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.

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

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.

Products of modal logics, part 1

1998 ◽  
Vol 6 (1) ◽  
pp. 73-146 ◽  
Author(s):  
D. Gabbay
Keyword(s):  
Modal Logics ◽  
Products Of Modal Logics

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.

Normal Products of Modal Logics

2002 ◽  
pp. 241-255
Author(s):  
YASUSI HASIMOTO
Keyword(s):  
Modal Logics ◽  
Products Of Modal Logics
Sign in / Sign up

Export Citation Format

Share Document