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1
Content available remote The quasi-relevant 3-valued logic RM3 and some of its sublogics lacking the variable-sharing property
Robles G.
Reports on Mathematical Logic
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2016
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Vol. 51
105--131
EN
The 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 first aim of this paper is to define 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.
2
Content available remote Paraconsistency and consistency understood as the absence of the negation of any implicative theorem
Robles G.
Reports on Mathematical Logic
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2012
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Vol. 47
147-171
EN
As 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 theo- rems are also proven. All logics in this paper are included in Lewis' S4. They are all paraconsistent, but none of them is relevant.
3
Content available remote Minimal non-relevant logics without the K axiom II. Negation introduced via the unary connective
Robles G.
Reports on Mathematical Logic
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2010
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Vol. 45
97-118
EN
In the first part of this paper (RML No. 42) a spectrum of constructive logics without the K axiom is defined. Nega- tion 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.
4
Content available remote Minimal non-relevant logics without the K axiom
Robles G., Mendez J. M.
Reports on Mathematical Logic
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2007
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No. 42
117-144
EN
The logic B+ is Routley and Meyer's basic positive logic. The logic BK+ is B+plus the K rule. We add to BK+ four intuitionistic-type negations. We show how to extend the resulting logics within the modal and relevance spectra. We prove that all the logics dened lack the K axiom.
5
Content available remote Minimal negation in the ternary relational semantics
Robles G., Méndez J. M., Salto F.
Reports on Mathematical Logic
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2005
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Nr 39
47-65
EN
Minimal Negation is defined within the basic positive relevance logic in the relational ternary semantics: B+. Thus, by defining a number of subminimal negations in the B+ context, principles of weak negation are shown to be isolable. Complete ternary semantics are offered for minimal negation in B+. Certain forms of reductio are conjectured to be undefinable (in ternary frames) without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive logic are offered.
6
Content available remote Anderson and Belnap's minimal positive logic with minimal negation
Mendez J. M., Salto F., Robles G.
Reports on Mathematical Logic
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2002
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No. 36
117-130
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