalued modal. Finite model properties for manyvalued modal logics. Possibilistic semantics for a modal kd45 extension of g. Introduction and motivation a manyvalued hybrid logic mvhl a tableau system for mvhl. The modal logic kx is characterized by the class of all models in. For example, the following are all modal propositions.
He was a professor at city university of new york, lehman college and the graduate center. Manyvalued modal logics, under different forms and contexts, have appeared in the literature for different reasoning modeling purposes. Let us refer to the papers 1, 2 and 3 containing very interesting semantics for modal language consisting of socalled many valued kripke models. Manyvalued hybrid logic journal of logic and computation. Now, manyvalued modal models where introduced by fitting in fit92b as an extension of heyting semantics for intuitionistic propositional logic to modal logic. Traditionally, in aristotles logical calculus, there were only two possible values i. At the graduate center he was in the departments of computer science, philosophy, and mathematics, and at lehman college he was in the. An introduction to modal logic 2009 formosan summer school on logic, language, and computation.
While presenting you the many applications of modal logic might convince you, it may be easier to indicate how modal logic was important in the development of the very field of mathematical logic that you seem to appreciate. He was a professor at city university of new york, lehman college and the graduate center 723724 from 1968 to 20. In this paper we introduce nonmonotonic modal logics based on many valued logics, rather than on classical logic. Basic concepts in modal logic1 stanford university. In logic, a many valued logic also multior multiple valued logic is a propositional calculus in which there are more than two truth values. Complexity of modal logic introduction ps pdf author. Melvin fitting 199192 considers systems that define such modalities by merging modal and manyvalued logic, with intended applications to problems of artificial intelligence.
The second family generalizes this to allow the accessibility relation between worlds also to be many valued. The most wellknown modal propositions are propositions about what is necessarily the case and what is possibly the case. Just as the notion of possible worlds in the semantics of modal logic can be. We note that in general this operator does not satisfy the k axiom. In this paper we introduce nonmonotonic modal logics based on manyvalued logics, rather than on. Manyvalued modal logics ii 3 values in a manyvalued logic at possible worlds, but otherwise the general structure of a kripke frame was not altered. Many valued modal logic can, for instance, be used for fuzzy similaritybased reasoning 21 or for reasoning about fuzzy beliefs 26. Rules and propositional rules are direct translations of fittings corre. Why did many valued logic fail in describing modal logic. Thomas bolander2 and torben brauner 1 manyvalued hybrid logic. We focus on three basic classes, according to the accessibility relation, of kripke frames. Semantically, one family is characterized using kripke models that allow formulas to take values in a finite manyvalued logic, at each possible world.
On the axiomatizability of modal manyvalued logics amanda vidal institute of computer science, czech academy of sciences. The above possibilities are the only operators used in modal logic in the narrow sense. Since from the beginning, the modal and manyvalued logic notions were part of jan. Dualities for algebras of fittings manyvalued modal logics dualities for algebras of fittings manyvalued modal logics maruyama, yoshihiro 20110101 00. 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. Manyvalued modal logics with the fmp finite model properties for manyvalued modal logics.
Two families of many valued modal logics are investigated. The second family generalizes this to allow the accessibility relation between worlds also. Semantically, one family is characterized using kripke models that allow formulas to take values in a. A modala word that expresses a modalityqualifies a statement. A manyvalued modal logic is introduced that combines the usual kripke frame semantics of the modal logic k with connectives interpreted locally at worlds by lattice and group operations over the real numbers. Tableaus for manyvalued modal logic melvin fitting. Axiomatizing a realvalued modal logic advances in modal logic. Notes on modal logic notes for phil370 eric pacuit october 22, 2012.
This text provides both a philosophical and technical. It elegantly straddles the line between philosophy and mathematics, without getting bogged down in the details of either as much of the rest of the modal logic literature seems to. Manyvalued modal logic can, for instance, be used for fuzzy similaritybased reasoning 21 or for reasoning about fuzzy beliefs 26. Willem conradie 1 wilmari morton 1 claudette robinson 2 1department of pure and applied mathematics, university of johannesburg. The equivalence between the many valued version and the multiple expert one will be formally established. On transitive modal manyvalued logics sciencedirect. Two families of manyvalued modal logics are investigated. The equivalence between the manyvalued version and the multiple expert one will be formally established. In this paper we provide a simpli ed semantics for the logic. Buehler based on first order modal logic by fitting and mendelsohn january 5, 2015. The most crucial criterion in classical logic is that logical reasoning. On algebras of latticevalued modal logic lml algebraic study of latticevalued logic and latticevalued modal logic yoshihiro maruyama faculty of integrated human studies, kyoto university, japan third indian conference on logic and its applications yoshihiro maruyama algebraic study of. A clear example of this are fuzzy description logics see e.
