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 Logics with Impossibility as the Negation and Regular Extensions of the Deontic Logic D2

2017 ◽  
Vol 46 (3/4) ◽  
Author(s):  
Krystyna Mruczek-Nasieniewska ◽  
Marek Nasieniewski
Keyword(s):  
Intuitionistic Logic ◽  
Deontic Logic ◽  
Modus Ponens ◽  
Modal Logics ◽  
Point Of View ◽  
Modal Operator ◽  
Modal Interpretation ◽  
Semantical Point ◽  
The Right ◽  
Normal Modal Logics

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics (see [8] and [9]). In (Došen; [2]) a logic N has been investigated in the language with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation (see [3; p. 300]). In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21 (see [4] and [13]). The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+.

scholarly journals Canonical formulas via locally finite reducts and generalized dualities

10.29007/hgbj ◽  
2018 ◽  
Author(s):  
Nick Bezhanishvili
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Algebraic Approach ◽  
Modal Logics ◽  
Substructural Logic ◽  
Locally Finite ◽  
Canonical Formulas

The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.

Normal monomodal logics can simulate all others

1999 ◽  
Vol 64 (1) ◽  
pp. 99-138 ◽  
Author(s):  
Marcus Kracht ◽  
Frank Wolter
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Normal Modal Logics

AbstractThis paper shows that non-normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic.

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.

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.

Canonical modal logics and ultrafilter extensions

1979 ◽  
Vol 44 (1) ◽  
pp. 1-8 ◽  
Author(s):  
J. F. A. K. van Benthem
Keyword(s):  
Modal Logic ◽  
Algebraic Theory ◽  
Main Tool ◽  
Modal Logics ◽  
Modal Operators ◽  
Modal Algebras ◽  
Lower Case ◽  
Elementary Classes ◽  
Preceding Discussion ◽  
Ultrafilter Extensions

In this paper thecanonicalmodal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of anultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable Σ⊿-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any complete modal logic defining a Σ⊿-elementary class of frames is canonical.The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in §2. The concept of a canonical modal logic is introduced and motivated in §3, which also contains the above-mentioned theorems. In §4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions (and some other relevant operations on frames) is discussed.The modal language to be considered here has an infinite supply of proposition letters (p, q, r, …), a propositional constant ⊥ (the so-calledfalsum, standing for a fixed contradiction), the usual Boolean operators ¬ (not), ∨ (or), ∨ (and), → (if … then …), and ↔ (if and only if)—with ¬ and ∨ regarded as primitives—and the two unary modal operators ◇ (possibly) and □ (necessarily)— ◇ being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals.

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.

Intuitionistic tense and modal logic

1986 ◽  
Vol 51 (1) ◽  
pp. 166-179 ◽  
Author(s):  
W. B. Ewald
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Tense Logic ◽  
List Type ◽  
Kripke Models ◽  
Classical Modal Logic

In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Since each classical modal logic can be gotten from some tense logic by one of the definitions(i) □ p ≡ p ∧ Gp ∧ Hp, ◇p ≡ p ∨ Fp ∨ Pp; or,(ii) □ p ≡ p ∧ Gp, ◇p = p ∨ Fp(see [5]), we shall find that our intuitionistic tense logics give us analogues to the classical modal logics as well.We shall not here discuss the philosophical issues raised by our logics. Readers interested in the intuitionistic view of time and modality should see [2] for a detailed discussion.In §2 we define the Kripke models for IKt, the intuitionistic analogue to Lemmon's system Kt. We then prove the completeness and decidability of this system (§§3–5). Finally, we extend our results to other sorts of tense logic and to modal logic.In the language of IKt, we have: sentence-letters p, q, r, etc.; the (intuitionistic) connectives ∧, ∨, →, ¬; and unary operators P (“it was the case”), F (it will be the case”), H (“it has always been the case”) and G (“it will always be the case”). Formulas are defined inductively: all sentence-letters are formulas; if X is a formula, so are ¬X, PX, FX, HX, and GX; if X and Y are formulas, so are X ∧ Y, X ∨ Y, and X → Y. We shall see that, in contrast to classical tense logic, F and P cannot be defined in terms of G and H.

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.

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