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##### The logic determined by Smiley’s matrix for Anderson and Belnap’s first-degree entailment logic with Gemma Robles Journal of Applied Non-Classical Logics 26 (1): 47-68. 2016.
The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and Belnap’s lo…Read more
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##### Strong paraconsistency and the basic constructive logic for an even weaker sense of consistency with Gemma Robles Journal of Logic, Language and Information 18 (3): 357-402. 2009.
In a standard sense, consistency and paraconsistency are understood as the absence of any contradiction and as the absence of the ECQ (‘E contradictione quodlibet’) rule, respectively. The concepts of weak consistency (in two different senses) as well as that of F -consistency have been defined by the authors. The aim of this paper is (a) to define alternative (to the standard one) concepts of paraconsistency in respect of the aforementioned notions of weak consistency and F -consistency; (b) to…Read more
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##### Relevance logics and intuitionistic negation with Gemma Robles Journal of Applied Non-Classical Logics 18 (1): 49-65. 2008.
The logic B+ is Routley and Meyer's basic positive logic. We show how to introduce a minimal intuitionistic negation and an intuitionistic negation in B+. The two types of negation are introduced in a wide spectrum of relevance logics built up from B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule.
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##### Curry’s Paradox, Generalized Modus Ponens Axiom and Depth Relevance with Gemma Robles Studia Logica 102 (1): 185-217. 2014.
“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox
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##### A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes with Gemma Robles Logica Universalis 9 (4): 501-522. 2015.
The aim of this paper is to introduce an alternative to Łukasiewicz’s 4-valued modal logic Ł. As it is known, Ł is afflicted by “Łukasiewicz type paradoxes”. The logic we define, PŁ4, is a strong paraconsistent and paracomplete 4-valued modal logic free from this type of paradoxes. PŁ4 is determined by the degree of truth-preserving consequence relation defined on the ordered set of values of a modification of the matrix MŁ characteristic for the logic Ł. On the other hand, PŁ4 is a rich logic i…Read more
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##### A Routley-Meyer semantics for relevant logics including TWR plus the disjunctive syllogism with Gemma Robles Logic Journal of the IGPL 19 (1): 18-32. 2011.
We provide Routley-Meyer type semantics for relevant logics including Contractionless Ticket Entailment TW (without the truth constant t and o) plus reductio R and Ackermann’s rule γ (i.e., disjunctive syllogism). These logics have the following properties. (i) All have the variable sharing property; some of them have, in addition, the Ackermann Property. (ii) They are stable. (iii) Inconsistent theories built upon these logics are not necessarily trivial.
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##### A constructive negation defined with a negation connective for logics including Bp+ with Gemma Robles and Francisco Salto Bulletin of the Section of Logic 34 (3): 177-190. 2005.
The concept of constructive negation we refer to in this paper is (minimally) intuitionistic in character (see [1]). The idea is to understand the negation of a proposition A as equivalent to A implying a falsity constant of some sort. Then, negation is introduced either by means of this falsity constant or, as in this paper, by means of a propositional connective defined with the constant. But, unlike intuitionisitc logic, the type of negation we develop here is, of course, devoid of paradoxes …Read more
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##### El sistema bp+ : Una lógica positiva mínima para la negación mínima (the system bp+: A minimal positive logic for minimal negation) with Francisco Salto and Gemma Robles Theoria 22 (1): 81-91. 2007.
Entendemos el concepto de “negación mínima” en el sentido clásico definido por Johansson. El propósito de este artículo es definir la lógica positiva mínima Bp+, y probar que la negación mínima puede introducirse en ella. Además, comentaremos algunas de las múltiples extensiones negativas de Bp+.“Minimal negation” is classically understood in a Johansson sense. The aim of this paper is to define the minimal positive logic Bp+ and prove that a minimal negation can be inroduced in it. In addition,…Read more
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##### A weak logic with the axiom Mingle lacking the variable-sharing property with Gemma Robles and Francisco Salto Bulletin of the Section of Logic 40 (3/4): 195-202. 2011.
As it is well known, Relevance Logic R plus the axiom mingle (R-Mingle) does not have the variable-sharing property (vsp). The aim of this paper is to improve this result by defining a weak logic with the axiom mingle and not included in minimal logic BM lacking the vsp.
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##### A General Characterization of the Variable-Sharing Property by Means of Logical Matrices with Gemma Robles Notre Dame Journal of Formal Logic 53 (2): 223-244. 2012.
As is well known, the variable-sharing property (vsp) is, according to Anderson and Belnap, a necessary property of any relevant logic. In this paper, we shall consider two versions of the vsp, what we label the "weak vsp" (wvsp) and the "strong vsp" (svsp). In addition, the "no loose pieces property," a property related to the wvsp and the svsp, will be defined. Each one of these properties shall generally be characterized by means of a class of logical matrices. In this way, any logic verified…Read more
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##### Restricting the contraction axiom in Dummett's LC: a sublogic of LC with the Converse Ackermann Property, the logic LCo with Francisco Salto and Gemma Robles Bulletin of the Section of Logic 30 (3): 139-146. 2001.
LCo with the Converse Ackermann Property is defined as the result of restricting Contraction in LC. Intuitionistic and Superintuitionistic Negation is shown to be compatible with the CAP.
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##### The basic constructive logic for a weak sense of consistency with Gemma Robles Journal of Logic, Language and Information 17 (1): 89-107. 2008.
In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.
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##### Relevance logics, paradoxes of consistency and the K rule II. A non-constructive negation with Gemma Robles Logic and Logical Philosophy 15 (3): 175-191. 2007.
The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency
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##### Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz's 3-valued Logic Ł3 with Gemma Robles and Francisco Salto Journal of Philosophical Logic (2-3): 1-30. 2013.
Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for …Read more
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##### Converse Ackermann croperty and semiclassical negation Studia Logica 47 (2). 1988.
A prepositional logic S has the Converse Ackermann Property (CAP) if (AB)C is unprovable in S when C does not contain . In A Routley-Meyer semantics for Converse Ackermann Property (Journal of Philosophical Logic, 16 (1987), pp. 65–76) I showed how to derive positive logical systems with the CAP. There I conjectured that each of these positive systems were compatible with a so-called semiclassical negation. In the present paper I prove that this conjecture was right. Relational Routley-Meyer typ…Read more