New Dimensions on Translations Between Logics

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: (conservative) translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another. Dedicated to the memory of Mário Tourasse Teixeira and Antonio Mário Sette     

Categorical abstract algebraic logic: Gentzen π ‐institutions and the deduction‐detachment property

Given a π ‐institution I , a hierarchy of π ‐institutions I ^{(n )} is constructed, for n ≥ 1. We call I ^{(n )} the n‐th order counterpart of I . The second‐order counterpart of a deductive π ‐institution is a Gentzen π ‐institution, i.e. a π ‐institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I ⁽²⁾ of I is also called the “Gentzenization” of I . In the main result of the paper, it is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction‐detachment property . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Light affine lambda calculus and polynomial time strong normalization

Light Linear Logic (LLL) and Intuitionistic Light Affine Logic (ILAL) are logics that capture polynomial time computation. It is known that every polynomial time function can be represented by a proof of these logics via the proofs-as-programs correspondence. Furthermore, there is a reduction strategy which normalizes a given proof in polynomial time. Given the latter polynomial time “weak” normalization theorem, it is natural to ask whether a “strong” form of polynomial time normalization theorem holds or not. In this paper, we introduce an untyped term calculus, called Light Affine Lambda Calculus (λLA), which corresponds to ILAL. λLA is a bi-modal λ-calculus with certain constraints, endowed with very simple reduction rules. The main property of LALC is the polynomial time strong normalization: any reduction strategy normalizes a given λLA term in a polynomial number of reduction steps, and indeed in polynomial time. Since proofs of ILAL are structurally representable by terms of λLA, we conclude that the same holds for ILAL. This is a full version of the paper [21] presented at LICS 2001.     

On contraction and the modal fragment

We observe that removing contraction from a standard sequent calculus for first‐order predicate logic preserves completeness for the modal fragment. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two fuzzier implication operators in the theory of fuzzy power sets

The theory of fuzzy power sets requires the use of an implication operator acting within the set of values taken by the membership functions of the fuzzy sets. Two such operators and resulting relationships between fuzzy sets are studied here, and the results compared with previous ones obtained with other implication operators.     

Handling the valuation of the predicates in a fuzzy model

Given a fuzzy model, and therefore a graded valuation of the predicates, several handlings of such a valuation are possible. For example, we can change the set of truth values, modify the scale, cut the predicates, and identify truth values. In this exploratory paper we analyze some of the properties preserved under these handlings. To do this we refer to an approach to fuzzy semantics in which the valuation structures can vary freely inside a given type.     

A survey of fuzzy implication algebras and their axiomatization

The theory of fuzzy implication algebras was proposed by Professor Wangming Wu in 1990. The present paper reviews the following two aspects of studies on FI-algebras: concepts, properties and some subclasses of FI-algebras; axiomatization of the class of FI-algebras and some of its important subclasses. The main results are summarized in the current paper, the relationships between FI-algebras and several classes of important fuzzy algebras are discussed, such as BL-algebras, MTL-algebras, and residuated lattices, and propositional calculus systems of several special classes of FI-algebras are shown.