scholarly journals Four-valued modal logic: Kripke semantics and duality

10.29007/12bb ◽  
2018 ◽  
Author(s):  
Achim Jung ◽  
Umberto Rivieccio
Keyword(s):  
Modal Logic ◽  
Independent Interest ◽  
Kripke Semantics ◽  
Kripke Models ◽  
Consequence Relations ◽  
Accessibility Relation ◽  
Kripke Frames ◽  
Modal Systems ◽  
Global Consequence

Along the lines of recent investigations combining many-valued and modal systems, we address the problem of defining and axiomatizing the least modal logic over the four-element Belnap lattice. By this we mean the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued. Our main result is the introduction of two Hilbert-style calculi that provide complete axiomatizations for, respectively, the local and the global consequence relations associated to the class of all four-valued Kripke models. Our completeness proofs make an extensive and profitable use of algebraic and topological techniques; in fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach.

scholarly journals Recursive enumerability and elementary frame definability in predicate modal logic

2019 ◽  
Vol 30 (2) ◽  
pp. 549-560 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
The Other ◽  
Kripke Semantics ◽  
First Order ◽  
Elementary Class ◽  
Recursive Enumerability ◽  
Kripke Frames ◽  
Recursively Enumerable ◽  
The Relationship

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

Canonical rules

2009 ◽  
Vol 74 (4) ◽  
pp. 1171-1205 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Alternative Proof ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Finite Model Property ◽  
Admissible Rules ◽  
Superintuitionistic Logics ◽  
Canonical Formulas ◽  
Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

A Modal Semantics of Negation in Logic Programming

1992 ◽  
Vol 16 (3-4) ◽  
pp. 231-262
Author(s):  
Philippe Balbiani
Keyword(s):  
Modal Logic ◽  
Logic Programming ◽  
Modal Logics ◽  
Kripke Models ◽  
Logic Programs ◽  
Modal Semantics ◽  
Soundness And Completeness ◽  
Modal Completion ◽  
Well Foundedness

The beauty of modal logics and their interest lie in their ability to represent such different intensional concepts as knowledge, time, obligation, provability in arithmetic, … according to the properties satisfied by the accessibility relations of their Kripke models (transitivity, reflexivity, symmetry, well-foundedness, …). The purpose of this paper is to study the ability of modal logics to represent the concepts of provability and unprovability in logic programming. The use of modal logic to study the semantics of logic programming with negation is defended with the help of a modal completion formula. This formula is a modal translation of Clack’s formula. It gives soundness and completeness proofs for the negation as failure rule. It offers a formal characterization of unprovability in logic programs. It characterizes as well its stratified semantics.

On finite and infinite modal systems

1938 ◽  
Vol 3 (2) ◽  
pp. 77-82 ◽  
Author(s):  
C. West Churchman
Keyword(s):  
Modal Logic ◽  
Symbolic Logic ◽  
Modal Operator ◽  
Image Position ◽  
Modal Systems ◽  
Complex Modal

In Oskar Becker's Zur Logik der Modalitäten four systems of modal logic are considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on A1–8 plus the postulate,From A7: ∼◊p⊰∼p we can prove the converse of C11 by writing ∼◊p for p, and hence deriveThe addition of this postulate to A1–8, as Becker points out, allows us to “reduce” all complex modal functions to six, and these six are precisely those which Lewis mentions in his postulates and theorems: p, ∼p, ◊p, ∼◊p, ∼◊∼p, and ◊∼p This reduction is accomplished by showingwhere ◊n means that the modal operator ◊ is repeated n times; e.g., ◊3p = ◊◊◊p. Then it is shown thatBy means of (1), (2), and (3) any complex modal function whatsoever may be reduced to one of the six “simple” modals mentioned above.It might be asked whether this reduction could be carried out still further, i.e., whether two of the six “irreducible” modals could not be equated. But such a reduction would have to be based on the fact that ◊p = p which is inconsistent with the set B1–9 of Lewis and Langford's Symbolic logic and independent of the set A1–8. Hence for neither set would such a reduction be possible.

