•  3
    Let MK3 I and MK3 II be Kleene's strong 3-valued matrix with only one and two designated values, respectively. Next, let MK3 G be defined exactly as MK3 I, except th...
  •  5
    The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of implicative expansions of Kleene’s 3-valued logic functionally including Łukasiewicz’s logic Ł3 is defined. Several properties of this class and/or some of its subclasses are investigated. Properties contemplated include functional completeness for the 3-element set of truth-values, presence of natural conditionals, variable-sharing property and vsp-related properties.
  •  2
    Basic Quasi-Boolean Expansions of Relevance Logics
    Journal 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
  •  13
    A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3DH
    Journal of Logic, Language and Information 30 (1): 117-138. 2021.
    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
  •  1
    A classical result by Słupecki states that a logic L is functionally complete for the 3-element set of truth-values THREE if, in addition to functionally including Łukasiewicz’s 3-valued logic Ł3, what he names the ‘$T$-function’ is definable in L. By leaning upon this classical result, we prove a general theorem for defining binary expansions of Kleene’s strong logic that are functionally complete for THREE.
  •  1
    Curry’s Paradox, Generalized Contraction Rule and Depth Relevance
    In Konstantinos Boudouris (ed.), Proceedings XXIII world Congress Philosophy, Philosophy Documentation Center. pp. 35-39. 2018.
    As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W causes. Among these, the Contraction rule or the Modus …Read more
  •  39
    Minimal Negation in the Ternary Relational Semantics
    with Gemma Robles and Francisco Salto
    Reports on Mathematical Logic 39 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 (in ternary frames) without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive lo…Read more
  •  7
    We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova.
  •  10
    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.
  •  7
    Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with only one designated value are provided.
  •  10
    Blocking the Routes to Triviality with Depth Relevance
    Journal of Logic, Language and Information 23 (4): 493-526. 2014.
    In Rogerson and Restall’s, the “class of implication formulas known to trivialize ” is recorded. The aim of this paper is to show how to invalidate any member in this class by using “weak relevant model structures”. Weak relevant model structures verify deep relevant logics only.
  •  11
    ABSTRACTA conditional is natural if it fulfils the three following conditions. It coincides with the classical conditional when restricted to the classical values T and F; it satisfies the Modus Ponens; and it is assigned a designated value whenever the value assigned to its antecedent is less than or equal to the value assigned to its consequent. The aim of this paper is to provide a ‘bivalent’ Belnap-Dunn semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with …Read more
  • Exhaustively axiomatizing S3-> and S4-> with a select list of representative theses
    Bulletin of the Section of Logic 17 (1): 15-20. 1988.
    This paper is a sequel to [2]. We extend Anderson and Belnap’s list with the characteristic axioms of S3→ and S4→ . Then we exhaustively axiomatize these systems with the list thus extended
  •  15
    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
  •  8
    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 dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic…Read more
  •  10
    Two versions of minimal intuitionism with the CAP. A note
    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.
  •  4
    Two Extensions of Lewis’ S3 with Peirce’s Law
    Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 14 (3): 407-411. 1999.
    We define two extensions of Lewis’ S3 with two versions of Peirce’s Law. We prove that both of them have the Ackermann Property.
  • Lógica intuicionista en tres horas
    with Francisco Alemany
    Laguna 9. 2001.
  •  8
    Exhaustively axiomatizing RMO with an appropiate extension of Anderson and Belnap’s “strong and natural list of valid entailments”
    Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 5 (1-2): 223-228. 1990.
    RMO -> is the result of adding the ‘mingle principle’ ) 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.
  •  10
    El sistema Bp+ : una lógica positiva mínima para la negación mínima
    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
  •  17
    Urquhart's C with Intuitionistic Negation: Dummett's LC without the Contraction Axiom
    Notre 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
  •  43
    Systems with the converse Ackermann property
    Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 1 (1): 253-258. 1985.
    A system S has the “converse Ackermann property” if -> C is unprovable in S whenever C is a propositional variable. In this paper we define the fragments with the C.A.P. of some well-know propositional systems in the spectrum between the minimal and classical logic. In the first part we succesively study the implicative and positive fragments and the full calculi. In the second, we prove by a matrix method that each one of the systems has the C.A.P. Thus, we think the problem proposed in Anderso…Read more
  • Relevance Logics, Paradoxes Of Consistency And The K Rule Ii
    Logic and Logical Philosophy 15 175-191. 2006.
    The logic B+ is Routley and Meyer’s basic positive logic. Wedefine the logics BK+ and BK′+ by adding to B+ the K rule and to BK+the characteristic S4 axiom, respectively. These logics are endowed witha relatively strong non-constructive negation. We prove that all the logicsdefined lack the K axiom and the standard paradoxes of consistency
  • Minimal Non-relevant Logics Without The K Axiom
    Reports on Mathematical Logic. 2007.
    The logic B$_{+}$ is Routley and Meyer's basic positive logic. The logic B$_{K+}$ is B$_{+}$ plus the $K$ rule. We add to B$_{K+}$ four intuitionistic-type negations. We show how to extend the resulting logics within the modal and relevance spectra. We prove that all the logics defined lack the K axiom
  •  30
    Constructive R
    Bulletin 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
  • Exhaustively Axiomatizing S3 (->) degrees and S4 (->) degrees
    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 S40 that can be formulated with a modification of Anderson and Belnap's list of valid entailments.
  •  28
    We provide all possible axiomatizations with independent axioms of E→ and R→ formulable with Anderson and Belnap’s list
  •  84
    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.
  •  19
    A Class of Simpler Logical Matrices for the Variable-Sharing Property
    with G. Robles
    Logic 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
  •  15
    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