An alternative justification of the axioms of fuzzy logics

Libor Behounek
Institute of Computer Science, Academy of Sciences of the Czech Republic
Faculty of Philosophy, Charles University in Prague
behounek@cs.cas.cz

Fuzzy logics are usually introduced by imposing some reasonable constraints on the [0, 1] truth functions of logical connectives, which lead to the notion of (left-)continuous t-norm [1]. However, the Hilbert-style calculi and algebraic semantics of the resulting logics (MTL, BL, Lukasiewicz logic, Goedel logic, etc.) make no reference to the [0, 1] interval, and their completeness w.r.t. the real-valued semantics is often hard to prove. Thus it is reasonable to search for a [0, 1]-independent justification of the axioms of these logics.
The present talk gives a justification which is based on two kinds of fuzzy consequence relations, viz. the ‘global’ (full-truth preserving) one and the ‘local’ (partial-truth preserving) one. The account proceeds in four steps of formalizing their pre-theoretical properties: (i) the intuitive properties of full-truth preservation are those of Tarski structural consequence relations; (ii) the properties of full partial-truth preservation lead to Cintula’s class of weakly implicative fuzzy logics [2]; (iii) the properties of partially valid partial-truth preservation, internalized by implication, generate the implicational fragment of MTL (possibly fine-tuned by additional assumptions); finally (iv) by adding further propositional connectives we arrive at usual fuzzy logics.
This account clarifies the position of fuzzy logics in the logical landscape: they can be characterized as intuitionistic non-contractive prelinear logics: intuitionistic and non-contractive, since the laws of double negation and contraction are often inappropriate for fuzzy propositions; and prelinear, since the property of prelinearity corresponds to ‘measuring’ truth along linear scales, which is the distinctive feature of fuzzy logics [3].

[1]	Hajek P.: Metamathematics of fuzzy logic. Kluwer, Dordrecht 1998.
[2]	Cintula P.: Weakly implicative (fuzzy) logics. To appear in Arch. Math. Logic.
[3]	Behounek L., Cintula P.: Fuzzy logics as the logics of chains. Fuzzy Sets Syst. 157 (2006) 604–610.