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Regular Faculty
León, CL, Spain
Areas of Specialization
Areas of Interest

• ##### Routley-Meyer ternary relational semantics for intuitionistic-type negations with José M. Méndez Elsevier, Academic Press. 2018.
Routley-Meyer Ternary Relational Semantics for Intuitionistic-type Negations examines how to introduce intuitionistic-type negations into RM-semantics. RM-semantics is highly malleable and capable of modeling families of logics which are very different from each other. This semantics was introduced in the early 1970s, and was devised for interpreting relevance logics. In RM-semantics, negation is interpreted by means of the Routley operator, which has been almost exclusively used for modeling De…Read more
• ##### Relational semantics for the 4-valued relevant logics BN4 and E4 with José M. Blanco, Sandra M. López, Jesús R. Paradela, and Marcos M. Recio Logic and Logical Philosophy 25 (2): 173-201. 2016.
The logic BN4 was defined by R.T. Brady in 1982. It can be considered as the 4-valued logic of the relevant conditional. E4 is a variant of BN4 that can be considered as the 4-valued logic of entailment. The aim of this paper is to define reduced general Routley-Meyer semantics for BN4 and E4. It is proved that BN4 and E4 are strongly sound and complete w.r.t. their respective semantics.
• ##### Two versions of minimal intuitionism with the CAP. A note with José M. Méndez Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 20 (2): 183-190. 2005.
Two versions of minimal intuitionism are defined restricting Contraction. Both are defined by means of a falsity constant F. The first one follows the historical trend, the second is the result of imposing specialconstraints on F. RelationaI ternary semantics are provided.
• ##### El sistema Bp+ : una lógica positiva mínima para la negación mínima with José M. Méndez and Francisco Salto Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 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
• ##### A simple Henkin-style completeness proof for Gödel 3-valued logic G3 Logic and Logical Philosophy 23 (4): 371-390. 2014.
A simple Henkin-style completeness proof for Gödel 3-valued propositional logic G3 is provided. The idea is to endow G3 with an under-determined semantics of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation”. It is shown that consistent prime theories built upon G3 can be understood as u-interpretations. In order to prove this fact we follow Brady by defining G3 as an extension of Anderson and Belnap’s positive fragment of First Degree Entailme…Read more
• ##### An Interpretation of Łukasiewicz’s 4-Valued Modal Logic with José M. Méndez and Francisco Salto Journal of Philosophical Logic 45 (1): 73-87. 2016.
A simple, bivalent semantics is defined for Łukasiewicz’s 4-valued modal logic Łm4. It is shown that according to this semantics, the essential presupposition underlying Łm4 is the following: A is a theorem iff A is true conforming to both the reductionist and possibilist theses defined as follows: rt: the value of modal formulas is equivalent to the value of their respective argument iff A is true, etc.); pt: everything is possible. This presupposition highlights and explains all oddities arisi…Read more
• ##### A Routley-Meyer semantics for relevant logics including TWR plus the disjunctive syllogism with José M. Méndez 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.
• ##### Two versions of minimal intuitionism with the cap. A note with José M. Méndez Theoria 20 (2): 183-190. 2005.
Two versions of minimal intuitionism are defined restricting Contraction. Both are defined by means of a falsity constant F. The first one follows the historical trend, the second is the result of imposing specialconstraints on F. RelationaI ternary semantics are provided
• ##### The basic constructive logic for negation-consistency Journal of Logic, Language and Information 17 (2): 161-181. 2008.
In this paper, consistency is understood in the standard way, i.e. as the absence of a contradiction. The basic constructive logic BKc4, which is adequate to this sense of consistency in the ternary relational semantics without a set of designated points, is defined. Then, it is shown how to define a series of logics by extending BKc4 up to minimal intuitionistic logic. All logics defined in this paper are paraconsistent logics.
• ##### Minimal negation in the ternary relational semantics with J. Mendez and F. Salto Reports on Mathematical Logic 47-65. 2005.
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 without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive logic are offered
• ##### Relevance logics, paradoxes of consistency and the K rule II. A non-constructive negation with José M. Méndez 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
• ##### Curry's Paradox, Generalized Modus Ponens Axiom and Depth Relevance with José M. Méndez 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
• ##### Strong paraconsistency and the basic constructive logic for an even weaker sense of consistency with José M. Méndez 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
• ##### A Routley-Meyer type semantics for relevant logics including B r plus the disjunctive syllogism with José M. Méndez Journal of Philosophical Logic 39 (2): 139-158. 2010.
Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π ′ are among the logics considered.
• ##### Extensions of the basic constructive logic for weak consistency BKc1 defined with a falsity constant Logic and Logical Philosophy 16 (4): 311-322. 2007.
The logic BKc1 is the basic constructive logic for weak consistency in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the contraposition axioms
• ##### A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes with José M. Méndez 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
• ##### A Routley-Meyer semantics for truth-preserving and well-determined Lukasiewicz 3-valued logics with J. M. Mendez Logic Journal of the IGPL 22 (1): 1-23. 2014.
Łukasiewicz 3-valued logic Ł3 is often understood as the set of all valid formulas according to Łukasiewicz 3-valued matrices MŁ3. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: ‘truth-preserving’ Ł3a and ‘well-determined’ Ł3b defined by two different consequence relations on the 3-valued matrices MŁ3. The aim of this article is to provide a Routley–Meyer ternary semantics for each one of these three versions of Łukasiewicz 3-valued logic: Ł3, Ł3a and Ł…Read more
• ##### Admissibility of Ackermann's rule δ in relevant logics Logic and Logical Philosophy 22 (4): 411-427. 2013.
It is proved that Ackermann’s rule δ is admissible in a wide spectrum of relevant logics satisfying certain syntactical properties