Manyvalued logics treat their truth degrees as technical tools, and intend to. Algebraic study of latticevalued logic and latticevalued. This paper is focused on the study of modal logics defined from valued kripke frames, and particularly, on computability and expressivity questions of modal logics of transitive kripke frames evaluated over certain residuated lattices. To this end, fitting introduces an adaptation of the canonical model. Many valued modal logics with the fmp finite model properties for many valued modal logics. First order modal logic by melvin fitting and elliot mehdelsohn. Modal decision problems introduction ps pdf authors. Jun, 2005 the presence of explicit modal operators allows flexibility in the embedding of other approaches. The logic presented is a hybrid extension of fittings manyvalued modal logic given in \tableaus for manyvalued modal logic, studia logica, 55. Pdf lukasiewiczs 4valued logic and normal modal logics. Frame constructions, truth invariance and validity. Manyvalued reflexive autoepistemic logic, logic journal of. A fixpoint 3valued semantics for autoepistemic logic our semantics for autoepistemic logic is defined in. Computational modal logic introduction ps pdf authors.
The focus here is on rstorder modal logic as opposed to propositional modal logic which is the focus of most of the other texts mentioned here. Syntax and semanticslogicsfiltrations on manyvalued kripke modelsapplications. We introduce a propositional manyvalued modal logic which is an extension. Its origins can be found in the analysis of modal logic but it was independently rediscovered by computer sci. Pdf on mar 5, 2015, siegfried gottwald and others published manyvalued logic find, read and cite all the research you need on researchgate.
On the minimum manyvalued modal logic over a finite. The other family considered in 3 allowed the accessibility relation itself to be manyvalued. This paper deals with many valued modal logics, based only on the necessity operator, over a residuated lattice. Frame constructions, truth invariance and validity preservation in manyvalued modal logic pantelis e. The sep article on many valued logic makes the following statement. A family of manyvalued modal logics which correspond to possibleworlds models with manyvalued accessibility relations, has been recently proposed by m. Willem conradie 1 wilmari morton 1 claudette robinson 2 1department of pure and.
On algebras of lattice valued modal logic lml algebraic study of lattice valued logic and lattice valued modal logic yoshihiro maruyama faculty of integrated human studies, kyoto university, japan third indian conference on logic and its applications yoshihiro maruyama algebraic study of lattice valued logics. This paper is intended to be a demonstration of this slogan. An extended gentzentype formulation of a manyvalued. Among many existing manyvalued logics, we employ a modi. A fuzzy modal logic is a combination of a fuzzy logic and a modal logic such that a formula at a given world may have a truth value other than true and false many valued modal logic 17. Its origins can be found in the analysis of modal logic but it was independently rediscovered by. Manyvalued logic stanford encyclopedia of philosophy. In addition to fittings multiexpert motivation, there are a number of applications of manyvalued modal logics, mostly in cases where fuzziness, uncertainty or truthlikeness play a role. Here is a quick presentation to establish notation, and to point out how one of freges proposals fits in. This is a great place to get a clear introduction to firstorder modal logic. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt.
This very extensive volume represents the current statofa airs in modal logic. This article deals with manyvalued modal logics, based only on the necessity operator, over a residuated lattice. The other family considered in 3 allowed the accessibility relation itself to be many valued. Such logics, in fact, have a long history, 14, 12, 5, 7, 6, 8. For this purpose we consider a manyvalued version of kripke semantics for modal logic where both propositions at each world and the accessibility relation are in. Also several theoretical results of interest have been established concerning these logics. Firstordermodallogic university of california, berkeley. Kripke semantics for propositional modal logic is, by now, a very familiar thing. Tableaus for manyvalued modal logic lehman college cuny. The second family generalizes this to allow the accessibility relation between worlds also to be manyvalued. Buehler based on firstorder modal logic by fitting and mendelsohn january 5, 2015.
966 1507 1455 179 29 1540 1408 806 890 1425 226 1307 1327 768 586 830 1097 693 231 1064 508 141 219 1479 56 377 1089 1197 305 976