12,714 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
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
A view of canonical extension
This is a short survey illustrating some of the essential aspects of the
theory of canonical extensions. In addition some topological results about
canonical extensions of lattices with additional operations in finitely
generated varieties are given. In particular, they are doubly algebraic
lattices and their interval topologies agree with their double Scott topologies
and make them Priestley topological algebras.Comment: 24 pages, 2 figures. Presented at the Eighth International Tbilisi
Symposium on Language, Logic and Computation Bakuriani, Georgia, September
21-25 200
A Topological Study of Contextuality and Modality in Quantum Mechanics
Kochen-Specker theorem rules out the non-contextual assignment of values to
physical magnitudes. Here we enrich the usual orthomodular structure of quantum
mechanical propositions with modal operators. This enlargement allows to refer
consistently to actual and possible properties of the system. By means of a
topological argument, more precisely in terms of the existence of sections of
sheaves, we give an extended version of Kochen-Specker theorem over this new
structure. This allows us to prove that contextuality remains a central feature
even in the enriched propositional system.Comment: 10 pages, no figures, submitted to I. J. Th. Phy
Characteristic formulas over intermediate logics
We expand the notion of characteristic formula to infinite finitely
presentable subdirectly irreducible algebras. We prove that there is a
continuum of varieties of Heyting algebras containing infinite finitely
presentable subdirectly irreducible algebras. Moreover, we prove that there is
a continuum of intermediate logics that can be axiomatized by characteristic
formulas of infinite algebras while they are not axiomatizable by standard
Jankov formulas. We give the examples of intermediate logics that are not
axiomatizable by characteristic formulas of infinite algebras. Also, using the
Goedel-McKinsey-Tarski translation we extend these results to the varieties of
interior algebras and normal extensions of S
Multiple Conclusion Rules in Logics with the Disjunction Property
We prove that for the intermediate logics with the disjunction property any
basis of admissible rules can be reduced to a basis of admissible m-rules
(multiple-conclusion rules), and every basis of admissible m-rules can be
reduced to a basis of admissible rules. These results can be generalized to a
broad class of logics including positive logic and its extensions, Johansson
logic, normal extensions of S4, n-transitive logics and intuitionistic modal
logics
Scopes and Limits of Modality in Quantum Mechanics
We develop an algebraic frame for the simultaneous treatment of actual and
possible properties of quantum systems. We show that, in spite of the fact that
the language is enriched with the addition of a modal operator to the
orthomodular structure, contextuality remains a central feature of quantum
systems.Comment: 9 pages, no figure
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