Summary

Chapter 4 of "Relevant logics and their rivals" (Routley, Meyer, Plumwood and Brady, Ridgeveiw, 1982) is an investigation on De Morgan negations (DM-negations) in Routley-Meyer ternary relational semantics (RM-semantics). "Routley-Meyer ternary relational semantics for intuitionistic-type negations" (Robles and Méndez, Elsevier, 1th January, 2018) is an investigation on intuitionistic-type negations in RMsemantics. The aim of the present proposal is to mirror the two works just cited from the perspective of quasi-Boolean negations (QBnegations), this time. 

Consider the following axioms and rule: a1 (~A&~B)->~(AvB); a2 ~(A&B)->(~Av~B); a3 A-> ~~A; a4 ~~A->A; a5 (A->B)->(~B->~A); a6 (A&~A)->B; a7 B->(Av~A); a8 Av~A; r1 A->B=> ~B->~A.

We define the QB-negations to be investigated. Then, we detail the aim of the proposal. 

We consider two fundamental types of QB-negations. The former is para-intuitionistic in nature; the latter, is dual-intuitionistic in character (H refers to Heyting). We study six varieties of the first type of QB-negations and three of the second one, which can be defined as follows. Let S be a logic endowed with an RM-semantics. QB-negations can be introduced in S by adding some of the axioms and rule recorded above. Then, an RM-semantics is supplied for each extension (or expansion, as the case may be) of S thus defined. The varieties referred to above are the following. 

QBH-negations:  "Minimal QBH-negation" (QBHm-negation): a1, a2, a6 y r1; "Basic QBH-negation" (QBHb-negation): a1, a2, a3, a6 and r1; "Strong QBH-negation" (QBHs-negation): a1, a2, a3, a5 and a6 (r1 is derivable). A strong version of each variety just defined is obtained by adding a8 to each one of them. Thus, we get SQBHm-negation, SQBHb-negation and SQBHs-negation. 

QBDH-negations: "Minimal QBDH-negation" (QBDHm-negation): a1, a2, a7 y r1; "Basic QBDH-negation" (QBDHb-negation): a1, a2, a4, a7 and r1; "Strong QBDH-negation" (QBDHs-negation): a1, a2, a4, a5 and a7 (r1 is derivable). 

The aims of the present proposal are: (1) To provide an RM-semantics for the nine varieties of QB-negations defined above. We give two alternative interpretations of negation. The first one uses the Routley operator, the tool customarily used for interpreting negation in RMsemantics. The second one leans on a falsity constant. (2) To investigate the relationship between QBH-negations, QBDH-negations and DM-negations within the context of RM-semantics.

This project was based at the Universidad de Salamanca and was funded by the Spanish Ministry of Economy, Industry and Competitiviness (MINEICO), [Project FFI2017-82878-P]. Duration: from 2018 to 2020.