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1.
Hiroakira Ono 《Studia Logica》2012,100(1-2):339-359
This paper discusses Crawley completions of residuated lattices. While MacNeille completions have been studied recently in relation to logic, Crawley completions (i.e. complete ideal completions), which are another kind of regular completions, have not been discussed much in this relation while many important algebraic works on Crawley completions had been done until the end of the 70’s. In this paper, basic algebraic properties of ideal completions and Crawley completions of residuated lattices are studied first in their conncetion with the join infinite distributivity and Heyting implication. Then some results on algebraic completeness and conservativity of Heyting implication in substructural predicate logics are obtained as their consequences. 相似文献
2.
Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss
the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and
establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain
interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
Dedicated to the memory of Willem Johannes Blok 相似文献
3.
Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL
+ is also finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product. 相似文献
4.
On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic 总被引:1,自引:0,他引:1
The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. 相似文献
5.
A Proof of Standard Completeness for Esteva and Godo's Logic MTL 总被引:7,自引:0,他引:7
In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo's logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in. 相似文献
6.
Josep Maria Font 《Studia Logica》2009,91(3):383-406
This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract
algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the
conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage
of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying
these in the particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices
are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the
need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the
idea of truth degree and to some mathematical difficulties. 相似文献
7.
Studia Logica - Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson... 相似文献
8.
Sato Kentaro 《Studia Logica》2008,88(2):295-324
We study filters in residuated structures that are associated with congruence relations (which we call -filters), and develop a semantical theory for general substructural logics based on the notion of primeness for those filters.
We first generalize Stone’s sheaf representation theorem to general substructural logics and then define the primeness of
-filters as being “points” (or stalkers) of the space, the spectrum, on which the representing sheaf is defined. Prime FL-filters
will turn out to coincide with truth sets under various well known semantics for certain substructural logics. We also investigate
which structural rules are needed to interpret each connective in terms of prime -filters in the same way as in Kripke or Routley-Meyer semantics. We may consider that the set of the structural rules that
each connective needs in this sense reflects the difficulty of giving the meaning of the connective. A surprising discovery
is that connectives , ⅋ of linear logic are linearly ordered in terms of the difficulty in this sense.
Presented by Wojciech Buszkowski 相似文献
9.
This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9]. Let L be a residuated lattice, and f : L k ?? L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P(x, y) and Q(x, y) on a residuated lattice L which imply that the function ${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators. 相似文献
10.
Minimal Varieties of Involutive Residuated Lattices 总被引:1,自引:0,他引:1
We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices.
The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the
fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated
lattice.
Dedicated to the memory of Willem Johannes Blok 相似文献
11.
Lou Goble 《Studia Logica》2007,85(2):171-197
The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain
substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In
particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms
to the basic positive relevant logic B∘T, then LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural
logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself.
Presented by Rob Goldblatt 相似文献
12.
We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests. 相似文献
13.
We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results with other works and show some applications of the equivalence.
相似文献14.
Kosta Došen 《Journal of Philosophical Logic》1992,21(3):283-336
Substructural logics are logics obtained from a sequent formulation of intuitionistic or classical logic by rejecting some structural rules. The substructural logics considered here are linear logic, relevant logic and BCK logic. It is proved that first-order variants of these logics with an intuitionistic negation can be embedded by modal translations into S40type extensions of these logics with a classical, involutive, negation. Related embeddings via translations like the double-negation translation are also considered. Embeddings into analogues of S4 are obtained with the help of cut climination for sequent formulations of our substructural logics and their modal extensions. These results are proved for systems with equality too. 相似文献
15.
The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of
propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class
of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed
under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological
Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators
in the above-mentioned classes.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
16.
《Journal of Applied Logic》2008,6(4):609-626
Extended-order algebras are defined, whose operation extends the order relation of a poset with a greatest element. Most implicative algebras, including Hilbert algebras and BCK algebras fall within this context. Several classes of extended-order algebras are considered that lead to most well known multiplicative ordered structures by means of adjunction, once the completion process due to MacNeille is applied. In particular, complete distributive extended-order algebras are considered as a generalization of complete residuated lattices, to provide a structure that suits quite well for many-valued mathematics. 相似文献
17.
A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR
+ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices. 相似文献
18.
Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic (Belnap, 1982). However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics. 相似文献
19.
We introduce modal propositional substructural logics with strong negation, and prove the completeness theorems (with respect to Kripke models) for these logics. 相似文献
20.
Synthese - Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named... 相似文献