Given a residuated lattice L, we prove that the subset M V (L) of complement elements x * of L generates an M V -algebra if, and only if L is semi-divisible. Riečan states on a semi-divisible residuated lattice L, and Riečan states on M V (L) are essentially the very same thing. The same holds for Bosbach states as far as L is divisible. There are semi-divisible residuated lattices that do not have Bosbach states.
The root of this work is on the one hand in Belnap's four valued paraconsistent logic, and on the other hand on Pavelka's papers further developed by Turunen. We do not introduce a new non-classical logic but, based on a related study of Perny and Tsoukiás, we introduce paraconsistent semantics of Pavelka style fuzzy sentential logic. Restricted to Lukasiewicz t-norm, our approach and the approach of Perny and Tsoukiás 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 atomic formula p two values a and b that can be interpreted as the degrees of pros and cons for p, respectively. Four values t, f, k, u, interpreted as the degrees of the truth, falsehood, contradiction and unknowness of p, respectively, can then be calculated by means of a and b and finally, the degrees of the truth, falsehood, contradiction and unknowness of any well formed formula α are available. The obtained logic is Pavelka style fuzzy sentential logic. In such an approach truth and falsehood are not each others complements. Moreover, we solve some open problems presented in by Perny and Tsoukiás.
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