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A Routley-Meyer semantics for Ackermann's logics of “strenge implication” Logic and Logical Philosophy 18 (3-4): 191-219. 2009.
The aim of this paper is to provide a Routley-Meyer 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
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A modal restriction of R-Mingle with the variable-sharing property with Gemma Robles and Francisco Salto Logic and Logical Philosophy 19 (4): 341-351. 2010.
A restriction of R-Mingle with the variable-sharing 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
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A Routley-Meyer semantics for relevant logics including TWR plus the disjunctive syllogism with Gemma Robles Logic Journal of the IGPL 19 (1): 18-32. 2011.
We provide Routley-Meyer 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.
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Two versions of minimal intuitionism with the cap. A note with Gemma Robles Theoria 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
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A Routley-Meyer type semantics for relevant logics including B r plus the disjunctive syllogism with Gemma Robles Journal of Philosophical Logic 39 (2): 139-158. 2010.
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.
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A natural negation completion of Urquhart's many-valued logic C with Francisco Salto Journal of Philosophical Logic 27 (1): 75-84. 1998.
Etude de l'extension par la negation semi-intuitionniste de la logique positive des propositions appelee logique C, developpee par A. Urquhart afin de definir une semantique relationnelle valable pour la logique des valeurs infinies de Lukasiewicz (Lw). Evitant les axiomes de contraction et de reduction propres a la logique classique de Dummett, l'A. propose une semantique de type Routley-Meyer pour le systeme d'Urquhart (CI) en tant que celle-la ne fournit que des theories consistantes pour la …Read more
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A constructive negation for logics including TW+ with Gemma Robles Journal of Applied Non-Classical Logics 15 (4): 389-404. 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
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The basic constructive logic for a weak sense of consistency with Gemma Robles Journal of Logic, Language and Information 17 (1): 89-107. 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.
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Curry’s Paradox, Generalized Modus Ponens Axiom and Depth Relevance with Gemma Robles Studia Logica 102 (1): 185-217. 2014.
“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
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Strong paraconsistency and the basic constructive logic for an even weaker sense of consistency with Gemma Robles 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
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A note on "Recent work in relevant logic"
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
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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
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Two extensions of Lewis' s3 with Peirce's law with Francisco Salto Theoria 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
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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
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The basic constructive logic for absolute consistency with Gemma Robles Journal of Logic, Language and Information 18 (2): 199-216. 2009.
In this paper, consistency is understood as absolute consistency (i.e. non-triviality). 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.
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Paraconsistent logics included in Lewis’ S4 with Gemma Robles Review of Symbolic Logic 3 (3): 442-466. 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
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Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators with Gemma Robles Journal of Logic, Language and Information 25 (2): 163-189. 2016.
Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4-valued 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 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy…Read more
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A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes with Gemma Robles Logica Universalis 9 (4): 501-522. 2015.
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
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El sistema bp+ : Una lógica positiva mínima para la negación mínima (the system bp+: A minimal positive logic for minimal negation) with Francisco Salto and Gemma Robles Theoria 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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Intuitionistic propositional logic without 'contraction' but with 'reductio' with F. Salto Studia Logica 66 (3): 409-418. 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
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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
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A General Characterization of the Variable-Sharing Property by Means of Logical Matrices with Gemma Robles Notre Dame Journal of Formal Logic 53 (2): 223-244. 2012.
As is well known, the variable-sharing property (vsp) is, according to Anderson and Belnap, a necessary property of any relevant logic. In this paper, we shall consider two versions of the vsp, what we label the "weak vsp" (wvsp) and the "strong vsp" (svsp). In addition, the "no loose pieces property," a property related to the wvsp and the svsp, will be defined. Each one of these properties shall generally be characterized by means of a class of logical matrices. In this way, any logic verified…Read more
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Relevance logics, paradoxes of consistency and the K rule II. A non-constructive negation with Gemma Robles Logic and Logical Philosophy 15 (3): 175-191. 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 non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency
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Converse Ackermann croperty and semiclassical negation Studia 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 Routley-Meyer 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 so-called semiclassical negation. In the present paper I prove that this conjecture was right. Relational Routley-Meyer typ…Read more
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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
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Axiomatizing E→ and R→ with Anderson and Belnap's 'strong and natural'list of valid entailments Bulletin 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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Exhaustively axiomatizing RMO with an appropiate extension of Anderson and Belnap's “strong and natural list of valid entailments” Theoria 5 (1): 223-228. 1990.
RMO -&gt; is the result of adding the ‘mingle principle’ (viz. A-&gt; (A -&gt; A)) to Anderson and Belnap’s implicative logic of relevance R-&gt;. The aim of this paper is to provide all possible axiomatizations with independent axioms of RMO -&gt; formulable with Anderson and Belnap’s list extended with three characteristic minglish principles