
2Reduced Routley–Meyer semantics for the logics characterized by natural implicative expansions of Kleene’s strong 3valued matrixLogic Journal of the IGPL. forthcoming.

Converse Ackermann Property and Minimal NegationTeorema: International Journal of Philosophy 24 (1). 2005.

3RoutleyMeyer ternary relational semantics for intuitionistictype negationsElsevier, Academic Press. 2018.RoutleyMeyer Ternary Relational Semantics for Intuitionistictype Negations examines how to introduce intuitionistictype negations into RMsemantics. RMsemantics 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 RMsemantics, negation is interpreted by means of the Routley operator, which has been almost exclusively used for modeling De…Read more

2Relational semantics for the 4valued relevant logics BN4 and E4Logic and Logical Philosophy 25 (2): 173201. 2016.The logic BN4 was defined by R.T. Brady in 1982. It can be considered as the 4valued logic of the relevant conditional. E4 is a variant of BN4 that can be considered as the 4valued logic of entailment. The aim of this paper is to define reduced general RoutleyMeyer semantics for BN4 and E4. It is proved that BN4 and E4 are strongly sound and complete w.r.t. their respective semantics.

1Two versions of minimal intuitionism with the CAP. A noteTheoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 20 (2): 183190. 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.

Dual Equivalent Twovalued Underdetermined and Overdetermined Interpretations for Łukasiewicz’s 3valued Logic Ł3Journal of Philosophical Logic 43 (23): 303332. 2014.

4El sistema Bp+ : una lógica positiva mínima para la negación mínimaTheoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 22 (1): 8191. 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

3A simple Henkinstyle completeness proof for Gödel 3valued logic G3Logic and Logical Philosophy 23 (4): 371390. 2014.A simple Henkinstyle completeness proof for Gödel 3valued propositional logic G3 is provided. The idea is to endow G3 with an underdetermined semantics of the type defined by Dunn. The key concept in usemantics is that of “underdetermined interpretation”. It is shown that consistent prime theories built upon G3 can be understood as uinterpretations. 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

9An Interpretation of Łukasiewicz’s 4Valued Modal LogicJournal of Philosophical Logic 45 (1): 7387. 2016.A simple, bivalent semantics is defined for Łukasiewicz’s 4valued 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

4Exhaustively Axiomatizing S3°→ and S4°→Teorema: International Journal of Philosophy 27 (2): 7989. 2008.

18A semantical proof of the admissibility of the rule assertion in some relevant and modal logicsBulletin of the Section of Logic 41 (1/2): 5160. 2012.

114A RoutleyMeyer semantics for relevant logics including TWR plus the disjunctive syllogismLogic Journal of the IGPL 19 (1): 1832. 2011.We provide RoutleyMeyer 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.


65Two versions of minimal intuitionism with the cap. A noteTheoria 20 (2): 183190. 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

6A binary Routley semantics for intuitionistic De Morgan minimal logic HM and its extensionsLogic Journal of the IGPL 23 (2): 174193. 2015.

20The basic constructive logic for negationconsistencyJournal of Logic, Language and Information 17 (2): 161181. 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.

17Relevance logics, paradoxes of consistency and the K rule II. A nonconstructive negationLogic and Logical Philosophy 15 (3): 175191. 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 nonconstructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency

Minimal negation in the ternary relational semanticsReports on Mathematical Logic 4765. 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

35Curry's Paradox, Generalized Modus Ponens Axiom and Depth RelevanceStudia Logica 102 (1): 185217. 2014.“Weak relevant model structures” (wrms) 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 wrms 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

19Strong paraconsistency and the basic constructive logic for an even weaker sense of consistencyJournal of Logic, Language and Information 18 (3): 357402. 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

61A RoutleyMeyer type semantics for relevant logics including B r plus the disjunctive syllogismJournal of Philosophical Logic 39 (2): 139158. 2010.RoutleyMeyer 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.

6A companion to Brady's 4valued relevant logic BN4: The 4valued logic of entailment E4Logic Journal of the IGPL 24 (5). 2016.


21The basic constructive logic for negationconsistency defined with a propositional falsity constantBulletin of the Section of Logic 36 (12): 4558. 2007.

Paraconsistency and consistency understood as the absence of the negation of any implicative theoremReports on Mathematical Logic 147171. 2012.

13A Strong and Rich 4Valued Modal Logic Without ŁukasiewiczType ParadoxesLogica Universalis 9 (4): 501522. 2015.The aim of this paper is to introduce an alternative to Łukasiewicz’s 4valued 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 4valued modal logic free from this type of paradoxes. PŁ4 is determined by the degree of truthpreserving 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

429Extensions of the basic constructive logic for weak consistency BKc1 defined with a falsity constantLogic and Logical Philosophy 16 (4): 311322. 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 negationconsistency or absolute consistency in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the contraposition axioms

19A weak logic with the axiom Mingle lacking the variablesharing propertyBulletin of the Section of Logic 40 (3/4): 195202. 2011.

9A RoutleyMeyer semantics for truthpreserving and welldetermined Lukasiewicz 3valued logicsLogic Journal of the IGPL 22 (1): 123. 2014.Łukasiewicz 3valued logic Ł3 is often understood as the set of all valid formulas according to Łukasiewicz 3valued matrices MŁ3. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: ‘truthpreserving’ Ł3a and ‘welldetermined’ Ł3b defined by two different consequence relations on the 3valued matrices MŁ3. The aim of this article is to provide a Routley–Meyer ternary semantics for each one of these three versions of Łukasiewicz 3valued logic: Ł3, Ł3a and Ł…Read more

8Admissibility of Ackermann's rule δ in relevant logicsLogic and Logical Philosophy 22 (4): 411427. 2013.It is proved that Ackermann’s rule δ is admissible in a wide spectrum of relevant logics satisfying certain syntactical properties
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Logic and Philosophy of Logic 
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