A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3 $$_{\text {DH}}$$ DH
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A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3DH Gemma Robles1
· José M. Méndez2
Accepted: 8 October 2020 © Springer Nature B.V. 2020
Abstract The logic DHb is the result of extending Sylvan and Plumwood’s minimal De Morgan logic BM with a dual intuitionistic negation of the type Sylvan defined for the extension CCω of da Costa’s paraconsistent logic Cω. We provide Routley–Meyer ternary relational semantics with a set of designated points for DHb and a wealth of its extensions included in G3DH , the expansion of G3+ with a dual intuitionistic negation of the kind considered by Sylvan (G3+ is the positive fragment of Gödelian 3-valued logic G3). All logics in the paper are paraconsistent. Keywords Dual intuitionistic logics · De Morgan logics · paraconsistent logics · Routley–Meyer ternary relational semantics
1 Introduction The aim of this paper is to introduce the logic DHb and a wealth of its extensions. The kind of negation these logics enjoy will be generally referred to as DH-negation, DH-logics being the general term used to mention DHb and its extensions defined in the sequel. DH-negation can be considered as a dual intuitionistic negation in some sense to be explained in what follows (H stands for Heyting; DH for dual H-negation). DHb is an extension of Sylvan and Plumwood’s minimal De Morgan logic BM , in its turn, an expansion of Routley and Meyer’s basic positive logic B+ (cf. Definitions
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Gemma Robles [email protected] http://grobv.unileon.es José M. Méndez [email protected] http://sites.google.com/site/sefusmendez
1
Dpto. de Psicología, Sociología y Filosofía, Universidad de León, Campus de Vegazana, s/n, 24071 León, Spain
2
Universidad de Salamanca, Edificio FES, Campus Unamuno, 37007 Salamanca, Spain
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G. Robles, J. M. Méndez
2.4 and 2.6). DH-logics are defined by using the list of theses and rules in Lemma 2.13. Of these, a32-a44 concern negation, while a1-a31 are formulated in the negationless logical language. All DH-logics are endowed with an unreduced Routley-Meyer ternary relational semantics (RM-semantics) with a set of designated points (cf. Routley et al. (1982) and Brady (2003)). Concerning da Costa’s paraconsistent logic Cω (cf. da Costa (1974)), system to which we will return below, Richard Sylvan (né Routley) notes that “Cω is in certain respects the dual of intuitionistic logic” (cf. Sylvan (1990), p. 48). In particular, if a semantical point of view is adopted, Sylvan notes (Sylvan 1990, p. 49) “whereas intuitionism is essentially focused on evidentially incomplete situations excluding inconsistent situations, the C-systems admit inconsistent situations but remove incomplete situations” . Well then, contrary to what happens in Kripke models for intuitionistic logic, the models in the RM-semantics we define for the DH-logics are composed exclusively of complete though not necessarily consistent elements, unlike it is the case in standard RM-semantics, where the elements can be incomplete, inconsistent or both. But this duality can also tak
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