On inter-expressibility of logical connectives in Gödel fuzzy logic

In Gödel fuzzy propositional logic, neither conjunction nor implication is expressible (definable) in terms of the remaining three logical connectives. In classical propositional logic, propositions are associated with two truth values 1 (truth) and 0 (falsity). In fuzzy logics further (intermediate) truth values lying between 0 and 1 are possible. Usually the truth values are linearly ordered and it is common to assume that they are elements of the real interval [0, 1]. In Gödel fuzzy propositional logic G (see Gödel [5] and/or Hájek [6]), we deal with propositional formulas built up from (say denumerable infinite set of) propositional atoms using the logical connectives & (conjunction), ∨ (disjunction), → (implication) and ¬ (negation). The meaning of the four logical connectives is determined by their truth functions defined as follows. The truth function of conjunction and disjunction are the functions min (minimum) and max (maximum). The truth function of implication is the residuum function ⇒, where x⇒y = y if y < x and x⇒y = 1 otherwise. The truth function of negation is the function − defined by −x = x ⇒ 0 (so, in this paper, the symbol − does not denote subtraction). The real interval [0, 1] equipped with the functions min, max, ⇒ and − is called the standard G-algebra and denoted [0, 1]G. A truth evaluation is a function from propositional atoms to [0, 1]. In full analogy to the classical case, any truth evaluation has a uniquely determined extension defined on all propositional formulas and respecting the truth functions of the standard G-algebra defined above. A propositional formula A is a G-tautology if v(A) = 1 for each truth evaluation v. An example ∗Charles University, Prague, and Institute of Computer Science of the Academy of Sciences of the Czech Republic, Prague, vitezslavdotsvejdaratcunidotcz, http://www1.cuni.cz/ ̃svejdar/. †This paper was supported by grant 401/98/0383 of the Grant Agency of the Czech Republic. ‡Charles University, Prague.