Another paraconsistent algebraic semantics for Lukasiewicz–Pavelka logic

As recently proved in a previous work of Turunen, Tsoukiàs and Öztürk, starting from an evidence pair ( a , b ) on the real unit square and associated with a propositional statement α, we can construct evidence matrices expressed in terms of four values t, f, k, u that respectively represent the logical valuations true, false, contradiction (both true and false) and unknown (neither true nor false) regarding the statement α. The components of the evidence pair ( a , b ) are to be understood as evidence for and against α, respectively. Moreover, the set of all evidence matrices can be equipped with an injective MV-algebra structure. Thus, the set of evidence matrices can play the role of truth-values of a Lukasiewicz–Pavelka fuzzy logic, a rich and applicable mathematical foundation for fuzzy reasoning, and in such a way that the obtained new logic is paraconsistent. In this paper we show that a similar result can be also obtained when the evidence pair ( a , b ) is given on the real unit triangle. Since the real unit triangle does not admit a natural MV-structure, we introduce some mathematical results to show how this shortcoming can be overcome, and another injective MV-algebra structure in the corresponding set of evidence matrices is obtained. Also, we derive several formulas to explicitly calculate the evidence matrices for the operations associated to the usual connectives.