Fleas and fuzzy logic: a survey

Keywords: mathematical fuzzy logic, basic fuzzy logic, monoidal t-norm logic, hoop logic, non-commutative fuzzy logic. This paper is a companion to my [12]. The latter is a rather technical paper developing a generalized (mathematical) fuzzy logic. Main definitions and results of [12] are reproduced here without any proofs; but this is preceded by a survey on the basic proposi-tional fuzzy logic BL and its generalizations and by an attempt to explain why these generalizations are worth of study. 1. Basic fuzzy logic. (See [8] for details.) The term " fuzzy logic " is understood in its narrow sense – symbolic logical systems, with a comparative notion of truth. We restrict ourselves to truth-functional logic, i.e. semantics of connectives is given by their truth functions. The standard domain of truth degrees is the real unit interval [0, 1] with its natural order. Each continuous t-norm * can be taken as the truth-function of conjunction; thus its residuum is taken to be the truth function of implication. Some other connectives are defined, notably ¬ϕ is ϕ → 0. The three most important continuous t-norms are Lukasiewicz (x * y = max(0, x + y − 1)), Gödel x * y = min(x · y), and product t-norm (x * y = x · y). A continuous t-norm * and an evaluation e of propositional variables unique defines the value e * (ϕ) for any propositional formula ϕ built from the variables using conections &, → and the truth constant 0. The formula ϕ is a t-tautology (tautology of the continuous t-norm logic) of e * (ϕ) = 1 for each e and each *. In general, truth derees need not to be linearly ordered ; the domain of truth degrees may just be a bounded lattice (partially ordered set in which each pair of elements has sup and inf and having a largest and a least element).) is a bounded lattice (with 0 L , 1 L as least and largest element, * is a binary operation which is associative, commutative and 1 * x = x for all x, → is its residuum (i.e. z ≤ x → y iff x * z ≤ y) and the following axioms of divisibility and prelinearity are satisfied: x ∧ y = x * (x → y), (x → y) ∨ (y → x) = 1. These are properties satisfied by each continuous t-norm …