Studia Logica最新文献

The aim of this paper is to combine several Ivlev-like modal systems characterized by 4-valued non-deterministic matrices (Nmatrices) with (mathcal {IDM}4), a 4-valued expansion of Belnap–Dunn’s logic (FDE) with an implication introduced by Pynko in 1999. In order to do this, we introduce a new methodology for combining logics which are characterized by means of swap structures, based on what we call superposition of snapshots. In particular, the combination of (mathcal {IDM}4) with (Tm), the 4-valued Ivlev’s version of KT, will be analyzed with more details. From the semantical perspective, the idea is to combine the 4-valued swap structures (Nmatrices) for (Tm) (and several of its extensions) with the 4-valued twist structure (logical matrix) for (mathcal {IDM}4). This superposition produces a universe of 6 snapshots, with 3 of them being designated. The multioperators over the new universe are defined by combining the specifications of the given swap and twist structures. This gives rise to 6 different paradefinite Ivlev-like modal logics, each one of them characterized by a 6-valued Nmatrix, and conservatively extending the original modal logic and (mathcal {IDM}4). This important feature allows to consider the proposed construction as a genuine technique for combining logics. In addition, it is possible to define in the combined logics a classicality operator in the sense of logics of evidence and truth (LETs). A sound and complete Hilbert-style axiomatization is also presented for the 6 combined systems, as well as a Prolog program which implements the swap structures semantics for the 6 systems, producing a decision procedure for satisfiability, refutability and validity of formulas in these logics.

The McKinsey axiom ((textrm{M}) Box Diamond prightarrow Diamond Box p) has a local first-order correspondent on the class of all weakly transitive frames ({{mathcal {W}}}{{mathcal {T}}}). It globally corresponds to Lemmon’s condition (({textsf{m}}^infty )) on ({{mathcal {W}}}{{mathcal {T}}}). The formula ((textrm{M})) is canonical over the weakly transitive modal logic (textsf{wK4}={textsf{K}}oplus pwedge Box prightarrow Box Box p). The modal logic (mathsf {wK4.1}=textsf{wK4}oplus textrm{M}) has the finite model property. The modal logics (mathsf {wK4.1T}_0^n) (( n>0)) form an infinite descending chain in the interval ([mathsf {wK4.1},mathsf {K4.1}]) and each of them has the finite model property. Thus all the modal logics (mathsf {wK4.1}) and (mathsf {wK4.1T}_0^n) ((n>0)) are decidable.

We present an extension of simple type theory that incorporates types for any kind of mathematical structure (of any order). We further extend this system allowing isomorphic structures to be identified within these types thanks to some syntactical restrictions; for this purpose, we formally define what it means for two structures to be isomorphic. We model both extensions in NFU set theory in order to prove their relative consistency.

The aim of this paper is to give the first steps towards the formal study of swap structures, which are non-deterministic matrices (Nmatrices) defined over tuples of 0–1 truth values generalizing the notion of twist structures. To do this, a precise notion of clauses which axiomatize bivaluation semantics is proposed. From this specification, a swap structure is naturally induced. This formalization allows to define the combination by fibring of two given logics described by swap structures generated by clauses in a very simple way, by gathering together the formal specifications of both swap structures. We provide simple sufficient conditions to guarantee the preservation by fibring of soundness and completeness w.r.t. Hilbert calculi naturally defined from the clauses, as well as to prove that the fibring is a conservative expansion of both logics. As application examples of this technique, the combination by fibring of some non-normal Ivlev-like modal logics with paraconsistent logics in the class of logics of formal inconsistency (LFIs) are obtained, producing so several paraconsistent modal logics, each of them decidable by a single 6-valued Nmatrix. As expected, the fibring (union) of the respective Hilbert calculi provides a sound and complete axiomatization of the combined logics. More than this, the fibring is the least conservative expansion of the given logics. This technique opens interesting perspectives for combining logics characterized by finite Nmatrices represented by swap structures.

Constructive dualities have recently been proposed for some lattice-based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining “choice-free spatial dualities for other classes of algebras [(ldots )], giving rise to choice-free completeness proofs for non-classical logics”. We present in this article a way to complete the Holliday–Bezhanishvili project (uniformly, for any normal lattice expansion). This is done by recasting in a choice-free manner recent relational representation and duality results by the author. These results addressed the general representation and duality problem for lattices with quasi-operators, extending the Jónsson–Tarski approach for BAOs, and Dunn’s follow-up approach for distributive generalized Galois logics, to contexts where distributivity may not be assumed. To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining correspondence results and canonical extensions in relational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases.

This paper introduces a comprehensive logical framework to reason about threshold-driven diffusion and threshold-driven link change in social networks. It considers both monotonic dynamics, where agents can only adopt new features and create new connections, and non-monotonic dynamics, where agents may also abandon features or cut ties. Three types of operators are combined: one capturing diffusion only, one capturing link change only, and one capturing both at the same time. We first characterise the models on which diffusion of a unique feature and link change stabilise, whilst discussing salient properties of stable models with multiple spreading features. Second, we show that our operators (and any combination of them) are irreplaceable, in the sense that the sequences of model updates expressed by a combination of operators cannot always be expressed using any other operators. Finally, we analyse classes of models on which some operators can be replaced.

In this paper, we propose a computational interpretation of the generalized Kreisel–Putnam rule, also known as the generalized Harrop rule or simply the Split rule, in the style of BHK semantics. We will achieve this by exploiting the Curry–Howard correspondence between formulas and types. First, we inspect the inferential behavior of the Split rule in the setting of a natural deduction system for intuitionistic propositional logic. This will guide our process of formulating an appropriate program that would capture the corresponding computational content of the typed Split rule. Our investigation can also be reframed as an effort to answer the following question: is the Split rule constructively valid in the sense of BHK semantics? Our answer is positive for the Split rule as well as for its newly proposed general version called the S rule.

Hintikka’s game theoretical approach to semantics has been successfully applied also to some non-classical logics. A recent example is Başkent (A game theoretical semantics for logics of nonsense, 2020. arXiv:2009.10878), where a game theoretical semantics based on three players and the notion of dominant winning strategy is devised to fit both Bochvar and Halldén’s logics of nonsense, which represent two basic systems of the family of weak Kleene logics. In this paper, we present and discuss a new game theoretic semantics for Bochvar and Halldén’s logics, GTS-2, and show how it generalizes to a broader family of logics of variable inclusions.

Mereology in its formal guise is usually couched in a language whose signature contains only one primitive binary predicate symbol representing the part of relation, either the proper or improper one. In this paper, we put forward an approach to mereology that uses mereological sum as its primitive notion, and we demonstrate that it is definitionally equivalent to the standard parthood-based theory of mereological structures.

This expository paper presents an application, to the modal logic S4, of the valuation semantics technique proposed by Loparić for the basic normal modal logic K. In previous works we presented a valuation semantics for the minimal temporal logic Kt and several other systems modal and temporal logic. How to deal with S4, however, was left as an open problem—although we arrived at a working definition of (A_1,ldots ,A_n)-valuations, we were not able to prove an important lemma for correctness. In this paper we solve this, presenting valuations for S4.