scholarly journals Useful Four-Valued Extension of the Temporal Logic KtT4

2018 ◽  
Vol 47 (1) ◽  
Author(s):  
Vincent Degauquier
Keyword(s):  
Modal Logic ◽  
Temporal Logic ◽  
Paraconsistent Logics ◽  
Accessibility Relation

The temporal logic KtT4 is the modal logic obtained from the minimal temporal logic Kt by requiring the accessibility relation to be reflexive (which corresponds to the axiom T) and transitive (which corresponds to the axiom 4). This article aims, firstly, at providing both a model-theoretic and a proof-theoretic characterisation of a four-valued extension of the temporal logic KtT4 and, secondly, at identifying some of the most useful properties of this extension in the context of partial and paraconsistent logics.

Modal logic, philosophical issues in

2018 ◽  
Author(s):  
Thomas J. McKay
Keyword(s):  
Modal Logic ◽  
Natural Language ◽  
Possible Worlds ◽  
Possible World ◽  
Modal Claim ◽  
Knowing That ◽  
Modal Semantics ◽  
Bull's Eye ◽  
Modal Systems ◽  
Richard Montague

In reasoning we often use words such as ‘necessarily’, ‘possibly’, ‘can’, ‘could’, ‘must’ and so on. For example, if we know that an argument is valid, then we know that it is necessarily true that if the premises are true, then the conclusion is true. Modal logic starts with such modal words and the inferences involving them. The exploration of these inferences has led to a variety of formal systems, and their interpretation is now most often built on the concept of a possible world. Standard non-modal logic shows us how to understand logical words such as ‘not’, ‘and’ and ‘or’, which are truth-functional. The modal concepts are not truth-functional: knowing that p is true (and what ‘necessarily’ means) does not automatically enable one to determine whether ‘Necessarily p’ is true. (‘It is necessary that all people have been people’ is true, but ‘It is necessary that no English monarch was born in Montana’ is false, even though the simpler constituents – ‘All people have been people’ and ‘No English monarch was born in Montana’– are both true.) The study of modal logic has helped in the understanding of many other contexts for sentences that are not truth-functional, such as ‘ought’ (‘It ought to be the case that p’) and ‘believes’ (‘Alice believes that p’); and also in the consideration of the interaction between quantifiers and non-truth-functional contexts. In fact, much work in modern semantics has benefited from the extension of modal semantics introduced by Richard Montague in beginning the development of a systematic semantics for natural language. The framework of possible worlds developed for modal logic has been fruitful in the analysis of many concepts. For example, by introducing the concept of relative possibility, Kripke showed how to model a variety of modal systems: a proposition is necessarily true at a possible world w if and only if it is true at every world that is possible relative to w. To achieve a better analysis of statements of ability, Mark Brown adapted the framework by modelling actions with sets of possible outcomes. John has the ability to hit the bull’s-eye reliably if there is some action of John’s such that every possible outcome of that action includes John’s hitting the bull’s-eye. Modal logic and its semantics also raise many puzzles. What makes a modal claim true? How do we tell what is possible and what is necessary? Are there any possible things that do not exist (and what could that mean anyway)? Does the use of modal logic involve a commitment to essentialism? How can an individual exist in many different possible worlds?

Weak liberated versions of T and S4

1975 ◽  
Vol 40 (1) ◽  
pp. 25-30 ◽  
Author(s):  
Charles G. Morgan
Keyword(s):  
Possible Worlds ◽  
Accessibility Relation ◽  
Independence Results ◽  
Modal Systems

AbstractThe usual semantics for the modal systems T, S4, and S5 assumes that the set of possible worlds contains at least one member. Recently versions of these modal systems have been developed in which this assumption is dropped. The systems discussed here are obtained by slightly weakening the liberated versions of T and S4. The semantics does not assume the existence of possible worlds, and the accessibility relation between worlds is only required to be quasi-reflexive instead of reflexive. Completeness and independence results are established.

New foundations for Lewis modal systems

1957 ◽  
Vol 22 (2) ◽  
pp. 176-186 ◽  
Author(s):  
E. J. Lemmon
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Modal System ◽  
New Foundations ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Modal Systems

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

Sign in / Sign up

Export Citation Format

Share Document