57 research outputs found
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
First-order Nilpotent Minimum Logics: first steps
Following the lines of the analysis done in [BPZ07, BCF07] for first-order
G\"odel logics, we present an analogous investigation for Nilpotent Minimum
logic NM. We study decidability and reciprocal inclusion of various sets of
first-order tautologies of some subalgebras of the standard Nilpotent Minimum
algebra. We establish a connection between the validity in an NM-chain of
certain first-order formulas and its order type. Furthermore, we analyze
axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The
introduction section has been rewritten, and many modifications have been
done to improve the readability; moreover, numerous references have been
added. Concerning the technical side, some proofs has been shortened or made
more clear, but the mathematical content is substantially the same of the
previous versio
Canonical formulas for k-potent commutative, integral, residuated lattices
Canonical formulas are a powerful tool for studying intuitionistic and modal
logics. Actually, they provide a uniform and semantic way to axiomatise all
extensions of intuitionistic logic and all modal logics above K4. Although the
method originally hinged on the relational semantics of those logics, recently
it has been completely recast in algebraic terms. In this new perspective
canonical formulas are built from a finite subdirectly irreducible algebra by
describing completely the behaviour of some operations and only partially the
behaviour of some others. In this paper we export the machinery of canonical
formulas to substructural logics by introducing canonical formulas for
-potent, commutative, integral, residuated lattices (-).
We show that any subvariety of - is axiomatised by canonical
formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
Logics of left variable inclusion and PĹ‚onka sums of matrices
The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic ⊢. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic ⊢ is related to the construction of Płonka sums of the matrix models of ⊢. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy
Polyatomic Logics and Generalised Blok-Esakia Theory
This paper presents a novel concept of a Polyatomic Logic and initiates its
systematic study. This approach, inspired by Inquisitive semantics, is obtained
by taking a variant of a given logic, obtained by looking at the fragment
covered by a selector term. We introduce an algebraic semantics for these
logics and prove algebraic completeness. These logics are then related to
translations, through the introduction of a number of classes of translations
involving selector terms, which are noted to be ubiquitous in algebraic logic.
In this setting, we also introduce a generalised Blok-Esakia theory which can
be developed for special classes of translations. We conclude by showing some
systematic connections between the theory of Polyatomic Logics and the general
Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure
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