Kripke models for classical logic

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

Information and Computation, 1998

We introduce e ectiveness considerations into model theory of intuitionistic logic. We investigate e ectiveness of completeness by Kripke results for intermediate logics such as for example, intuitionistic logic, classical logic, constant domain logic, directed frames logic, Dummett's logic, etc.

2013

iii 0 Vers une théorie des preuves pour la logique classique v 0.1 Catégories des preuves.............................. vi

By introducing the intensional mappings and their properties, we establish a semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_ complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ' weakest characterization property' of intensional mappings. In particular, we show that classical logic has the weakest characterization property cl, which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property cl.

Journal of Logic and Computation, 2001

In this paper we define cut-free hypersequent calculi for some intermediate logics semantically characterized by bounded Kripke models. In particular we consider the logics characterized by Kripke models of bounded width Bw k , by Kripke models of bounded cardinality Bc k and by linearly ordered Kripke models of bounded cardinality G k . The latter family of logics coincides with finite-valued Gödel logics. Our calculi turn out to be very simple and natural. Indeed, for each family of logics (respectively, Bw k , Bc k and G k ), they are defined by adding just one structural rule to a common system, namely the hypersequent calculus for Intuitionistic Logic. This structural rule reflects in a natural way the characteristic semantical features of the corresponding logic. * Partly supported by EC Marie Curie fellowship HPMF-CT-1999-00301

2019

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In partic- ular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to be identified with an infinite sequence of conse- quence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting con- sequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences.

Proceedings of the Japan Academy, 1973

2007

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.

This paper introduces LK, a cut-free sequent calculus able to faithfully characterize classical (propositional) non-theorems, in the sense that a formula α is provable in LK if, and only if, α is not provable in LK (i.e. α is not a tautology) [Tiomkin(1988), Bonatti(1993)]. This calculus is enriched with a set of admissible cut rules, which provide a cut-elimination algorithm. The calculus is also proved to be strong normalizing and strongly confluent.