A modala word that expresses a modalityqualifies a statement. The n dimensional relational structures described in 2 are examples of products of kripke frames, a notion introduced in 38, 39. L 2 has the finite model property as well actually, theorem 4. Product frames have been widely used for modelling interactions between modal operators. Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multidimensional. On the complexity of modal axiomatisations over many. In this chapter we recollect some basic facts concerning modal logic, concentrating on completeness theory. In the case of modal logics with kripke semantics the notions of simulation and bisimulation prove far more practical and interesting than isomorphism.
The processes are linked by accessibility relations of the kind studied in manydimensional modal logic, modulo complications from constraints between translation and entailment violations in which may trigger reattunement and from refinement and underspecification. The processes are linked by accessibility relations of the kind studied in manydimensional modal logic, modulo compli. Multidimensional modal logic applied logic series pdf. Twodimensional semantics stanford encyclopedia of philosophy. Modal logic is a well behaved subset of firstorder logic. While this is faithful to the field as a whole technically, modal predicate logic is just one of many system combinations, it is a serious omission for many purposes, and we will only. The category of objective modality let zn name the actual number of inhabited planets. In particular, we show that the satisfaction problem and so many other reasoning problems as well for the logics with modal operators applicable to arbitrary concepts, local and global roles. Thus far, many linguists and computer scientists have proposed the temporal relations in occurrences. 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. Towards a manydimensional modal logic for semantic processing tim fernando abstract. Modal logic is a branch of logic with applications in many related disciplines such as computer science, philosophy, linguistics and artificial intelligence.
As these formalisms have a lot in common from a technical point of view, one aim of this dissertation is to study manydimensional modal logics. We tested the claim that the modal word is the crucial factor using a new paradigm that isolates its effects. Two dimensional semantics has also been applied to thought contents. Two dimensional 2d semantics is a formal framework that is used to characterize the meaning of certain linguistic expressions and the entailment relations among sentences containing them. The n dimensional relational structures described in 2 are. Request pdf many dimensional modal logic of tense and temporal interval and its decidability linear tense logics are widely accepted for structural temporal representation, where the basic kt. A twodimensional logic for diagonalization and the a priori.
Representable cylindric algebras and manydimensional modal. In twodimensional modal logic, the truth of a formula. After seeing so many multidimensional modal logics, the reader might wonder if not every modal logic is or can be seen as a multidimensional one. The book treats modal logic as a theory, with several subtheories, such as. Modal languages and bounded fragments of predicate logic 219 this paper is the.
Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi dimensional. However, the nice computational properties can drastically change if we combine some of these formalisms into a manydimensional system, say, to reason about knowledge bases developing in time or moving objects. However, the nice computational properties can drastically change if we combine some of these formalisms into a many dimensional system, say, to reason about knowledge bases developing in time or moving objects. We will show that for every modal similarity type s, the basic derivation system k s is strongly sound and complete with respect to a multidimensional class. There are many sentences that you cant express in classic logic that can be expressed in modal logic. The modal analysis of the mathematical modal logic to the foundations of mathematics. The discovery of topological and algebraic semantics for modal logic connected it with general topology and. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. Modal science to appear in canadian journal of philosophy. However, the term modal logic is used more broadly to cover a family of logics with similar rules and a variety of different symbols. Tools and techniques in modal logic marcus kracht ii. This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. The processes are linked by accessibility relations of the kind studied in many dimensional modal logic, modulo compli.
A manydimensional approach to simulations in modal logic. On the degree of incompleteness in modal logic and the covering relations in the lattice of modal logics. The standard philosophy curriculum therefore includes a healthy dose of logic. There are many textbooks that you can consult for more information. The processes are linked by accessibility relations of the kind studied in many dimensional modal logic, modulo complications from constraints between translation and entailment violations in which may trigger reattunement and from refinement and underspecification. There are exactly n inhabited planets, as our stipulation guarantees. Keywordsdecidability, manydimensional modal logic, sequent system, temporal logic. We show that every modal logic with arbitrary many modalities of arbitrary arity can be seen as a multi dimensional modal logic in the sense of venema. Modal predicate logic an important topic in philosophical applications of modal logic that we have mostly ignored in this survey is modal predicate logic. We show that every modal logic with arbitrary many modalities of arbitrary arity can be seen as a multidimensional modal logic in the sense of venema.
The processes are linked by accessibility relations of the kind studied in many dimensional modal logic, modulo compli cations from constraints between translation and entailment viola tions in. In this way wewillintroduce two novelties to algebraic logic. In studies in logic and the foundations of mathematics, 2003. If youre looking for a free download links of multidimensional modal logic applied logic series pdf, epub, docx and torrent then this site is not for you. Towards a manydimensional modal logic for semantic. Twodimensional 2d semantics is a formal framework that is used to characterize the meaning of certain linguistic expressions and the entailment relations among sentences containing them. Temporal and spatial obda with manydimensional halpern. They find substantial applications in theoretical computer science a modal logic with n primitive unary modal operators. Multidimensional modal logic maarten marx springer. Twodimensional semantics has also been applied to thought contents. The following is a list of some texts this is not a complete list, but a pointer to books that i have found particularly useful.
