Intermediate Logics Research Articles

A note on admissible rules and the disjunction property in intermediate logics

With any structural inference rule A/B, we associate the rule $${(A \lor p)/(B \lor p)}$$ , providing that formulas A and B do not contain the variable p. We call the latter rule a join-extension ( $${\lor}$$ -extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a $${\lor}$$ -extension of any admissible rule is also admissible in this logic. We investigate intermediate logics, in which the $${\lor}$$ -extension of each admissible rule is admissible. We prove that any structural finitary consequence operator (for intermediate logic) can be defined by a set of $${\lor}$$ -extended rules if and only if it can be defined through a set of well-connected Heyting algebras of a corresponding quasivariety. As we exemplify, the latter condition is satisfied for a broad class of algebraizable logics.

Modal Logics of Stone Spaces

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axiomatization of the intermediate logic of each metrizable Stone space and of each extremally disconnected Stone space. In particular, we obtain that the intuitionistic logic is the logic of the Pelczynski compactification of the natural numbers and the logic of weak excluded middle is the logic of the Gleason cover of the Cantor space.

New constants in two pretabular superintuitionistic logics

We give a semantic description of all Novikov complete extensions of Dummett’s logic in a language with several extra constants, and of all Novikov complete extensions of a logic L2 in a language with one extra constant.

Unification and projectivity in Fregean varieties

In some varieties of algebras one can reduce the question of finding most general unifiers (mgus) to the problem of the existence of unifiers that fulfill the additional condition called projectivity. In this paper we study this problem for Fregean (1-regular and orderable) varieties that arise from the algebraization of fragments of intuitionistic or intermediate logics. We investigate properties of Fregean varieties, guaranteeing either for a given unifiable term or for all unifiable terms, that projective unifiers exist. We indicate the identities which fully characterize congruence permutable Fregean varieties having projective unifiers. In particular, we show that for such a variety there exists the largest subvariety that have projective unifiers.

On Lob algebras, II

We study the variety of Lob algebras, the algebraic structures associated with Formal Propositional Calculus. Among other things, we show that there exist only two maximal intermediate logics in the lattice of intermediate logics over Basic Propositional Calculus. We introduce countably many locally finite sub-varieties of the variety of Lob algebras and show that their corresponding intermediate logics have the interpolation property. Finally, we characterize all chain basic algebras with empty set of generators, and show that there are continuum many such chain basic algebras.

Omitting Types in an Intermediate Logic

We prove an omitting types theorem and one direction of the related Ryll-Nardzewski theorem for semi-classical theories introduced in [2].

Connected modal logics

We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R.

Admissible rules in the implication–negation fragment of intuitionistic logic

Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication–negation fragments of intuitionistic logic IPC and its consistent axiomatic extensions (intermediate logics). A Kripke semantics characterization is given for the (hereditarily) structurally complete implication–negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of IPC form a PSPACE-complete set and have no finite basis.

Logics without the contraction rule and residuated lattices

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.

Inquisitive Logic

This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).

Joint consistency in extensions of the minimal logic

Analogs of Robinson’s theorem on joint consistency are found which are equivalent to the weak interpolation property (WIP) in extensions of Johansson’s minimal logic J. Although all propositional superintuitionistic logics possess this property, there are J-logics without WIP. It is proved that the problem of the validity of WIP in J-logics can be reduced to the same problem over the logic Gl obtained from J by adding the tertium non datur. Some algebraic criteria for validity of WIP over J and Gl are found.

Canonical rules

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules overK4 orIPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

A Remark on Superintuitionistic Predicate Logics of Kripke Frames with Constant and with Nested Domains

The superintuitionistic predicate logics (without or with equality) of all predicate Kripke frames with nested domains over a fixed poset W (a set of possible worlds) are embeddable in the logic (without equality) of all Kripke frames with constant domains over W. Therefore, Takano's result [13] on finite axiomatizability of the logic of Kripke frames with constant domains over the set of real numbers implies the recursive axiomatizability of the corresponding logics with nested domains. Other consequences are mentioned as well.

AN ALGEBRAIC APPROACH TO CANONICAL FORMULAS: INTUITIONISTIC CASE

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧,→) homomorphisms, (∧,→, 0) homomorphisms, and (∧,→,∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s theorem that each intermediate logic can be axiomatized by canonical formulas. §

Problem of restricted interpolation in superintuitionistic and some modal logics

A restricted interpolation property IPR is investigated in modal and superintuitionistic logics. The problem of description of logics with IPR over the intuitionistic logic Int and the modal Grzegorczyk logic Grz is solved. It is proved that in extensions of Int or Grz IPR is equivalent to the projective Beth property PB2. It follows that IPR is decidable over Int and strongly decidable over Grz.

Table admissible inference rules

A recursive basis of inference rules is described which are instantaneously admissible in all table (residually finite) logics extending one of the logics Int and Grz. A rather simple semantic criterion is derived to determine whether a given inference rule is admissible in all table superintuitionistic logics, and the relationship is established between admissibility of a rule in all table (residually finite) superintuitionistic logics and its truth values in Int.

Interpolation Properties, Beth Definability Properties and Amalgamation Properties for Substructural Logics

This article develops a comprehensive study of various types of interpolation properties and Beth definability properties (BDPs) for substructural logics, and their algebraic characterizations through amalgamation properties (APs) and epimorphisms surjectivity. In general, substructural logics are algebraizable but lack many of the basic logical properties that modal and superintuitionistic logics enjoy [Gabbay and Maksimova (2005, Oxford Logic Guides, Vol. 46)]. In this case, careful examination is necessary to see how these logical and algebraic properties are related. To describe these relations exactly, many variants of interpolation properties and BDPs, and also corresponding algebraic properties, are introduced. Because of their generality, the results reported here hold not only for substructural logics, but can also be extended to a more general setting such as abstract algebraic logic [Andreka, Nemeti and Sain (Handbook of Philosophical Logic, Vol. 2, 2nd edn, pp. 133–247) and Czelakowski and Pigozzi (1999, Vol. 203 of Lecture Notes in Pure and Applied Mathematics, pp. 187–265)].

Restricted interpolation property in superintuitionistic logics

The restricted interpolation property IPR in modal and superintuitionistic logics is investigated. It is proved that in superintuitionistic logics of finite slices and in finite-slice extensions of the Grzegorczyk logic, the property IPR is equivalent to the projective Beth property PB2.

Completeness and incompleteness for intuitionistic logic

AbstractWe call a logicregularfor a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmeticHASand the intuitionistic set theoryIZFprove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive thetertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski, Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle.

Elementary Amalgamation and Joint Embedding Property for Intermediate Logics

In this paper we study the elementary amalgamation property (AP) and the joint embedding property (JEP) for intermediate logics. We point out the class of Kripke structures with elementary embedding can be viewed within abstract elementary class framework. Following this approach, both elementary AP and JEP can be considered quite naturally for intermediate logics. The main method for our investigations is the extension of Morleyization method from classical model theory to Kripke model theory. The almost-classical logic and almost-classical models have been defined. After verifying that the class of almost-classical models forms an abstract elementary class, we, furthermore, prove that the almost-classical logic neither has the elementary AP nor JEP property. We finally give an example of a non-classical intermediate logic which extends the almost-classical logic and has both elementary AP and JEP. MSC 2000: 03C95, 03C90, 03B55.