Intermediate Logics Research Articles

Computable Kripke Models and Intermediate Logics

We introduce effectiveness considerations into model theory of intuitionistic logic. We investigate effectiveness of completeness (by Kripke) results for intermediate logics such as intuitionistic logic, classical logic, constant domain logic, directed frames logic, and Dummett's logic.

On Extensions of Intermediate Logics by Strong Negation

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n (Λ). This summarizes results that can be found already in [13,14] and [4]. Furthermore, we determine the structure of the lattice of extensions of n(LC).

Intermediate predicate logic without the beth property

It is shown that a logic J fd * characterized by all Kripke frames the domains of all nonmaximal worlds of which are finite lacks the Beth property. The logic is the first example of an intermediate superintuitionistic logic without the Beth property. The interpolation and the Beth properties are also proved missing in all predicate superintuitionistic logics which contain J * and are contained in a logic characterized by frames of the form〈N n , ≤,{Dk}k∈N n〉.

On Some Kripke Complete and Kripke Incomplete Intermediate Predicate Logics

The Kripke-completeness and incompleteness of some intermediate predicate logics is established. In particular, we obtain a Kripke-incomplete logic (H* +A+D+K) where H* is the intuitionistic predicate calculus, A is a disjunction-free propositional formula, D = ∀x(P(x) V Q) ⊃ ∀xP(x) V Q, K = ¬¬∀x(P(x) V ¬P(x)) (the negative answer to a question of T. Shimura).

Basic Propositional Calculus I

AbstractWe present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T → ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If B is a formula such that (T → B) → B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula (T → B) → B is derivable exactly when B is provably equivalent to a formula of the form ((T → A) → A) → (T → A).

Interpolation in superintuitionistic predicate logics with equality

It is shown that the Craig interpolation property and the Beth property are preserved under passage from a superintuitionistic predicate logic to its extension via standard axioms for equality, and under adding formulas of pure equality as new axioms. We find an infinite independent set of formulas which, though not equivalent to formulas of pure equality, may likewise be added as new axiom schemes without loss of the interpolation, or Beth, property. The formulas are used to construct a continuum of logics with equality, which are intermediate between the intuitionistic and classical ones, having the interpolation property. Moreover, an equality-free fragment of the logics constructed is an intuitionistic predicate logic, and formulas of pure equality satisfy all axioms of the classical predicate logic.

Propositional Lax Logic

We investigate a peculiar intuitionistic modal logic, called Propositional Lax Logic (PLL), which has promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness up to behavioural constraints—a central notion in hardware verification—as a logical modality. As a modal logic it is special since it features a single modal operator ○ that has a flavour both of possibility and of necessity. In the paper we provide the motivation for PLL and present several technical results. We investigate some of its proof-theoretic properties, presenting a cut-elimination theorem for a standard Gentzen-style sequent presentation of the logic. We go on to define a new class of fallible two-frame Kripke models for PLL. These models are unusual since they feature worlds with inconsistent information; furthermore, the only frame condition imposed is that the ○-frame be a subrelation of the ⊃-frame. We give a natural translation of these models into Goldblatt's J -space models of PLL. Our completeness theorem for these models yields a Gödel-style embedding of PLL into a classical bimodal theory of type (S4, S4) and underpins a simple proof of the finite model property. We proceed to prove soundness and completeness of several theories for specialized classes of models. We conclude with a brief exploration of two concrete and rather natural types of model from hardware verification for which the modality ○ models correctness up to timing constraints. We obtain decidability of ○-free fragment of the logic of the first type of model, which coincides with the stable form of Maksimova's intermediate logic LΠ .

The relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇.

Superintuitionistic Companions of Classical Modal Logics

This paper investigates partitions of lattices of modal logics based on superintuitionistic logics which are defined by forming, for each superintuitionistic logic L and classical modal logic Θ, the set L[Θ] of L-companions of Θ. Here L[Θ] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize Θ if the law of excluded middle p V ⌍p is added. Questions addressed are, for instance, whether there exist logics with the disjunction property in L[Θ], whether L[Θ] contains a smallest element, and whether L[Θ] contains lower covers of Θ. Positive solutions as concerns the last question show that there are (uncountably many) superclean modal logics based on intuitionistic logic in the sense of Vakarelov [28]. Thus a number of problems stated in [28] are solved. As a technical tool the paper develops the splitting technique for lattices of modal logics based on superintuitionistic logics and ap plies duality theory from [34].

