Kripke Semantics for Logics with BCK (original) (raw)
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Kripke models and intermediate logics
Publications of the Research Institute for Mathematical Sciences, 1970
In [10], Kripke gave a definition of the semantics of the intuitionistic logic. Fitting [2] showed that Kripke's models are equivalent to algebraic models (i.e., pseudo-Boolean models) in a certain sense. As a corollary of this result, we can show that any partially ordered set is regarded as a (characteristic) model of a intermediate logic ^ We shall study the relations between intermediate logics and partially ordered sets as models of them, in this paper. We call a partially ordered set, a Kripke model. 2^ At present we don't know whether any intermediate logic 'has a Kripke model. But Kripke models have some interesting properties and are useful when we study the models of intermediate logics. In §2, we shall study general properties of Kripke models. In §3, we shall define the height of a Kripke model and show the close connection between the height and the slice, which is introduced in [7]. In §4, we shall give a model of LP» which is the least element in n-ih slice S n (see [7]). §1. Preliminaries We use the terminologies of [2] on algebraic models, except the use of 1 and 0 instead of V and /\, respectively. But on Kripke models, we give another definition, following Schiitte [13]. 3) Definition 1.1. If M is a non-empty partially ordered set, then
Simplified Kripke style semantics for some very weak modal logics
Logic and Logical Philosophy, 2010
In the present paper 1 we examine very weak modal logics ½, ½, ½, ˼. • , ˼. • +(D), ˼. and some of their versions which are closed under replacement of tautological equivalents (rte-versions). We give semantics for these logics, formulated by means of Kripke style models of the form w, A, V , where w is a «distinguished» world, A is a set of worlds which are alternatives to w, and V is a valuation which for formulae and worlds assigns the truth-vales such that: (i) for all formulae and all worlds, V preserves classical conditions for truth-value operators; (ii) for the world w and any formula ϕ, V (ϕ, w) = 1 iff ∀x∈A V (ϕ, x) = 1; (iii) for other worlds formula ϕ has an arbitrary value. Moreover, for rte-versions of considered logics we must add the following condition: (iv) V (χ, w) = V (χ[ ϕ / ψ ], w), if ϕ and ψ are tautological equivalent. Finally, for ½, ½ and ½ we must add queer models of the form w, V in which: (i) holds and (ii ′) V (ϕ, w) = 0, for any formula ϕ. We prove that considered logics are determined by some classes of above models.
2022
A non-distributive two-sorted hypersequent calculus PDBL and its modal extension MPDBL are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for PDBL is next proposed, where any formula is interpreted as a semiconcept of a context. For MPDBL, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding appropriate sequents to MPDBL results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. One of these systems is a logic for topological pure double Boolean algebras. It is demonstrated that, using PDBL, the basic notions and relations of conceptual knowledge can be expressed and inferences involving negations can be obtained. Further, drawing a connection with rough set theory, lower and upper approximations of semiconcepts of a context are defined. It is then shown that, using the formulae and sequents involving modal operators in MPDBL, these approximation operators and their properties can be captured.
Relational semantics for full linear logic
Journal of Applied Logic, 2014
Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [5] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form. In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [4,5] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms. Traditionally, so-called phase semantics are used as models for (provability in) linear logic . These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.
Kripke models for classical logic
Annals of Pure and Applied Logic, 2010
We introduce a notion of Kripke model for classical logic for which we constructively prove soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications.
Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics
Studia Logica, 2012
This paper presents an algebraic approach of some many-valued generalizations of modal logic. The starting point is the denition of the [0, 1]-valued Kripke models, where [0, 1] denotes the well known MV-algebra. Two types of structures are used to dene validity of formulas: the class of frames and the class of n-valued frames. The latter structures are frames in which we specify in each world u the set (a subalgebra of n) of the allowed truth values of the formulas in u. We apply and develop algebraic tools (namely, canonical and strong canonical extensions) to generate complete modal n + 1-valued logics and we obtain many-valued counterparts of Shalqvist canonicity result.