On product logic with truth constants

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Authors:
Petr Savický
Roberto Cignoli
Francesc Esteva
Lluís Godo
Carles Noguera
Title: On product logic with truth constants
Journal: Journal of Logic and Computation
Volume 16
Number 2
Pages: 205-225
Year: 2006
Preprint



Abstract

Product Logic is an axiomatic extension of Hájek's Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication are interpreted by the product of reals in [0, 1] and its residuum respectively. In this paper we investigate expansions of Product Logic by adding into the language a countable set of truth-constants (one truth-constant for each r in a countable -subalgebra of [0, 1]) and by adding the corresponding book-keeping axioms for the truthconstants. We first show that the corresponding logics Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Pi(\mathcal{C})} are algebraizable, and hence complete with respect to the variety of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Pi(\mathcal{C})} -algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Pi(\mathcal{C})} are exactly the 1-tautologies of the algebra defined over the real unit interval where the truth-constants are interpreted as their own values. It is also shown that they do not enjoy the canonical strong standard completeness, but they enjoy it for finite theories when restricted to evaluated -formulas of the kind , where is a truth-constant and a formula not containing truth-constants. Finally we consider the logics Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Pi_\Delta(\mathcal{C})} , the expansion of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Pi(\mathcal{C})} with the well-known Baaz's projection connective , and we show canonical finite strong standard completeness for them.