Antoni Torrens

Basic Fuzzy Logic is the logic of continuous t-norms and their residua

Abstract In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0, 1], solving the open problem posed by Hájek in [4]. In fact, Hájek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Gödel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.

Logics preserving degrees of truth from varieties of residuated lattices

A wrong argument in the proof of one of the main results in the paper is corrected. The result itself remains true. The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.

Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic

Abstract. Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras.

On the infinite-valued Łukasiewicz logic that preserves degrees of truth

AbstractŁukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.

On Gentzen Systems Associated with the Finite Linear MV-algebras

In this paper we obtain a characterization of the algebraizability of an m-dimensional Gentzen system in line with the characterization obtained for m-dimensional deductive systems and the characterization of 2-dimensional Gentzen systems. We also prove that if S(m) is the finite linear MV-algebraof m elements, then the m-dimensional Gentzen system obtained by using the sequent calculi associated with S(m) is equivalent to the m-valued Łukasiewicz logic Ł m and to the equational consequence relation associated with S(m). Taking the two-element Boolean algebra we obtain the expected result concerning the relationship between the sequent calculus LK, the Classical Prepositional Calculus and the variety of Boolean algebras.

W-algebras which are Boolean products of members of SR[1] and CW-algebras

Preprint enviat per a la seva publicacio en una revista cientifica: Stud Logica 46, 265–274 (1987). [https://doi.org/10.1007/BF00372551]We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.

Synthetic Applications of 2-(1,3-Dithian-2-yl)indoles. A New Synthetic Approach to Strychnos Alkaloids

The first synthesis of Na-methyl-20-hydroxydasycarpidone and the tetracyclic systems are reported. The synthesis involves an acid cyclization of an appropriate 4-indolyl-carbonyl-2-piperidinecarbonitrile which in turn is obtained regioselectively from the corresponding piperidinol by a modified Polonovski reaction

Cyclic Elements in MV‐Algebras and Post Algebras

In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras. Mathematics Subject Classiflcation: 03G20, 03625, 06D25, 06D30, 06F15, 06F35.

Standard completeness of Hájek basic logic and decompositions of BL-chains

The aim of this paper is to survey the tools needed to prove the standard completeness of Hájek Basic Logic with respect to continuous t-norms. In particular, decompositions of totally ordered BL-algebras into simpler components are considered in some detail.

Bounded BCK‐algebras and their generated variety

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)