An Approach to Paraconsistent Multivalued Logic: Evaluation by Complex Truth Values

The main purpose of the paper is to connect some kind of dialetheism to the use of complex truth values, with new definitions of basic truth-functional connectives that allow for p, \not p to both be true. 'True' is interpreted as vertical bar p vertical bar = 1, 'False' as vertical bar p vertical bar = 0; other values are dispensed with. New definitions of basic truth-functional connectives then allow for "p and not p" to be true. A propositional logic is discussed with the set of connectives including negation, conjunction, disjunction, implication, concordance, discordance, complementary, and equivalence. The authors introduce truth values of propositions, which belong to a subset E, of an uncountable semi-ring F and valuations of propositions, which can be obtained from truth values with the help of a function V : E -> [0, 1] satisfying simple properties. Finally, a paraconsistent Boolean logic is introduced.