Modal logics and varieties of modal algebras: The beth properties, interpolation, and amalgamation

1992 ◽  
Vol 31 (2) ◽  
pp. 90-105 ◽  
Author(s):  
L. L. Maksimova
Keyword(s):  
Modal Logics ◽  
Modal Algebras

An abstract algebraic logic approach to tetravalent modal logics

2000 ◽  
Vol 65 (2) ◽  
pp. 481-518 ◽  
Author(s):  
Josep Maria Font ◽  
Miquel Rius
Keyword(s):  
Algebraic Logic ◽  
Modal Logics ◽  
The Other ◽  
Abstract Algebraic Logic ◽  
Algebraic Semantics ◽  
Modal Algebras ◽  
Logic Approach ◽  
Abstract Logics ◽  
Joint Study

AbstractThis paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

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.

scholarly journals Products of modal logics and tensor products of modal algebras

2014 ◽  
Vol 12 (4) ◽  
pp. 570-583
Author(s):  
Dov Gabbay ◽  
Ilya Shapirovsky ◽  
Valentin Shehtman
Keyword(s):  
Tensor Products ◽  
Modal Logics ◽  
Modal Algebras ◽  
Products Of Modal Logics

The lattice of modal logics: an algebraic investigation

1980 ◽  
Vol 45 (2) ◽  
pp. 221-236 ◽  
Author(s):  
W. J. Blok
Keyword(s):  
Modal Logic ◽  
Natural Number ◽  
Classical Logic ◽  
Finite Algebra ◽  
Modal Logics ◽  
Kripke Semantics ◽  
Modal Algebras ◽  
Algebraic Investigation

AbstractModal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jónsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □p→P or containing an axiom of the form □mp↔□m+1p for some natural number m is . Furthermore, we show that there exists an immediate predecessor of classical logic (axiomatized by p↔□p) which is not characterized by any finite algebra. The existence of modal logics having immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra.

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 Tensor products of modal logics

10.29007/mtw5 ◽  
2018 ◽  
Author(s):  
Ilya Shapirovskiy ◽  
Valentin Shehtman
Keyword(s):  
Tensor Products ◽  
Modal Logics ◽  
Boolean Algebras ◽  
Modal Algebras ◽  
Products Of Modal Logics ◽  
Usual Product ◽  
Filtration Technique

We consider shifted products of modal algebras and logics first introduced by Y. Hasimoto in 2000. For logics this operation is similar to the well-known usual product but it is logically invariant. We prove the conjecture of D. Gabbay that shifted products act on Boolean algebras exactly as tensor products, so we call them tensor products of modal algebras. We also propose a filtration technique for models based on tensor products and obtain some decidability results.

Modal logics with Belnapian truth values

2010 ◽  
Vol 20 (3) ◽  
pp. 279-304 ◽  
Author(s):  
Serge P Odintsov ◽  
Heinrich Wansing
Keyword(s):  
Modal Logics ◽  
Truth Values

scholarly journals On modal logics arising from scattered locally compact Hausdorff spaces

2019 ◽  
Vol 170 (5) ◽  
pp. 558-577
Author(s):  
Guram Bezhanishvili ◽  
Nick Bezhanishvili ◽  
Joel Lucero-Bryan ◽  
Jan van Mill
Keyword(s):  
Modal Logics ◽  
Hausdorff Spaces ◽  
Locally Compact
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