•  15
    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
  •  1
    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.
  •  2
    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.
  •  6
    ABSTRACTThis 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 semant...
  •  5
    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.
  •  8
    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.
  •  10
    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
  • Minimal negation in the ternary relational semantics
    with G. Robles, 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
  •  11
    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
    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.
  •  5
    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
  •  7
    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.
  •  7
    El sistema Bp+ : una lógica positiva mínima para la negación mínima
    with Francisco Salto and Gemma Robles
    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
  •  16
    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
  • Lógica intuicionista en tres horas
    with Francisco Alemany
    Laguna 9. 2001.
  • 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
  •  22
    An Interpretation of Łukasiewicz’s 4-Valued Modal Logic
    with Gemma Robles 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
  •  39
    In his paper “Recent work in relevant logic”, Jago includes a section on Disjunctive Syllogism . The content of the section essentially consists of (a) a valuation of some work by Robles and Méndez on the topic as “not particularly interesting in itself”; (b) a statement establishing that “What would be interesting is to discover just how weak a relevant logic needs to be before disjunctive syllogism becomes inadmissible”. The main problem with this section of Jago’s paper on DS is that the auth…Read more
  •  18
    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
  •  24
    Relevance logics and intuitionistic negation
    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.
  •  43
    Strong paraconsistency and the basic constructive logic for an even weaker sense of consistency
    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
  •  30
    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
  •  50
    “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