Modal logic, fundamentally

Notre Dame Journal of Formal Logic, 2022

A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI , LV or LF , and every antitone logic, to LN , LV or LF , where LI , LN , LV and LF are logics axiomatized, respectively, by the schemas □α ↔ α, □α ↔ ¬α, □α ↔ ⊤ and □α ↔ ⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV , LF and the minimum amphitone logic AM axiomatized by the schema □α → □β. These logics, along with LI , LN and a wider class of "extensional" logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized ("multiple-conclusion") consequence relations on languages that may lack some or all of the non-modal connectives. We close by discussing these divergences and conditions under which our results do carry over.

Journal of Logic and Computation, 2001

We study two-dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like à ¿, Ë ¿, Ä ¿, ÖÞ ¿, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman . We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square à ¿ ¢ à ¿ of the minimal liner logic using non-structural Gabbay-type inference rules.

Metascience, 2014

The volume under review contains work dedicated to the memory of Leo Esakia, who died in 2010, after having worked for over 40 years towards developing duality theory for modal and intuitionistic logics. The collection comprises ten technical contributions that follow the first chapter, in which the reader can find information on Esakia's studies and career, as well as a complete list of his research publications. In the sequel, we will refer briefly to each of these ten chapters, following the order in the list of contents. B. Jónsson and A. Tarski, in two papers they published in the early 1950s in the American Journal of Mathematics, initiated the study of duality for Boolean algebras with additional operations, via the theory of canonical extensions. Esakia was among the first researchers who studied duality for lattices with additional operations [Topological Kripke models. Soviet Math. Dokl. 15 (1974), 147-151], in particular for Heyting algebras and S4 modal algebras. M. Gehrke, author of the second chapter, shows how distributive lattices, Heyting algebras and S4 modal algebras can be viewed as certain maps between distributive lattices and Boolean algebras. Furthermore, he shows how Stone duality follows from the canonical extension results and how both Priestley and Esakia duality can be derived from Stone duality. In the third chapter, N. Bezhanishvili, S. Ghilardi and M. Jibladze discuss the step-by-step method, i.e. how duality theory can be used to arrive at descriptions of finitely generated free algebras, thus shedding light on issues concerning modal propositional logics. The authors begin by recalling how this method works for free rank one modal logics and then, exploiting the method developed by D. Coumans and S. Van Gool [On generalizing free algebras for a functor. J. Logic Comput. 23 (2012), 645-672], show how it can be extended to work for logics of rank greater than one, such as T, K4 and S4. The paper ends with

A structure here is a non-empty universe together with a collection of functions and predicates. Such a structure is considered as a generalized Kripke frame whose set of possible worlds is endowed with a specific algebraic structure. Thus, a class of similar structures induces a certain multimodal logic. The authors axiomatize the basic modal logic of the class of algebras of arbitrary signature and give universal schemes for axiomatization of modal logic for universal and for Π 2 0 -classes of structures. They also discuss the connections of their approach with modal languages for complex algebras and promise to discuss a number of logical and algebraical consequences in a forthcoming full paper.

The Review of Symbolic Logic, 2011

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra-Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics. §1. Yet another semantics for modal logic? Besides its indisputable success, there may be reasons to doubt about the universal acceptability of the possible-worlds semantics (often called relational semantics or Kripke semantics) for modal logics, basically because they reduce (or equate) meaning to extensions of worlds. Indeed, criticisms against possible-worlds semantics abound. For instance, Kuczynski (2007) purports to show that the very assumptions of possible-worlds semantics lead to the absurd conclusion that all propositions are necessarily true. Even if this critique would be perhaps circumscribed to uses of possible-worlds semantics in quantified modal logic, it is not easy to rebut the well-known Quine's criticisms about the difficulties in separating the notion of possibility within a world from the notion of consistency of the world's description (cf., Section 1 of Quine, 1953). But there are other issues with regard to the possible-worlds account of modal notions. As argued in , the mathematics associated with such semantics may be quite complicated, besides the inappropriateness of the possible-worlds to formalizing knowledge, belief, and other nonalethic modal concepts. Other interesting semantical approaches like algebraic semantics, neighborhood semantics, and topological semantics have been employed in the characterization of modal logics (see Blackburn &, which by itself suggests that the

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.

Theoria a Swedish Journal of Philosophy, 2008

MISCELLANEA LOGICA

Bulletin of the Section of Logic, 2021

This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.

arXiv (Cornell University), 2021

Gödel modal logics can be seen as extenions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety GAO of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper.