
430A RoutleyMeyer semantics for Ackermann's logics of “strenge implication”Logic and Logical Philosophy 18 (34): 191219. 2009.The aim of this paper is to provide a RoutleyMeyer semantics for Ackermann’s logics of “strenge Implikation” Π ′ and Π ′′ . Besides the Disjunctive Syllogism, this semantics validates the rules Necessitation and Assertion. Strong completeness theorems for Π ′ and Π ′′ are proved. A brief discussion on Π ′ , Π ′′ and paraconsistency is included

429A modal restriction of RMingle with the variablesharing propertyLogic and Logical Philosophy 19 (4): 341351. 2010.A restriction of RMingle with the variablesharing property and the Ackermann properties is defined. From an intuitive semantical point of view, this restriction is an alternative to Anderson and Belnap’s logic of entailment E

124A 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.

96Two 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

73A 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.

66A constructive negation for logics including TW+Journal of Applied NonClassical Logics 15 (4): 389404. 2005.The logic TW+ is positive Ticket Entailment without the contraction axiom. Constructive negation is understood in the intuitionistic sense but without paradoxes of relevance. It is shown how to introduce a constructive negation of this kind in positive logics at least as strong as TW+. Special attention is paid to the reductio axioms. Concluding remarks about relevance, modal and entailment logics are stated. Complete relational ternary semantics are provided for the logics introduced in this pa…Read more

59The basic constructive logic for a weak sense of consistencyJournal of Logic, Language and Information 17 (1): 89107. 2008.In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

50Curry’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

43Strong 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

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

39In 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

38De Rijke, M., 109 Di Maio, MC, 435 Doria, FA, 553 French, S., 603Journal of Philosophical Logic 27 (661). 1998.

38The basic constructive logic for absolute consistencyJournal of Logic, Language and Information 18 (2): 199216. 2009.In this paper, consistency is understood as absolute consistency (i.e. nontriviality). The basic constructive logic BKc6, 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 BKc6 up to contractionless intuitionistic logic. All logics defined in this paper are paraconsistent logics.

36Two extensions of Lewis' s3 with Peirce's lawTheoria 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

35Paraconsistent logics included in Lewis’ S4Review of Symbolic Logic 3 (3): 442466. 2010.As is known, a logic S is paraconsistent if the rule ECQ (E contradictione quodlibet) is not a rule of S. Not less well known is the fact that Lewis’ modal logics are not paraconsistent. Actually, Lewis vindicates the validity of ECQ in a famous proof currently known as the “Lewis’ proof” or “Lewis’ argument.” This proof essentially leans on the Disjunctive Syllogism as a rule of inference. The aim of this paper is to define a series of paraconsistent logics included in S4 where the Disjunctive …Read more

35A natural negation completion of Urquhart's manyvalued logic CJournal of Philosophical Logic 27 (1): 7584. 1998.

30A 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

30Entendemos 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

30Relevance 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

28Intuitionistic propositional logic without 'contraction' but with 'reductio'Studia Logica 66 (3): 409418. 2000.Routley Meyer type relational complete semantics are constructed for intuitionistic contractionless logic with reductio. Different negation completions of positive intuitionistic logic without contraction are treated in a systematical, unified and semantically complete setting

27Axiomatizing 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

27RMO > 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

25The basic constructive logic for negationconsistency defined with a propositional falsity constantBulletin of the Section of Logic 36 (12): 4558. 2007.

25Strengthening Brady’s Paraconsistent 4Valued Logic BN4 with TruthFunctional Modal OperatorsJournal of Logic, Language and Information 25 (2): 163189. 2016.Łukasiewicz presented two different analyses of modal notions by means of manyvalued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type paradoxes”. On the other hand, Brady’s 4valued logic BN4 is the basic 4valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy…Read more

24Relevance logics and intuitionistic negationJournal of Applied NonClassical Logics 18 (1): 4965. 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.

24A weak logic with the axiom Mingle lacking the variablesharing propertyBulletin of the Section of Logic 40 (3/4): 195202. 2011.

24Dual Equivalent Twovalued Underdetermined and Overdetermined Interpretations for Łukasiewicz's 3valued Logic Ł3Journal of Philosophical Logic (23): 130. 2013.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 (by using Dunn semantics) dual equivalent twovalued underdetermined and overdetermined interpretations for …Read more

24Converse Ackermann croperty and semiclassical negationStudia Logica 47 (2). 1988.A prepositional logic S has the Converse Ackermann Property (CAP) if (AB)C is unprovable in S when C does not contain . In A RoutleyMeyer semantics for Converse Ackermann Property (Journal of Philosophical Logic, 16 (1987), pp. 65–76) I showed how to derive positive logical systems with the CAP. There I conjectured that each of these positive systems were compatible with a socalled semiclassical negation. In the present paper I prove that this conjecture was right. Relational RoutleyMeyer typ…Read more

23Axiomatizing s4+ and j+ without the suffixing, prefixing and selfdistribution of the conditional axiomsBulletin of the Section of Logic 39 (1/2): 7991. 2010.

Universidad de SalamancaRegular Faculty