555 research outputs found
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
Singly generated quasivarieties and residuated structures
A quasivariety K of algebras has the joint embedding property (JEP) iff it is
generated by a single algebra A. It is structurally complete iff the free
countably generated algebra in K can serve as A. A consequence of this demand,
called "passive structural completeness" (PSC), is that the nontrivial members
of K all satisfy the same existential positive sentences. We prove that if K 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 K is
relatively semisimple then it is generated by one K-simple algebra. It is a
minimal quasivariety if, moreover, it is PSC but fails to unify some finite set
of equations. We also prove that a quasivariety of finite type, with a finite
nontrivial member, is PSC iff 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
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
A note on drastic product logic
The drastic product is known to be the smallest -norm, since whenever . This -norm is not left-continuous, and hence it
does not admit a residuum. So, there are no drastic product -norm based
many-valued logics, in the sense of [EG01]. However, if we renounce standard
completeness, we can study the logic whose semantics is provided by those MTL
chains whose monoidal operation is the drastic product. This logic is called
in [NOG06]. In this note we justify the study of this
logic, which we rechristen DP (for drastic product), by means of some
interesting properties relating DP and its algebraic semantics to a weakened
law of excluded middle, to the projection operator and to
discriminator varieties. We shall show that the category of finite DP-algebras
is dually equivalent to a category whose objects are multisets of finite
chains. This duality allows us to classify all axiomatic extensions of DP, and
to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure
Uniform interpolation and coherence
A variety V is said to be coherent if any finitely generated subalgebra of a
finitely presented member of V is finitely presented. It is shown here that V
is coherent if and only if it satisfies a restricted form of uniform deductive
interpolation: that is, any compact congruence on a finitely generated free
algebra of V restricted to a free algebra over a subset of the generators is
again compact. A general criterion is obtained for establishing failures of
coherence, and hence also of uniform deductive interpolation. This criterion is
then used in conjunction with properties of canonical extensions to prove that
coherence and uniform deductive interpolation fail for certain varieties of
Boolean algebras with operators (in particular, algebras of modal logic K and
its standard non-transitive extensions), double-Heyting algebras, residuated
lattices, and lattices
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