First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

Abstract

Dedicated to Franco Montagna in the occasion of his 60th birthday. MSC: 03B50 03B52 06B99Keywords: Algebraic logic Mathematical fuzzy logic First-order predicate non-classical logics Residuated lattices T-norm based fuzzy logics Truth-constants a b s t r a c t This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous tnorms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind r → ϕ, where ϕ has no additional truth-constants.

From rational Gödel logic to ultrametric logic

“…First, we prove the weak completeness of RGL * . A different but similar proof can be found in [9,Theorem 4.7]. Proof.…”

On some questions concerning the axiomatisation of WNM-algebras and their subvarieties

“…Nevertheless, several other extensions have been subjected to investigations in the last few years. To name just a few examples, states and connection with probability theory as studied for NM in [7]; connections with others non-classical logics, for instance Nelson's constructive logic, are dealt with in [18]; alternative, temporal semantics, are explored in [13]; extensions with truth constants, as well as the first-order case have been subjected to investigations [12,[26][27][28][29], too. The paper [31] classifies all the subvarieties of nilpotent minimum algebras.…”

Semantic theory of finite lattice-valued propositional logic

“…Furthermore, there are only inference rules by which we can deduce the formulas with truth value 1 from the formulas with truth value 1. The most of fuzzy logics on [0, 1] discussed up to now is of this type [7][8][9][10], which have been designed to formalize the logical background for fuzzy sets. In addition, some important fuzzy logics developed by Wang et al [11][12][13][14][15][16][17] are also based on traditional syntax.…”