scholarly journals On Formalizing Logical Modalities

2021 ◽  
Vol 21 (63) ◽  
pp. 419-430
Author(s):  
Luigi Pavone
Keyword(s):  
Modal Logic ◽  
Logical Truth ◽  
Substitution Rule ◽  
Normal Modal Logic ◽  
The One ◽  
Modal Systems ◽  
Logic System ◽  
Uniform Substitution

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that formalization purpose, of some modal schemes, such as S4 and S5 . It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon.

scholarly journals STABLE MODAL LOGICS

2018 ◽  
Vol 11 (3) ◽  
pp. 436-469 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
JULIA ILIN
Keyword(s):  
Modal Logic ◽  
General Concept ◽  
Modal Logics ◽  
Normal Modal Logic ◽  
Superintuitionistic Logics ◽  
Modal Systems

AbstractStable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

A Defence of Information Closure

2019 ◽  
pp. 149-161
Author(s):  
Luciano Floridi
Keyword(s):  
Modal Logic ◽  
Good Reason ◽  
Modal Logics ◽  
Epistemic Closure ◽  
Normal Modal Logic ◽  
The One ◽  
Normal Modal Logics

In this chapter, the principle of information closure (PIC) is defined and defended against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If successful, given that PIC is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, one potentially good reason to look for a formalization of the logic of ‘S is informed that p’ among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of ‘S is informed that p’ should be a normal modal logic, but that it could still be, insofar as the objection that it could not be, based on the sceptical objection against PIC, has been removed. In other words, this chapter argues that the sceptical objection against PIC fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of ‘S is informed that p’, which remains plausible insofar as this specific obstacle is concerned.

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.

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?

Georg Henrik von Wright. Form and content in logic. A revised reprint of XV 58(2), 199(2), 280(2). Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 1–21. - Georg Henrik von Wright. On the idea of logical truth (I). A revised reprint of XV 58(1), 199(1), 280(1). Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 22–43. - Georg Henrik von Wright. On double quantification. A revised reprint of XVII 201. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 44–57. - Georg Henrik von Wright. Deontic logic. A revised reprint of XVII 140. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 58–74. - Georg Henrik von Wright. Interpretations of modal logic. A revised reprint of XVIII 176. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 75–88. - Georg Henrik von Wright. A new system of modal logic. A revised version of XIX 66. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 89–126. - Georg Henrik von Wright. On conditionals. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 127–165. - Georg Henrik von Wright. The concept of entailment. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 166–191.

1970 ◽  
Vol 35 (3) ◽  
pp. 460-462
Author(s):  
Timothy Smiley
Keyword(s):  
New York ◽  
Modal Logic ◽  
Scientific Method ◽  
Deontic Logic ◽  
Logical Truth ◽  
Double Quantification ◽  
New System

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.

Liberated Brouwerian Modal Logic

Dialogue ◽  
1974 ◽  
Vol 13 (3) ◽  
pp. 505-514 ◽  
Author(s):  
Charles G. Morgan
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Possible World ◽  
Modal Systems ◽  
Relationship Of ◽  
The Relationship

In an attempt to “purify” logic of existential presuppositions, attention has recently focused on modal logics, where one usually assumes that at least one possible world exists. Systems very analogous to some of the standard modal systems have been developed which drop this presupposition. We will here treat the removal of the existential assumption from Brouwerian modal logic and discuss the relationship of the system so derived to other modal systems.

Finite Kripke models and predicate logics of provability

1990 ◽  
Vol 55 (3) ◽  
pp. 1090-1098 ◽  
Author(s):  
Sergei Artemov ◽  
Giorgie Dzhaparidze
Keyword(s):  
Modal Logic ◽  
Kripke Model ◽  
Finite Model ◽  
Kripke Models ◽  
Model Property ◽  
Logical Formula ◽  
Finite Model Property ◽  
Modal Predicate Logics ◽  
The One ◽  
Arithmetical Interpretation

AbstractThe paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic:If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that PA ⊬ fR.This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding “the predicate part” as a specific addition to the standard Solovay construction.

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