Ideal related algebras and their logics

2021 ◽  
Author(s):  
Ivo DÜntsch ◽  
Wojciech Dzik
Keyword(s):  
Modal Algebras

Abstract We investigate modal algebras that generalize the unary discriminator into two directions related to an ideal of the algebra. It turns out that some classes lead to well-known logics, while others have not yet been explored.

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.

scholarly journals Dualities for modal algebras from the point of view of triples

2015 ◽  
Vol 73 (3-4) ◽  
pp. 297-320 ◽  
Author(s):  
Dirk Hofmann ◽  
Pedro Nora
Keyword(s):  
Point Of View ◽  
Modal Algebras

Simple weakly transitive modal algebras

2010 ◽  
Vol 49 (3) ◽  
pp. 233-245 ◽  
Author(s):  
A. V. Karpenko ◽  
L. L. Maksimova
Keyword(s):  
Modal Algebras

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.

Temporal Algebras, Pretemporal Algebras, and Modal Algebras: A Relation between Time and Necessity

1995 ◽  
Vol 41 (1) ◽  
pp. 24-38
Author(s):  
Francisco M. García Olmedo ◽  
Antonio J. Rodríguez Salas
Keyword(s):  
Modal Algebras

scholarly journals Simple and subdirectly irreducibles bounded distributive lattices with unary operators

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Sergio Arturo Celani
Keyword(s):  
Distributive Lattices ◽  
Heyting Algebras ◽  
Subdirectly Irreducible ◽  
Modal Algebras ◽  
Heyting Algebras With Operators ◽  
Subdirectly Irreducibles ◽  
Subdirectly Irreducible Algebras

We characterize the simple and subdirectly irreducible distributive algebras in some varieties of distributive lattices with unary operators, including topological and monadic positive modal algebras. Finally, for some varieties of Heyting algebras with operators we apply these results to determine the simple and subdirectly irreducible algebras.

scholarly journals Duality for $$\kappa $$-additive complete atomic modal algebras

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Yoshihito Tanaka
Keyword(s):  
Modal Algebras

Kripke models for linear logic

1993 ◽  
Vol 58 (2) ◽  
pp. 514-545 ◽  
Author(s):  
Gerard Allwein ◽  
J. Michael Dunn
Keyword(s):  
Representation Theorem ◽  
Linear Logic ◽  
Kripke Model ◽  
Relevance Logic ◽  
Distributive Lattices ◽  
Kripke Models ◽  
Binary Operations ◽  
Modal Algebras ◽  
Boolean Lattices ◽  
Gaggle Theory

AbstractWe present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutativity and associativity are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, similar to Curry's subtraction operator, which is residuated with Linear Logic's cotensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic.

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