Let MK3 be Kleene's strong three-valued matrix (with only one as well as with two designated values). An implicative expansion of MK3 is natural if the following properties are predicable of the function defining the conditional: (1) it satisfies the modus ponens; (2) it assigns a designated value to a conditional if the value assigned to its antecedent is lesser than or equal to that assigned to its consequent; (3) it coincides with (the function defining) the classical conditional when restricted to the classical values. Dunn type bivalent semantics is the semantics originally defined by Dunn for interpreting Anderson and Belnap’s "First Degree Entailment Logic"; and Routley-Meyer type ternary relational semantics is the semantics introduced by the said authors in order to model the logic of relevance. We shall consider the following logics definable upon each EMK3: (1) the set of all valid formulas; (2) the logic determined by the truth-preserving relation; (3) the logic determined by the degree of truth-preserving relation.
The aim of the present proposal is twofold: (a) to define a Dunn type bivalent semantics for all the logics mentioned above in (1), (2) and (3); (b) to define a Routley-Meyer ternary relational semantics for all the logics mentioned above in (1), (2) and (3), provided that they include Routley and Meyer's basic positive logic B+.
This project was based at the Universidad de Salamanca and was funded by the Spanish Ministry of Economy and Competitiveness(MINECO) [Project FFI2014-53919-P]. Duration: from 2015 to 2017.