Note on inter-expressibility of logical connectives in finitely-valued Gödel-Dummett logics

Let Gm be the m-valued Gödel-Dummett fuzzy logic. If m ≥ 3 then neither conjunction nor implication is in Gm expressible in terms of the remaining connectives. This fact remains true even if the propositional language is enriched by propositional constants for all truth values. Gödel-Dummett fuzzy propositional logic can be defined as an extension of the intuitionistic propositional logic by the prelinearity schema (A→B)∨ (B→A). This logic is known to be complete w.r.t. Kripke semantics with linearly ordered frames. Alternatively, it is (even better) known to be complete w.r.t. fuzzy semantics where the truth values can be numbers from the real interval [0, 1], truth functions of conjunction and disjunction are the functions min and max, and the truth function of implication is the function ⇒ defined by a ⇒ b = 1 if a ≤ b and a ⇒ b = b otherwise. Negation ¬ can be considered a basic symbol in Gödel-Dummett logic, or the formula ¬A can be understood as a shorthand for A→⊥, where ⊥ is the symbol for falsity. In any case the truth function of negation is the function a 7→ a⇒ 0; its value is 1 for a = 0 and its value is 0 for any other a. The m-valued Gödel-Dummett logic Gm for m ≥ 2 is defined as Gödel-Dummett logic with an additional restriction that, besides the extremal truth values 0 and 1, only m− 2 intermediate truth values are possible. By convention, the set {1− 1 k ; 2 ≤ k < m} is usually taken for the set of intermediate truth values. If it is the case and if m > 2 then 1 2 is the least intermediate value. Fig. 1 shows the truth functions of &, ∨, →, ¬, in respective order, in the logic G3 having ∗This work is a part of the research plan MSM 0021620839 that is financed by the Ministry of Education of the Czech Republic. †Charles University, Prague, vitezslavdotsvejdaratcunidotcz, http://www1.cuni.cz/ ̃svejdar/. Palachovo nám. 2, 116 38 Praha 1, Czech Republic.