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 Ha Âjek in [4]. In fact, Ha Âjek 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 Go Èdel 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.Key words many-valued logic (basic, èukasiewicz, product and Go Èdel logics), equational classes, fuzzy logic, t-norms and standard completeness IntroductionBasic Logic (BL for short) and the corresponding algebras (BL-algebras) were introduced by Ha Âjek (see [3] and the references given there) as an attempt to axiomatize the many-valued semantics induced by continuous t-norms on the unit real interval [0, 1].As a ®rst step, Ha Âjek showed that a propositional formula is provable in BL if and only if it is a tautology in any linearly ordered BL-algebra (BL-chain for short). However, completeness of BL w.r.t. the BL algebras in [0, 1] induced by continuous t-norms was left as an open problem in [3].In the recent paper [4], Ha Âjek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axiomatic system of BL. These new axioms are rather unnatural and without a clear logical meaning. Hence it is natural to ask, as Ha Âjek did, whether they are necessary or can be derived from the original axioms. The aim of this paper is to solve this problem, showing that the new axioms are indeed redundant, i.e., that the original axiomatic of BL is complete w.r.t. the continuous t-norms on [0, 1].After a summary in Sect. 2 of main results of [4], Sect. 3 deals with the structure of BL-chains. Axiom B1 was introduced to prevent the existence of some``pathological triples'' in saturated and irreducible BL-chains. We show (Theorem 3.1) that these pathological triples cannot exist in any BL-chain, without using axiom B1. The role of Axiom B2 was to guarantee that a saturated and irreducible BL-chain with zero divisors is an MV-algebra. We present two proofs of this fact (Lemma 3.3 and Remark 2), neither of them depending on B2. In fact, in Theorem 3.4 we show that any saturated and irreducible BL-chain is either a MV chain or a product chain. In this way we obtain an improvement of the decomposition theorem for saturated Theorem 4]: any saturated BL-chain is an ordinal sum whose components are either MV, product or Go Èdel chains. The analogy between this representation and the well-known decomposition theorem for continuous t-no...

Advances in the theory of L algebras

“…BL was indeed shown to be the logic of continuous tnorms and their residua [9,6]. Important many-valued logics, previously studied in mathematical logic in independent settings, such as the Lukasiewicz infinitelyvalued logic or the Gödel-Dummett logic, were then proven to be extensions of BL.…”

On similarity in fuzzy description logics

“…A finite t-norm satisfying the divisibility condition is called a divisible finite t-norm. In [23,Corollary 3.7] it is shown that finite BL-chains are also ordinal sums of finite Łukasiewicz and minimum, and their finite ordinal sums.…”

“…By adding to the axiomatization of MTL the axiom ϕ ∧ ψ → ϕ&(ϕ → ψ) we obtain the logic BL, studied in [35] and proved in [23] to be the logic of all the ordered algebraic structures defined by continuous t-norms and their residua over the unit real interval [0, 1].…”

Fuzzy-Set Based Logics — an History-Oriented Presentation of their Main Developments

“…Moreover McNaughton theorem [McNaughton, 1951] provides a functional description of its logical functions. Many results about Lukasiewicz logic and MV-algebras can be found in the book [Cignoli et al, 1999]. On the other hand, a completeness theorem for Gödel logic was already given in the fifties by Dummett [1959].…”

“…These are the well-known infinitely-valued Lukasiewicz [1930] and Gödel [1932] logics 4 which are the logical systems corresponding to the so-called Lukasiewicz and minimum t-norms and their residuated implications respectively (see, for example, [Cignoli et al, 1999;Gottwald, 2001] for excellent descriptions of these logics). Much later, already motivated by research on fuzzy logic, Product logic, the many-valued logic corresponding to Product t-norm and its residuum, was also axiomatized in [Hájek et al, 1996].…”