Not Every "Tabular" Predicate Logic is Finitely Axiomatizable

An example of finite tree Mo is presented such that its predicate logic (i.e. the intermediate predicate logic characterized by the class of all predicate Kripke frames based on Mo) is not finitely axiomatizable. Hence it is shown that the predicate analogue of de Jongh - McKay - Hosoi's theorem on the finite axiomatizability of every finite intermediate propositional logic is not true.

Incompleteness Results in Kripke Bundle Semantics

AbstractKripke bundle and C‐set semantics are known as semantics which generalize standard Kripke semantics. In [4] and in [1, 2] it is shown that Kripke bundle and C‐set semantics are stronger than standard Kripke semantics. Also it is true that C‐set semantics for superintuitionistic logics is stronger than Kripke bundle semantics ([6]). Modal predicate logic Q‐S4.1 is not Kripke bundle complete ([3] ‐ it is also yielded as a corollary to Theorem 6.1(a) of the present paper). This is shown by using difference of Kripke bundle semantics and C‐set semantics. In this paper, by using the same idea we show that incompleteness results in Kripke bundle semantics which are extended versions of [2].

Kripke Bundle Semantics and C-set Semantics

Kripke bundle [3] and C-set semantics [1] [2] are known as semantics which generalize standard Kripke semantics. In [3] and in [1], [2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics [5].

Effectiveness of the Completeness Theorem for an Intermediate Logic

Effectiveness of the Completeness Theorem for an Intermediate Logic

Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics

AbstractThis is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systemsLi,i∈I, to form a new systemLI. The methodology ‘fibres’ the semanticsiofLiinto a semantics forLI, and ‘weaves’ the proof theory (axiomatics) ofLiinto a proof system ofLI. There are various ways of doing this, we distinguish by different names such as ‘fibring’, ‘dovetailing’ etc, yielding different systems, denoted byetc. Once the logics are ‘weaved’, further ‘interaction’ axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics.The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property.Some results on combining logics (normal modal extensions ofK) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20–30 years. We hope our methodology will help organise the field systematically.

ON THE PROPOSITIONAL SLDNF-RESOLUTION

We consider propositional logic programs with negations. We define notions of constructive transformation and constructive completion of a program. We use these notions to characterize SLDNF-resolution in classical, intuitionistic and intermediate logics, and also to derive a characterization in modal logics of knowledge. We show that the three-valued and four-valued fix-point or declarative semantics for program P are equivalent to the two-valued semantics for the constructive version of P. We argue that it would be beneficial to replace Negation as Failure by constructive transformation, and it would be beneficial to use the semantics for the constructive version of the program instead of multivalued semantics for the original program.

Predicate superintuitionistic logics without interpolation

Predicate superintuitionistic logics are considered. We prove that all such logics that contain a logic characterized by frames whose domains are all finite and are contained in the classical logic of finite domains do not have the interpolation and Beth properties. It is also established that the interpolation property is not shared by all predicate superintuitionistic logics which contain a logic characterized by frames whose domains of nonfinal worlds are all finite and which are contained in a logic characterized by all two-element frames with finite constant domains.

On variable separation in modal and superintuitionistic logics

In this paper we find an algebraic equivalent of the Hallden property in modal logics, namely, we prove that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras. The joint embedding property of a class of algebras is equivalent to the Pseudo-Relevance Property. We consider connections of the above-mentioned properties with interpolation and amalgamation. Also an algebraic equivalent of of the principle of variable separation in superintuitionistic logics will be found.

On the Independent Axiomatizability of Modal and Intermediate Logics

This paper gives a solution to the old independent axiomatizability problem by presenting normal modal logics above K4 and Grz and an intermediate logic without independent axiomatizations. Incidentally Blot's problem is solved: the lattices of varieties of topological Boolean and pseudo-Boolean algebras are not strongly atomic. We also study the relationship between independent axiomatizability of intermediate logics and their modal companions above S4.

A NEW INCOMPLETENESS RESULT IN KRIPKE SEMANTICS

We prove that a natural and simple predicate modal logic with the Barcan formula, namely QBF-KD plus the axiom of density, is not Kripke complete. Although incompleteness results are known in Kripke semantics, most of the methods used can only apply to logics stronger than QBF-S4 as they are based in a translation from intermediate logics. We give here an original proof of this incompleteness result.

Constructing a continuum of predicate extensions of each intermediate propositional logic

Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose “equality-free fragment” is just the intuitionistic predicate logic.