scholarly journals A Resolution Method for Modal Logic S5

10.29007/1zgr ◽  
2018 ◽  
Author(s):  
Yakoub Salhi ◽  
Michael Sioutis
Keyword(s):  
Normal Form ◽  
Modal Logic ◽  
Graph Coloring ◽  
Propositional Logic ◽  
Possible Worlds ◽  
Conjunctive Normal Form ◽  
Clause Structure ◽  
Possible Worlds Semantics ◽  
Resolution Method ◽  
Resolution Rule

The aim of this work is to define a resolution method for the modal logic S5. Wefirst propose a conjunctive normal form (S5-CNF) which is mainly based on using labelsreferring to semantic worlds. In a sense, S5-CNF can be seen as a generalization of theconjunctive normal form in propositional logic by using in the clause structure the modalconnective of necessity and labels. We show that every S5 formula can be transformedinto an S5-CNF formula using a linear encoding. Then, in order to show the suitabilityof our normal form, we describe a modeling of the problem of graph coloring. Finally, weintroduce a simple resolution method for S5, composed of three deductive rules, and weshow that it is sound and complete. Our deductive rules can be seen as adaptations ofRobinson’s resolution rule to the possible-worlds semantics.

scholarly journals Solving the Satisfiability Problem of Modal Logic S5 Guided by Graph Coloring

2019 ◽  
Author(s):  
Pei Huang ◽  
Minghao Liu ◽  
Ping Wang ◽  
Wenhui Zhang ◽  
Feifei Ma ◽  
...  
Keyword(s):  
Normal Form ◽  
Modal Logic ◽  
Graph Coloring ◽  
Upper Bound ◽  
Possible Worlds ◽  
Satisfiability Problem ◽  
Conflict Graph ◽  
Sat Solvers ◽  
Search Spaces ◽  
Sat Encoding

Modal logic S5 has found various applications in artificial intelligence. With the advances in modern SAT solvers, SAT-based approach has shown great potential in solving the satisfiability problem of S5. The scale of the SAT encoding for S5 is strongly influenced by the upper bound on the number of possible worlds. In this paper, we present a novel SAT-based approach for S5 satisfiability problem. We show a normal form for S5 formulas. Based on this normal form, a conflict graph can be derived whose chromatic number provides an upper bound of the possible worlds and a lot of unnecessary search spaces can be eliminated in this process. A heuristic graph coloring algorithm is adopted to balance the efficiency and optimality. The number of possible worlds can be significantly reduced for many practical instances. Extensive experiments demonstrate that our approach outperforms state-of-the-art S5-SAT solvers.

Language, Meaning, and Information

2014 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Modal Logic ◽  
Philosophy Of Language ◽  
Possible Worlds ◽  
Philosophy Of Mathematics ◽  
Scientific Study ◽  
Bertrand Russell ◽  
Gottlob Frege ◽  
Possible Worlds Semantics ◽  
Language Meaning

This chapter is a case study of the process by which the attempt to solve philosophical problems sometimes leads to the birth of new domains of scientific inquiry. It traces how advances in logic and the philosophy of mathematics, starting with Gottlob Frege and Bertrand Russell, provided the foundations for what became a rigorous and scientific study of language, meaning, and information. After sketching the early stages of the story, it explains the importance of modal logic and “possible worlds semantics” in providing the foundation for the last half century of work in linguistic semantics and the philosophy of language. It argues that this foundation is insufficient to support the most urgently needed further advances. It proposes a new conception of truth-evaluable information as inherently representational cognitive acts of certain kinds. The chapter concludes by explaining how this conception of propositions can be used to illuminate the notion of truth; vindicate the connection between truth and meaning; and fulfill a central, but so far unkept, promise of possible worlds semantics.

scholarly journals EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS

2019 ◽  
Vol 12 (2) ◽  
pp. 255-270 ◽  
Author(s):  
PAVEL NAUMOV ◽  
JIA TAO
Keyword(s):  
Modal Logic ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Completeness Theorem ◽  
Kripke Semantics ◽  
Possible Worlds Semantics ◽  
Distributed Knowledge ◽  
First Order ◽  
Soundness And Completeness ◽  
First Order Modal Logic

AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

Metaphysics of knowledge

2018 ◽  
Author(s):  
Jesper Kallestrup
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Modal Epistemology ◽  
Physical Realization ◽  
Possible Worlds Semantics ◽  
Justified Belief ◽  
Essential Nature ◽  
The World ◽  
Structure Of Knowledge ◽  
Shed Light

Epistemology has traditionally been concerned with the scope, sources and structure of knowledge and other epistemic statuses such as justified belief. Metaphysics of knowledge seeks to answer metaphysical questions about knowledge and its place in the world as they arise in such epistemological pursuits. More generally, metaphysics of epistemology can be understood to include metaphysical questions about a broader range of epistemic statuses. In either case, answers to such questions may help solve distinctive problems in epistemology or neighbouring fields, or they may be of independent theoretical importance. Since modal logic, possible worlds semantics and related formal frameworks are frequently brought to bear in traditional metaphysics, for example Williamson 2013, metaphysical inquiries about knowledge may also utilize such modal tools. But metaphysics of knowledge is distinct from both modal epistemology, which concerns the necessary links between belief and truth that supposedly constitute knowledge, and epistemology of modality, which concerns knowledge of modalities – that is, of what is necessary, possible, contingent and so on. Metaphysics of knowledge sits at the intersection of epistemology and metaphysics, whereas modal epistemology is an externalist branch of epistemology, and epistemology of modality is a sub-discipline within philosophy of modality. Both modal epistemology and epistemology of modality may of course shed light on metaphysical aspects of knowledge, for example, if it turns out knowledge is essentially a modally sensitive or safe belief. However, the intended scope of metaphysics of knowledge is much broader, encompassing not just the essential nature of knowledge, but also its metaphysical ground and its physical realization.

