Minimal Polynomial Logic: Properties and Extensions

Minimal Polynomial Logic (MPL) is a generalisation of classical propositional logic which allows truth values in the continuous interval [0; 1] and in which propositions are represented by multi-variate polynomials. In previous work [7] we have introduced some properties of MPL and shown that it has two possible semantics: a) it can be considered as a logic for expressing, handling and computing the probability of being true of complex expressions, like Nilsson's probabilistic logic, or b) it can be seen as a logic for expressing, handling and computing degrees of truth, as fuzzy logic. In this paper we report on new important properties of MPL, among which the isomorphism between MPL and propositional logic and a proof theory parallelling natural deduction. We also present a further generalisation of MPL in which the above-mentioned semantics are extended to express and handle probabilities in belief nets and degrees of truth in fuzzy expressions which do not collapse to standard prop..