Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

Abstract

In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with respect to chains defined over the rational unit interval with a special attention to the completeness with respect to the canonical chain, i.e. the algebra over ½0; 1 \ Q where each truth-constant is interpreted in its corresponding rational truth-value. Finally, we study rational completeness results when we restrict ourselves to deductions between the so-called evaluated formulae.

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

“…We denote by RC ev , FSRC ev , SRC ev , QC ev , FSQC ev , SQC ev the restriction of the properties we have been studying in the previous section to evaluated formulae (in the case of continuous t-norm based logics), or to positively evaluated formulae (in the case of WNM-t-norm based logics). 19 These completeness properties are straightforwardly refuted in many cases. Namely, for each * ∈ CONT-fin \ { * G } L * ∀ has not the RC and thus there is a constant-free formula ϕ such that L * ∀ ϕ and |= [0,1] * ϕ, and hence, since ϕ is equivalent to the evaluated formula 1 → ϕ and L * ∀(C) is a conservative expansion of L * ∀, we also have a counterexample to the RC ev of L * ∀(C).…”

“…As it regards to canonical strong completeness for evaluated formulae, both for the real and rational semantics, in [19] it is shown that it almost always fails. Indeed, it is only true for expansions of G, NM, L ⊗ and L when the algebra C of truthconstants satisfies the following topological property: it has no positive sup-accessible points, i.e.…”

“…19 Notice that the convention is sound since in the case of Gödel t-norm (the only t-norm which is continuous and WNM at the same time) the completeness properties restricted to evaluated formulae or to positively evaluated formulae are equivalent. The reason is that 0 is the only negative element.…”

“…The rational completeness properties have been studied in [19]. The results for general formulae are summarized in Table 6.…”

“…First, since the intended semantics for fuzzy logic systems has been that of the algebras defined over the real unit interval, the socalled standard semantics, the emphasis has been put on completeness results with respect to these semantics. This limitation has already disappeared in a few papers (see [12,20,8,19]) where alternative semantics and their associated completeness properties have been explored, in particular, algebras over the rational unit interval and finite chains. Second, the literature on fuzzy logic systems is usually related to what we may call truth-preserving deductive systems, i.e.…”