The aim of this paper is to investigate a Curry-Howard interpretation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable disjunctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that --typed (...) combinatory logic terms are proved to completely codify the associated logical proofs. (shrink)

We extend to the predicate frame a previous characterization of the maximal intermediate propositional constructive logics. This provides a technique to get maximal intermediate predicate constructive logics starting from suitable sets of classically valid predicate formulae we call maximal nonstandard predicate constructive logics. As an example of this technique, we exhibit two maximal intermediate predicate constructive logics, yet leaving open the problem of stating whether the two logics are distinct. Further properties of these logics will be also investigated.

A term td is called a ternary deductive term for a variety of algebras V if the identity td≈r holds in V and ∈θ yields td≈td for any A∈V and any principal congruence θ on A. A connective f is called td-distributive if td)≈ f,…,td). If L is a propositional logic and V is a corresponding variety that has a TD term td, then any admissible in L rule, the premises of which contain only td-distributive operations, is derivable, and the (...) substitution r↦td is a projective L-unifier for any formula containing only td-distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic. (shrink)

Several proof-theoretic notions of validity have been proposed in the literature, for which completeness of intuitionistic logic has been conjectured. We define validity for intuitionistic propositional logic in a way which is common to many of these notions, emphasizing that an appropriate notion of validity must be closed under substitution. In this definition we consider atomic systems whose rules are not only production rules, but may include rules that allow one to discharge assumptions. Our central result shows that Harrop’s rule (...) is valid under substitution, which refutes the completeness conjecture for intuitionistic logic. (shrink)

Metacompleteness is used to prove properties such as the disjunction property and the existence property in the area of relevant logics. On the other hand, the disjunction property of several basic propositional substructural logics over FL has been proved using the cut elimination theorem of sequent calculi and algebraic characterization. The present paper shows that Meyer’s metavaluational technique and Slaney’s metavaluational technique can be applied to basic predicate intuitionistic substructural logics and basic predicate involutive substructural logics, respectively. As a corollary (...) of metacompleteness, the disjunction property, the existence property, and the admissibility of certain rules in such logics can be proved. (shrink)

With each superintuitionistic propositional logic L with a disjunction property we associate a set of modal logics the assertoric fragment of which is L . Each formula of these modal logics is interdeducible with a formula representing a set of rules admissible in L . The smallest of these logics contains only formulas representing derivable in L rules while the greatest one contains formulas corresponding to all admissible in L rules. The algebraic semantic for these logics is described.

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).

Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined by \ is derivable in \. It is known that any (...) logic can be defined by a structurally complete deductive system—its structural completion. The main goal of the paper is to study the following problem: given a superintuitionistic logic L, is the structural completion of L hereditarily structurally complete? It is shown that, on the one hand, there is continuum many of such logics, including \, and many of its standard extensions. On the other hand, there is continuum many superintutitionistic logics structural completion of which is not hereditarily structurally complete. It is observed that the class of hereditarily structurally complete superintuitionistic consequence relations does not have the smallest element and it contains continuum many members lacking the finite model property. The following statement is instrumental in obtaining negative results: if a Lindenbaum algebra of formulas on one variable is finite and has more than 15 elements, then a structural completion of such a logic is not hereditarily structurally complete. (shrink)

The Visser rules form a basis of admissibility for the intuitionistic propositional calculus. We show how one can characterize the existence of covers in certain models by means of formulae. Through this characterization, we provide a new proof of the admissibility of a weak form of the Visser rules. Finally, we use this observation, coupled with a description of a generalization of the disjunction property, to provide a basis of admissibility for the intermediate logics BD2 and GSc.

This paper discusses the relation between the minimal positive relevant logic B and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking BT of B, which saves implication and conjunction but drops disjunction . The filter models of the -calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models (...) of these calculi. (The logician's "theory" is the algebraist's "filter".) The coincidence extends to a dual interpretation of key particles—the subtype translates to provable , type intersection to conjunction , function space to implication, and whole domain to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper, and to be expected that type union should correspond to the logical disjunction of B. But the simulation of functional application by a fusion (or modus ponens product) operation on theories leaves the key Bubbling lemma of work on ITD unprovable for the -prime theories now appropriate for the modeling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of and CL. (shrink)

Conventionalism about logic is the view that logical principles hold in virtue of some linguistic conventions. According to explicit conventionalism, these conventions have to be stipulated explicitly. Explicit conventionalism is subject to a famous criticism by Quine, who accused it of leading to an infinite regress. In response to the criticism, several authors have suggested reconstructing conventionalism as implicit in our linguistic behaviour. In this paper, drawing on a distinction from proof theory between derivable and admissible rules, I argue that (...) implicit conventionalism has not been stated in a sufficiently precise way, as it leaves open what one needs to say about admissible yet underivable rules. Moreover, it turns out that this challenge cannot be easily met, and that any attempt to meet the challenge makes conventionalism much less attractive a thesis. (shrink)

While specifications and verifications of concurrent systems employ Linear Temporal Logic , it is increasingly likely that logical consequence in image will be used in the description of computations and parallel reasoning. Our paper considers logical consequence in the standard image with temporal operations image and image . The prime result is an algorithm recognizing consecutions admissible in image, so we prove that image is decidable w.r.t. admissible inference rules. As a consequence we obtain algorithms verifying the validity of consecutions (...) in image and solving the satisfiability problem. We start by a simple reduction of logical consecutions of image to equivalent ones in the reduced normal form . Then we apply a semantic technique based on image-Kripke structures with formula definable subsets. This yields necessary and sufficient conditions for a consecution to be not admissible in image. These conditions lead to an algorithm which recognizes consecutions admissible in image by verifying the validity of consecutions in special finite Kripke structures of size square polynomial in reduced normal forms of the consecutions. As a consequence, this also solves the satisfiability problem for image. (shrink)

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one (...) of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC. (shrink)