41 research outputs found
Unifiability and Structural Completeness in Relation Algebras and in Products of Modal Logic S5
Unifiability of terms (and formulas) and structural completeness in the variety of relation algebras RA and in the products of modal logic S5 is investigated. Nonunifiable terms (formulas) which are satisfiable in varieties (in logics) are exhibited. Consequently, RA and products of S5 as well as representable diagonal-free n-dimensional cylindric algebras, RDfn, are almost structurally complete but not structurally complete. In case of S5ⁿ a basis for admissible rules and the form of all passive rules are provided
Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras
We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, and G operations as well as expansions of some commutative integral residuated lattices with successor operations
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
Intuitionistic logic with a Galois connection has the finite model property
We show that the intuitionistic propositional logic with a Galois connection
(IntGC), introduced by the authors, has the finite model property.Comment: 6 page
Closure algebras of depth two with extremal relations: Their frames, logics, and structural completeness
We consider varieties generated by finite closure algebras whose canonical
relations have two levels, and whose restriction to a level is an "extremal"
relation, i.e. the identity or the universal relation. The corresponding logics
have frames of depth two, in which a level consists of a set of simple clusters
or of one cluster with one or more elements
Remarks on projective unifiers
A projective unifier for a unifiable formula a in a logic L is a unifier a for a (i.e. a substitution making a a theorem of L) such that a —L a(x) o x. Using the result of Burris [3] we observe that every discriminator variety has projective unifiers. Several examples of projective unifiers both in discriminator and in non-discriminator varieties are given. As an application we show that logics with projective unifiers are almost structurally complete, i.e. every admissible rule with unifiable premises is derivable
Unitary unification of S5 modal logic and its extensions
It is shown that all extensions of S5 modal logic, both in the standard formalization and in the formalization with strict implication, as well as all varieties of monadic algebras have unitary unification