The polytheistic approach to modal logics alethic modal logic. Modal predicate logic university of california, davis. Our arguments had the same form as those used in previous research, except that we placed a modal word of interest in the conclusion. College park with a study guide that will complement the lectures on modal logic. Towards a many dimensional modal logic for semantic processing tim fernando abstract. Keywordsdecidability, many dimensional modal logic, sequent system, temporal logic. A logic with infinitely many modalities is dynamic logic, introduced in 1976 and having a separate modal operator for every regular expression.
It was baptized in philosophy, and for a long time it was known as the logic of necessity and possibility. In contrast with standard possible worlds semantics, 2d semantics assigns extensions. L 2 of l 1 and l 2 is the smallest multimodal logic l in the language l 1. Temporal and spatial obda with manydimensional halpernshoham logic r. Logic is part of our shared language and inheritance. Zakharyaschev1 1department of computer science, birkbeck, university of london, uk 2dibris department, university of genoa, italy 3polcoming department, university of sassari, italy.
Towards a manydimensional modal logic for semantic processing. Representable cylindric algebras and manydimensional. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many dimensional systems e. Manydimensional modal logic of tense and temporal interval. 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. For example, the statement john is happy might be qualified by saying that john is usually happy, in which.
To study the computational behaviour of many dimensional modal logics is the main aim of this book. We present and study these various notions, followed by a discussion of shehtmans frame product as semantics for certain manydimensional modal logics. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied manydimensional systems e. To study the computational behaviour of manydimensional modal logics is the main aim of this book. The choice of logical connectives depends on the development of propositional logic one wants to follow. A multimodal logic is a modal logic that has more than one primitive modal operator. Probabilistic description logics probdls are an extension of description logics that are designed to capture uncertainty. If both l 1 and l 2 are multimodal logics having the finite model property, then their fusion l 1. A modal a word that expresses a modalityqualifies a statement.
A note on axiomatisations of twodimensional modal logics. To study the computational behaviour of manydimensional modal logics is. Zakharyaschev1 1department of computer science, birkbeck, university of london, uk 2dibris department, university of genoa, italy 3polcoming department, university of sassari, italy 4faculty of computer science, free university of bozenbolzano, italy. Narrowly construed, modal logic studies reasoning that involves the use of the expressions necessarily and possibly. Modal logic is a branch of logic with applications in many related disciplines such as. Similarly, one can look at interesting bidimensional topological spaces.
Notions of context for natural language interpretation are factored in terms of three processes. Technical report 7807, department of mathematics, university of amsterdam, 1978. A version of temporal logic introduced in 1977 and intended for program verification has two modalities, corresponding to dynamic logic s a and a modalities for a single program a, understood. Description logic, uncertainty, manydimensional modal logics, reasoning, computational complexity, probabilistic logics. One of the several possible connections between propositional multimodal logics and classical firstorder logic is to consider finite variable fragments of the. Let l 1 and l 2 be two multimodal 1 logics formulated in languages l 1 and l 2, both containing the language l of classical propositional logic, but having disjoint sets of modal operators. We show that the existence of a model for a given set of formulas is equivalent to the existence of a. Temporal and spatial obda with many dimensional halpernshoham logic r. Request pdf manydimensional modal logic of tense and temporal interval and its decidability linear tense logics are widely accepted for structural temporal representation, where the basic kt.
Manydimensignal modallogic institute for logic, language. I get burned if i lie in the sun for too long, but you cant express the possibility of getting burned. Multidimensional semantics for modal logics project euclid. Notes on modal logic notes for phil370 eric pacuit. Basic concepts in modal logic1 stanford university. The ndimensional relational structures described in 2 are examples of products of kripke frames, a notion introduced in 38, 39. A list describing the best known of these logics follows. A manydimensional approach to simulations in modal logic core. An introduction to modal logic 2009 formosan summer school on logic, language, and computation 29 june10 july, 2009.
Our investigations can also be motivated from a purely modal logic point of view. Higher dimensional automata hda are a model of concurrency that can express most of the traditional partial order models like mazurkiewicz traces, pomsets, event structures, or petri nets. Further offspring of this amsterdambudapest collaboration in the. On the mosaic method for manydimensional modal logics. Description logic, uncertainty, many dimensional modal logics, reasoning, computational complexity, probabilistic logics. K m is the simplest normal multi modal logic interpreted over kripke structures. A catalog ofweak manyvalued modal axioms and their corresponding frame classes. There are many aspects of the semantics for predicate logic and for modal logic which allow for variation when they are combined. Modal logics, interpreted over kripke structures, are the logics for reasoning about sequential behavior and interleaved concurrency. Secondly, the semantics for modal and predicate logics have complicating features not found in. Chapter 1 modal logics of space institute for logic. Modal languages and bounded fragments of predicate logic. In each case, the semantics is twodimensional in the sense that formulas are evaluated at pairs of worlds. The processes are linked by accessibility relations of the kind studied in manydimensional modal logic, modulo compli cations from constraints between translation and entailment viola tions in.
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