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35Urquhart's C with Intuitionistic Negation: Dummett's LC without the Contraction AxiomNotre Dame Journal of Formal Logic 36 (3): 407-413. 1995.This paper offers a particular intuitionistic negation completion of Urquhart's system C resulting in a super-intuitionistic contractionless propositional logic equivalent to Dummett's LC without contraction
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35Constructive RBulletin of the Section of Logic 16 (4): 167-173. 1987.Let R+ be the positive fragment of Anderson and Belnap’s Logic of Relevance, R. And let RMO+ be the result of adding the Mingle principle ) to R+. We have shown in [2] that either a minimal negation or else a semiclassical one can be added to RMO+ preserving the variable-sharing property. Moreover, each of there systems is given a semantics in the Routley-Meyer style. In describing in [2] the models for RMO+ plus minimal negation, we noted that a similar strategy would give us a semantics for R+…Read more
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34Restricting the contraction axiom in Dummett's LC: a sublogic of LC with the Converse Ackermann Property, the logic LCoBulletin 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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33The logic determined by Smiley’s matrix for Anderson and Belnap’s first-degree entailment logicJournal 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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33Axiomatizing E→ and R→ with Anderson and Belnap's 'strong and natural'list of valid entailmentsBulletin of the Section of Logic 16 (1): 2-7. 1987.We provide all possible axiomatizations with independent axioms of E→ and R→ formulable with Anderson and Belnap’s list
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32A Routley-Meyer semantics for truth-preserving and well-determined Lukasiewicz 3-valued logicsLogic 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
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32Routley-Meyer ternary relational semantics for intuitionistic-type negationsElsevier, 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
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32RMO -> is the result of adding the ‘mingle principle’ (viz. A-> (A -> A)) to Anderson and Belnap’s implicative logic of relevance R->. The aim of this paper is to provide all possible axiomatizations with independent axioms of RMO -> formulable with Anderson and Belnap’s list extended with three characteristic minglish principles
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31Axiomatizing s4+ and j+ without the suffixing, prefixing and self-distribution of the conditional axiomsBulletin of the Section of Logic 39 (1/2): 79-91. 2010.
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31A weak logic with the axiom Mingle lacking the variable-sharing propertyBulletin 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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31Basic Quasi-Boolean Expansions of Relevance LogicsJournal of Philosophical Logic 50 (4): 727-754. 2021.The basic quasi-Boolean negation expansions of relevance logics included in Anderson and Belnap’s relevance logic R are defined. We consider two types of QB-negation: H-negation and D-negation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with H-negation are paracomplete; logics with D-negation are paraconsistent. All logics defined in the paper are given a Routley-Meyer ternary relational se…Read more
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31A paraconsistent 3-valued logic related to Godel logic G3Logic Journal of the IGPL 22 (4): 515-538. 2014.
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30Erratum to: The compatibility of relevance and Mingle (review)Journal of Philosophical Logic 39 (3): 339-339. 2010.
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30Converse Ackermann property and constructive negation defined with a negation connectiveLogic and Logical Philosophy 15 (2): 113-130. 2006.The Converse Ackermann Property is the unprovability of formulas of the form (A -> B) -> C when C does contain neither -> nor ¬. Intuitively, the CAP amounts to rule out the derivability of pure non-necessitive propositions from non-necessitive ones. A constructive negation of the sort historically defined by, e.g., Johansson is added to positive logics with the CAP in the spectrum delimited by Ticket Entailment and Dummett’s logic LC
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30Ticket Entailment plus the mingle axiom has the variable-sharing propertyLogic Journal of the IGPL 20 (1): 355-364. 2012.The logic TM is the result of adding the mingle axiom, M to Ticket Entailment logic, T. In the present study, it is proved that TM has the variable-sharing property . Ternary relational semantics for TM is provided. Finally, an interesting extension of TM with the vsp is briefly discussed
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28Generalizing the Depth Relevance Condition: Deep Relevant Logics Not Included in R-MingleNotre Dame Journal of Formal Logic 55 (1): 107-127. 2014.
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27The basic constructive logic for weak consistency and the reductio axiomsBulletin of the Section of Logic 38 (1/2): 61-76. 2009.
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27Relational ternary semantics for a logic equivalent to Involutive Monoidal t-norm based logic IMTLBulletin of the Section of Logic 34 (2): 101-116. 2005.
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27A binary Routley semantics for intuitionistic De Morgan minimal logic HM and its extensionsLogic Journal of the IGPL 23 (2): 174-193. 2015.
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26The non-relevant De Morgan minimal logic in Routley-Meyer semantics with no designated pointsJournal of Applied Non-Classical Logics 24 (4): 321-332. 2014.Sylvan and Plumwood’s is the relevant De Morgan minimal logic in the Routley-Meyer semantics with a set of designated points. The aim of this paper is to define the logic and some of its extensions. The logic is the non-relevant De Morgan minimal logic in the Routley-Meyer semantics without a set of designated points
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26A Variety of DeMorgan Negations in Relevant LogicsAustralasian Journal of Logic 20 (2): 348-374. 2023.The present paper is inspired by Sylvan and Plumwood’s logicBM defined in “Non-normal relevant logics” and by their treatmentof negation with the ∗-operator in “The semantics of first-degree en-tailment”. Given a positive logic L including Routley and Meyer’sbasic positive logic and included in either the positive fragment of Eor in that of RW, we investigate the essential De Morgan negation ex-pansions of L and determine all the deductive relations they maintainto each other. A Routley-Meyer se…Read more
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25Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix II. Only one designated valueJournal of Applied Non-Classical Logics 29 (3): 307-325. 2019.This paper is a sequel to ‘Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values’, where a ‘bivalent’ Belnap-Dunn semantics is provided for all the expansions referred to in its title. The aim of the present paper is to carry out a parallel investigation for all natural implicative expansions of Kleene's strong 3-valued matrix now with only one designated value.
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25A Class of Simpler Logical Matrices for the Variable-Sharing PropertyLogic and Logical Philosophy 20 (3): 241-249. 2011.In our paper “A general characterization of the variable-sharing property by means of logical matrices”, a general class of so-called “Relevant logical matrices”, RMLs, is defined. The aim of this paper is to define a class of simpler Relevant logical matrices RMLs′serving the same purpose that RMLs, to wit: any logic verified by an RML′has the variable-sharing property and related properties predicable of the logic of entailment E and of the logic of relevance R
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24El sistema Bp+ : una lógica positiva mínima para la negación mínima (The system Bp+: a minimal positive logic for minimal negation)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
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24A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3DHJournal of Logic, Language and Information 30 (1): 117-138. 2020.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 lo…Read more
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23Exhaustively Axiomatizing S3°→ and S4°→Teorema: International Journal of Philosophy 27 (2): 79-89. 2008.S3o and S4o are the restrictions with the Converse Ackermann Property of the implicative fragments of Lewis' S3 and S4 respectively. The aim of this paper is to provide all possible axiomatizations with independent axioms of S3o and S4o that can be formulated with a modification of Anderson and Belnap's list of valid entailments.
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22A constructive negation defined with a negation connective for logics including Bp+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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21Exhaustively axiomatizing rmo→ with a select list of representative theses including restricted Mingle principlesBulletin of the Section of Logic 28 (4): 195-206. 1999.
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Universidad de SalamancaRegular Faculty