J. Tinguaro Rodríguez, E. Turunen, D. Ruan, J. Montero:
"Another paraconsistent algebraic semantics for Lukasiewicz-Pavelka logic";
Fuzzy Sets and Systems,
As recently proved in a previous work of Turunen, Tsoukiàs and Öztürk, starting from an evidence pair (a,ba,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,ba,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,ba,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.
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