Logics of Kripke meta-models (original) (raw)

Simplified Kripke-Style Semantics for Some Normal Modal Logics

Studia Logica, 2019

proved that the normal logics K45, KB4 (= KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form W, A , where A ⊆ W. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of K45. Furthermore, a modal logic is a normal extension of K45 (resp. KD45; KB4; S5) if and only if it is determined by a set consisting of finite simplified frames (resp. such frames with A = ∅; such frames with A = W or A = ∅; such frames with A = W). Secondly, for all normal extensions of K45, KB4, KD45 and S5, in particular for extensions obtained by adding the so-called "verum" axiom, Segerberg's formulas and/or their T-versions, we prove certain versions of Nagle's Fact (

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.

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

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.

Products of ‘transitive’modal logics. Part I:‘negative’results

2004

We solve a major open problem concerning algorithmic properties of products of 'transitive' modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if C1 and C2 are classes of transitive frames such that their depth cannot be bounded by any fixed n < ω, then the logic of the class {F1 × F2 | F1 ∈ C1, F2 ∈ C2} is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n < ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π 1 1 -complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics. £ are interpreted by R h , while ¡ and ¤ are interpreted by R v .)

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

Journal of Logic and Computation, 2020

We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, varPi0_1\varPi ^0_1varPi0_1-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics textbfGL\textbf{GL}textbfGL, textbfGrz\textbf{Grz}textbfGrz and textbfKTB\textbf{KTB}textbfKTB—among them, textbfK\textbf{K}textbfK, textbfT\textbf{T}textbfT, textbfD\textbf{D}textbfD, textbfKB\textbf{KB}textbfKB, textbfK4\textbf{K4}textbfK4 and textbfS4\textbf{S4}textbfS4.

Kripke Models of Transfinite Provability Logic

2012

For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no non-trivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variable-free fragment with natural number modalities, denoted GLPω. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each ordinals Θ,Λ we build a Kripke model IΛ and show that GLP 0 Λ is sound for this structure. In our notation, Ignatiev’s original model becomes I0 ω .