We investigate algebraically the fragment of the intuitionistic propositional calculus consisting of equivalence together with conjunction on the intuitionistic regularizations. We find that this fragment is strongly algebraizable with the equivalent algebraic semantics being the variety of equivalential algebras with an additional binary operation that can be interpreted as the meet on regular elements. We give a finite equational base for this variety, and investigate its properties, in particular the commutator. As applications, we prove that the fragment is hereditarily structurally (...) complete and we describe the top of the lattice of its axiomatic extensions. (shrink)

Positive logics are $\{ \wedge, \vee, \to \}$-fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational (...) fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules. (shrink)

We study the variety of equivalential algebras with zero and its subquasivariety that gives the equivalent algebraic semantics for the ‐fragment of intuitionistic propositional logic. We prove that this fragment is hereditarily structurally complete. Moreover, we effectively construct the finitely generated free equivalential algebras with zero.

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.

Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.

A quasivariety of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the JEP and (...) its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if is relatively semisimple then it is generated by one ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics. (shrink)

This paper contains a brief overview of the area of admissible rules with an emphasis on results about intermediate and modal propositional logics. No proofs are given but many references to the literature are provided.

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules (...) in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal. (shrink)

The paper studies admissibility of multiple-conclusion rules in positive logics. Using modification of a method employed by M. Wajsberg in the proof of the separation theorem, it is shown that the problem of admissibility of multiple-conclusion rules in the positive logics is equivalent to the problem of admissibility in intermediate logics defined by positive additional axioms. Moreover, a multiple-conclusion rule \ follows from a set of multiple-conclusion rules \ over a positive logic \ if and only if \ follows from (...) \ over \. (shrink)

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)

Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.

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

In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer (...) a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics. (shrink)

In this paper, we show that the implication fragment of classical propositional logic is finitary for unification with parameters.