Gemma Robles
Reports on Mathematical Logic, Number 45, 2010, pp. 97-118
In the first part of this paper (RML No. 42) a spectrum of constructive logics without the K axiom is defined. Negation is introduced with a propositional falsity constant. The aim of this second part is to build up logics definitionally equivalent to those displayed in the first part, negation being now introduced as a primitive unary connective. Relational ternary semantics is provided for all logics defined in the paper.
Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem
Gemma Robles
Reports on Mathematical Logic, Number 47, 2012, pp. 147-171
https://doi.org/10.4467/20842589RM.12.007.0688As is stated in its title, in this paper consistency is understood as the absence of the negation of any implicative theorem. Then, a series of logics adequate to this concept of consistency is defined within the context of the ternary relational semantics with a set of designated points, negation being modelled with the Routley operator. Soundness and completeness theorems are provided for each one of these logics. In some cases, strong (i.e., in respect of deducibility) soundness and completeness theorems are also proven. All logics in this paper are included in Lewis’ S4. They are all paraconsistent, but none of them is relevant.
Gemma Robles
Reports on Mathematical Logic, Number 51, 2016, pp. 105-131
https://doi.org/10.4467/20842589RM.16.008.5285The logic RM3 is the 3-valued extension of the logic R-Mingle (RM). RM (and so, RM3) does not have the variable- sharing property (vsp), but RM3 (and so, RM) lacks the more "offending" paradoxes of relevance", such as A → (B → A) or A → (A → B). Thus, RM and RM3 can be useful when some relevance", but not the full vsp, is needed. Sublogics of RM3 with the vsp are well known, but this is not the case with those lacking this property. The rst aim of this paper is to dene an ample family of sublogics of RM3 without the vsp. The second one is to provide these sublogics and RM3 itself with a general Routley-Meyer semantics, that is, the semantics devised for relevant logics in the early seventies of the past century.