
3Natural implicative expansions of variants of Kleene's strong 3valued logic with Gödeltype and dual Gödeltype negationJournal of Applied NonClassical Logics 31 (2): 130153. 2021.Let MK3 I and MK3 II be Kleene's strong 3valued matrix with only one and two designated values, respectively. Next, let MK3 G be defined exactly as MK3 I, except th...

5A Class of Implicative Expansions of Kleene’s Strong Logic, a Subclass of Which Is Shown Functionally Complete Via the Precompleteness of Łukasiewicz’s 3Valued Logic Ł3Journal of Logic, Language and Information 30 (3): 533556. 2021.The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849019093062). A class of implicative expansions of Kleene’s 3valued 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 3element set of truthvalues, presence of natural conditionals, variablesharing property and vsprelated properties.

2Basic QuasiBoolean Expansions of Relevance LogicsJournal of Philosophical Logic 50 (4): 727754. 2021.The basic quasiBoolean negation expansions of relevance logics included in Anderson and Belnap’s relevance logic R are defined. We consider two types of QBnegation: Hnegation and Dnegation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with Hnegation are paracomplete; logics with Dnegation are paraconsistent. All logics defined in the paper are given a RoutleyMeyer ternary relational se…Read more

13A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3DHJournal of Logic, Language and Information 30 (1): 117138. 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 3valued lo…Read more

1A remark on functional completeness of binary expansions of Kleene’s strong 3valued logicLogic Journal of the IGPL. forthcoming.A classical result by Słupecki states that a logic L is functionally complete for the 3element set of truthvalues THREE if, in addition to functionally including Łukasiewicz’s 3valued 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.

1Curry’s Paradox, Generalized Contraction Rule and Depth RelevanceIn Konstantinos Boudouris (ed.), Proceedings XXIII world Congress Philosophy, Philosophy Documentation Center. pp. 3539. 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

39Minimal Negation in the Ternary Relational SemanticsReports on Mathematical Logic 39 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 (in ternary frames) without extending the positive logic. Complete semantics for such kinds of reductio in a properly extended positive lo…Read more

7The Class of All Natural Implicative Expansions of Kleene’s Strong Logic Functionally Equivalent to Łkasiewicz’s 3Valued Logic Ł3Journal of Logic, Language and Information 29 (3): 349374. 2020.We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3valued matrix and select the class of all logics functionally equivalent to Łukasiewicz’s 3valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova.

10BelnapDunn semantics for natural implicative expansions of Kleene's strong threevalued matrix II. Only one designated valueJournal of Applied NonClassical Logics 29 (3): 307325. 2019.This paper is a sequel to ‘BelnapDunn semantics for natural implicative expansions of Kleene's strong threevalued matrix with two designated values’, where a ‘bivalent’ BelnapDunn 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 3valued matrix now with only one designated value.

7Partiality and its dual in natural implicative expansions of Kleene’s strong 3valued matrix with only one designated valueLogic Journal of the IGPL 27 (6): 910932. 2019.Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3valued matrix with only one designated value are provided.

10Blocking the Routes to Triviality with Depth RelevanceJournal of Logic, Language and Information 23 (4): 493526. 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.

11BelnapDunn semantics for natural implicative expansions of Kleene's strong threevalued matrix with two designated valuesJournal of Applied NonClassical Logics 29 (1): 3763. 2019.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’ BelnapDunn semantics for all natural implicative expansions of Kleene's strong 3valued matrix with …Read more

Exhaustively axiomatizing S3> and S4> with a select list of representative thesesBulletin of the Section of Logic 17 (1): 1520. 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

15RoutleyMeyer 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

8Dual Equivalent Twovalued Underdetermined and Overdetermined Interpretations for Łukasiewicz’s 3valued Logic Ł3Journal of Philosophical Logic 43 (23): 303332. 2014.Łukasiewicz threevalued logic Ł3 is often understood as the set of all 3valued valid formulas according to Łukasiewicz’s 3valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “welldetermined” Ł3a and “truthpreserving” Ł3b defined by two different consequence relations on the 3valued matrices. The aim of this paper is to provide dual equivalent twovalued underdetermined and overdetermined interpretations for Ł3, Ł3a and Ł3b. The logic…Read more

10Two 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.

4Two Extensions of Lewis’ S3 with Peirce’s LawTheoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 14 (3): 407411. 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.

8Exhaustively 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 (12): 223228. 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.

10El 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

17Urquhart's C with Intuitionistic Negation: Dummett's LC without the Contraction AxiomNotre Dame Journal of Formal Logic 36 (3): 407413. 1995.This paper offers a particular intuitionistic negation completion of Urquhart's system C resulting in a superintuitionistic contractionless propositional logic equivalent to Dummett's LC without contraction

43Systems with the converse Ackermann propertyTheoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 1 (1): 253258. 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 wellknow 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 IiLogic and Logical Philosophy 15 175191. 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 nonconstructive negation. We prove that all the logicsdeﬁned lack the K axiom and the standard paradoxes of consistency

Minimal Nonrelevant Logics Without The K AxiomReports 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 intuitionistictype 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

30Constructive RBulletin of the Section of Logic 16 (4): 167173. 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 variablesharing property. Moreover, each of there systems is given a semantics in the RoutleyMeyer 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 (>) degreesTeorema: International Journal of Philosophy 27 (2): 7989. 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.

28Axiomatizing E→ and R→ with Anderson and Belnap's 'strong and natural'list of valid entailmentsBulletin of the Section of Logic 16 (1): 27. 1987.We provide all possible axiomatizations with independent axioms of E→ and R→ formulable with Anderson and Belnap’s list

84A 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.

19A Class of Simpler Logical Matrices for the VariableSharing PropertyLogic and Logical Philosophy 20 (3): 241249. 2011.In our paper “A general characterization of the variablesharing property by means of logical matrices”, a general class of socalled “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 variablesharing property and related properties predicable of the logic of entailment E and of the logic of relevance R

15Ticket Entailment plus the mingle axiom has the variablesharing propertyLogic Journal of the IGPL 20 (1): 355364. 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 variablesharing property . Ternary relational semantics for TM is provided. Finally, an interesting extension of TM with the vsp is briefly discussed

Universidad de SalamancaRegular Faculty