A resolution calculus for MinSAT

2019 ◽  
Author(s):  
Chu-Min Li ◽  
Fan Xiao ◽  
Felip Manyà
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Logical Calculus ◽  
Elimination Algorithm ◽  
Variable Elimination ◽  
Resolution Rule ◽  
Resolution Calculus

AbstractThe logical calculus for SAT are not valid for MaxSAT and MinSAT because they preserve satisfiability but not the number of unsatisfied clauses. To overcome this drawback, a MaxSAT resolution rule preserving the number of unsatisfied clauses was defined in the literature. This rule is complete for MaxSAT when it is applied following a certain strategy. In this paper we first prove that the MaxSAT resolution rule also provides a complete calculus for MinSAT if it is applied following the strategy proposed here. We then describe an exact variable elimination algorithm for MinSAT based on that rule. Finally, we show how the results for Boolean MinSAT can be extended to solve the MinSAT problem of the multiple-valued clausal forms known as signed conjunctive normal form formulas.

FORMULAS IN MODAL LOGIC S4

2010 ◽  
Vol 3 (4) ◽  
pp. 600-627
Author(s):  
KATSUMI SASAKI
Keyword(s):  
Normal Form ◽  
Modal Logic ◽  
Propositional Logic ◽  
Normal Forms ◽  
Classical Propositional Logic ◽  
Mutual Relation ◽  
Grzegorczyk Logic ◽  
Modal Logic S4 ◽  
Normal Modal Logics ◽  
Finite Method

Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables p1, …, pm in modal logic S4. Our description contains more information on S4 than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to find a normal form equivalent to a given formula A by clarifying the behavior of connectives and giving a finite method to list all exact models.

scholarly journals Philosophical Issues from Kripke’s ‘Semantical Considerations on Modal Logic’

2016 ◽  
Vol 20 (1) ◽  
pp. 01
Author(s):  
John Divers
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Ontological Commitment ◽  
Semantic Theory ◽  
Kripke Semantics ◽  
Crucial Importance ◽  
Possible Worlds Semantics ◽  
Philosophical Discussion ◽  
Modal Space ◽  
Clear Conception

http://dx.doi.org/10.5007/1808-1711.2016v20n1p1In ‘Semantical Considerations on Modal Logic’, Kripke articulates his project in the discourse of “possible worlds”. There has been much philosophical discussion of whether endorsement of the Kripke semantics brings ontological commitment to possible worlds. However, that discussion is less than satisfactory because it has been conducted without the necessary investigation of the surrounding philosophical issues that are raised by the Kripke semantics. My aim in this paper is to map out the surrounding territory and to commence that investigation. Among the surrounding issues, and my attitudes to them, are these: (1) the potential of the standard distinction between pure and impure versions of the semantic theory has been under-exploited; (2) there has been under-estimation of what is achieved by the pure semantic theory alone; (3) there is a methodological imperative to co-ordinate a clear conception of the purposes of the impure theory with an equally clear conception of the content the theory; (4) there is a need to support by argument claims about how such a semantic theory, even in an impure state, can fund explanations in the theory of meaning and metaphysics; (5) greater attention needs to be paid to the crucial advance that Kripke makes on the precursors of possible-worlds semantics proper (e.g. Carnap 1947) in clearly distinguishing variation across the worlds within a model of modal space from variation across such models and, finally, (6) the normative nature of the concept of applicability, of the pure semantic theory, is both of crucial importance and largely ignored.

scholarly journals Revisiting McKinsey's 'Syntactical' Construction of Modality

2020 ◽  
Vol 17 (2) ◽  
pp. 123
Author(s):  
Max Cresswell
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Kripke Models ◽  
Possible Worlds Semantics ◽  
Completeness Proof

In 1945 J.C.C. McKinsey produced a ‘semantics’ for modal logic based on necessity defined in terms of validity. The present papers looks at how to update F.R. Drake’s completeness proof for McKinsey’s semantics by comparing McKinsey ‘models’ with the now standard Kripke models. It also looks at the motivation behind the system McKinsey called S4.1, but which we now call S4M; and use this motivation to produce a McKinsey semantics for that system. One lesson which emerges from this work is an appreciation of the superiority of the current possible worlds semantics based on frames and models, both in terms of an intuitive understanding of modality, and also in terms of the ease of working with particular systems.

scholarly journals Understanding Necessity Causally

MANUSYA ◽  
2006 ◽  
Vol 9 (1) ◽  
pp. 26-37
Author(s):  
Puttawit Bunnag
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Central Issue ◽  
Possible Worlds Semantics ◽  
Modal Language

The paper concerns a central issue of alethic modality. It attempts to provide a criterion of meaning for statements with modal words: ‘necessary’, ‘possibly’, ‘must’, ‘can’, ‘could’, etc. By considering the main problems concerning modal logic, logic dealing specifically with modal language, the paper chooses to understand the meanings of modal language by ‘possible worlds’ semantics, and tries to make it more credible by employing a concept of causality which underline most of our normal modal language. Furthermore, the paper attempts to answer the following questions: Why are formal expositions essential to philosophically understanding problematic modal discourses?; What is the conceptual burden they impose on us which needs to be overcome?; Why are existing philosophical endeavours taking part in this semantic contest ‘unsatisfactory’ or ‘inadequate’?; How can we understand modal discourses causally?

Sign in / Sign up

Export Citation Format

Share Document