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.

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.

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES

2019 ◽  
Vol 13 (2) ◽  
pp. 416-435 ◽  
Author(s):  
SERGEI P. ODINTSOV ◽  
STANISLAV O. SPERANSKI
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Kripke Semantics ◽  
Strong Negation ◽  
Truth Values ◽  
Normal Modal Logic ◽  
Original Language ◽  
Normal Modal Logics

AbstractWe shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

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.

scholarly journals Remarks about the unification type of several non-symmetric non-transitive modal logics

2018 ◽  
Vol 27 (5) ◽  
pp. 639-658 ◽  
Author(s):  
Philippe Balbiani
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Normal Modal Logic ◽  
Complete Set ◽  
Unification Type

Abstract The problem of unification in a normal modal logic $L$ can be defined as follows: given a formula $\varphi$, determine whether there exists a substitution $\sigma$ such that $\sigma (\varphi )$ is in $L$. In this paper, we prove that for several non-symmetric non-transitive modal logics, there exists unifiable formulas that possess no minimal complete set of unifiers.

Algebraic semantics for quasi-classical modal logics

1983 ◽  
Vol 48 (4) ◽  
pp. 941-964 ◽  
Author(s):  
W.J. Blok ◽  
P. Köhler
Keyword(s):  
Modal Logic ◽  
Algebraic Approach ◽  
Modus Ponens ◽  
Modal Logics ◽  
Point Of View ◽  
The Road ◽  
Modal Algebras ◽  
Classical Modal Logic ◽  
Modal Systems ◽  
Normal Modal Logics

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈, F〉, where is an algebra of the appropriate type, and F a subset of the domain of , called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logic E, which is closed under the inference rules of substitution and modus ponens—is characterized by such a matrix, where now is a modal algebra, and F is a filter of . If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general.

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 Theorem Provers For Every Normal Modal Logic

10.29007/jsb9 ◽  
2018 ◽  
Author(s):  
Tobias Gleißner ◽  
Alexander Steen ◽  
Christoph Benzmüller
Keyword(s):  
Modal Logic ◽  
Higher Order ◽  
Modal Logics ◽  
Order Logic ◽  
Theorem Prover ◽  
Higher Order Logic ◽  
Normal Modal Logic ◽  
Theorem Provers

We present a procedure for algorithmically embedding problems formulated in higher- order modal logic into classical higher-order logic. The procedure was implemented as a stand-alone tool and can be used as a preprocessor for turning TPTP THF-compliant the- orem provers into provers for various modal logics. The choice of the concrete modal logic is thereby specified within the problem as a meta-logical statement. This specification for- mat as well as the underlying semantics parameters are discussed, and the implementation and the operation of the tool are outlined.By combining our tool with one or more THF-compliant theorem provers we accomplish the most widely applicable modal logic theorem prover available to date, i.e. no other available prover covers more variants of propositional and quantified modal logics. Despite this generality, our approach remains competitive, at least for quantified modal logics, as our experiments demonstrate.

scholarly journals Modal logic with non-deterministic semantics: Part I—Propositional case

2019 ◽  
Vol 28 (3) ◽  
pp. 281-315
Author(s):  
Marcelo E Coniglio ◽  
Fariñas Del Cerro Luis ◽  
Marques Peron Newton
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Modal Systems ◽  
Logical Matrices ◽  
Deterministic Matrix

Abstract Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. In this paper, we propose even weaker systems, by eliminating both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them.

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.

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