A Para Consistent Fuzzy Logic

The root of this work is in Belnap's four valued para consistent logic [2]. Based on a related study of Perny and Tsoukias [11], we introduce para consistent Pavelka style fuzzy sentential logic. Restricted to Lukasiewicz t-norm, our approach and the approach in [11] partly overlap; the main difference lies in the interpretation of the logical connectives implication and negation. The essential mathematical tool proved in this paper is a one-one correspondence between evidence couples and evidence matrices that holds in all injective MV-algebras. Evidence couples associate to each formula a two values a and b that can be interpreted as the degrees of pros and cons for alpha, respectively. Four values t, f, k, u, interpreted as the degrees of truth, falsehood, contradiction and unknowness of a, respectively, can be calculated. In such an approach truth and falsehood are not each others complements. This paper can be seen as a solution to some open problems presented in [11].