Maximality in finite-valued Łukasiewicz logics defined by order filters

Abstract

In this paper we consider the logics L i n obtained from the (n + 1)valued Lukasiewicz logics Ln+1 by taking the order filter generated by i/n as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that L i n is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality between the logics L i n (that is, maximality w.r.t. rules instead of axioms), we provide algebraic arguments in order to show that the logics L i n are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one extension between L i n and CPL obtained by adding to L i n a kind of graded explosion rule. Finally, using these results, we show that the logics L i n with n prime and i/n < 1/2 are ideal paraconsistent logics. arXiv:1803.09815v2 [math.LO]

On the expressive power of Lukasiewicz's square operator

“…Indeed, within the degree-preserving consequence relations all the truth-values play an equally important role. As an intermediate alternative, it is possible to consider matrix logics in which the designated truth-values are given by (products of) order filters, see for instance [12] and [13] for the case of (products of) Lukasiewicz logics or Gödel's logics (possibly expanded with an involution) respectively.…”

“…, (n − 1)/n, 1}. By means of a general result concerning equivalences between logics, based on translations presented in [3], the logic L i n characterized by the logical matrix L n+1 , F i/n (where F i/n is the order filter generated by i/n) is also algebraizable by means of the variety MV n+1 , see [12] 1 .…”

“…Because of this, in [12] we proposed an axiomatization of L 1 3 and L 2 3 in terms of another signature Σ given by ¬, ∨ and * , where * represents the square (w.r.t. the strong Lukasiewicz conjunction ) in L 3+1 , namely * x := x x.…”

“…In turn, this induces a 'classical' (deductive) implication x → i/3 y := ∼ i/3 x ∨ y, obtaining in this way a suitable and very natural language for axiomatizing the logics L i 3 , for i = 1, 2. Despite the success in axiomatizing L 1 3 and L 2 3 in the signature Σ = {∨, ¬, * }, it was observed in [12] that the issue of obtaining a 'natural' axiomatization defined over such signature for every L i n with n > 3 is a problem which "appears to be much more complicated, and certainly it lies outside the scope of this paper" ([12, p. 150]). A crucial feature for the case n = 3 mentioned in [12] is that Lukasiewicz implication can be recovered from such signature.…”

“…Despite the success in axiomatizing L 1 3 and L 2 3 in the signature Σ = {∨, ¬, * }, it was observed in [12] that the issue of obtaining a 'natural' axiomatization defined over such signature for every L i n with n > 3 is a problem which "appears to be much more complicated, and certainly it lies outside the scope of this paper" ([12, p. 150]). A crucial feature for the case n = 3 mentioned in [12] is that Lukasiewicz implication can be recovered from such signature. This feature does not hold for any n, not even for any prime number, as it is the case e.g.…”