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Gemma Robles
Universidad de León
  1. Extensions of the Basic Constructive Logic for Weak Consistency BKc1 Defined with a Falsity Constant.Gemma Robles - 2007 - Logic and Logical Philosophy 16 (4):311-322.
    The logic BKc1 is the basic constructive logic for weak consistency in the ternary relational semantics without a set of designated points. In this paper, a number of extensions of B Kc1 defined with a propositional falsity constant are defined. It is also proved that weak consistency is not equivalent to negation-consistency or absolute consistency in any logic included in positive contractionless intermediate logic LC plus the constructive negation of BKc1 and the contraposition axioms.
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  2. A Modal Restriction of R-Mingle with the Variable-Sharing Property.Gemma Robles, José M. Méndez & Francisco Salto - 2010 - Logic and Logical Philosophy 19 (4):341-351.
    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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  3.  48
    The Basic Constructive Logic for a Weak Sense of Consistency.Gemma Robles & José M. Méndez - 2008 - Journal of Logic, Language and Information 17 (1):89-107.
    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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  4.  64
    A Routley-Meyer Type Semantics for Relevant Logics Including B R Plus the Disjunctive Syllogism.Gemma Robles & José M. Méndez - 2010 - Journal of Philosophical Logic 39 (2):139-158.
    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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  5.  32
    A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart.Gemma Robles - 2013 - Logica Universalis 7 (4):507-532.
    Routley–Meyer semantics (RM-semantics) is defined for Gödel 3-valued logic G3 and some logics related to it among which a paraconsistent one differing only from G3 in the interpretation of negation is to be remarked. The logics are defined in the Hilbert-style way and also by means of proof-theoretical and semantical consequence relations. The RM-semantics is defined upon the models for Routley and Meyer’s basic positive logic B+, the weakest positive RM-semantics. In this way, it is to be expected that the (...)
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  6.  5
    Relational Semantics for the 4-Valued Relevant Logics BN4 and E4.Gemma Robles, José M. Blanco, Sandra M. López, Jesús R. Paradela & Marcos M. Recio - 2016 - Logic and Logical Philosophy 25 (2):173-201.
    The logic BN4 was defined by R.T. Brady in 1982. It can be considered as the 4-valued logic of the relevant conditional. E4 is a variant of BN4 that can be considered as the 4-valued logic of entailment. The aim of this paper is to define reduced general Routley-Meyer semantics for BN4 and E4. It is proved that BN4 and E4 are strongly sound and complete w.r.t. their respective semantics.
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  7.  14
    Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators.José M. Méndez & Gemma Robles - 2016 - Journal of Logic, Language and Information 25 (2):163-189.
    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 for defining truth-functional (...)
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  8.  13
    An Interpretation of Łukasiewicz’s 4-Valued Modal Logic.José M. Méndez, Gemma Robles & Francisco Salto - 2016 - Journal of Philosophical Logic 45 (1):73-87.
    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 arising in Łm4.
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  9.  7
    The Logic Determined by Smiley’s Matrix for Anderson and Belnap’s First-Degree Entailment Logic.José M. Méndez & Gemma Robles - 2016 - Journal of Applied Non-Classical Logics 26 (1):47-68.
    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 (...)
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  10.  22
    The Basic Constructive Logic for Negation-Consistency.Gemma Robles - 2008 - Journal of Logic, Language and Information 17 (2):161-181.
    In this paper, consistency is understood in the standard way, i.e. as the absence of a contradiction. The basic constructive logic BKc4, 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 BKc4 up to minimal intuitionistic logic. All logics defined in this paper are paraconsistent logics.
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  11.  18
    A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes.José M. Méndez & Gemma Robles - 2015 - Logica Universalis 9 (4):501-522.
    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 (...)
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  12. A Routley-Meyer Semantics for Relevant Logics Including TWR Plus the Disjunctive Syllogism.Gemma Robles & José M. Méndez - 2011 - Logic Journal of the IGPL 19 (1):18-32.
    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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  13.  5
    A Simple Henkin-Style Completeness Proof for Gödel 3-Valued Logic G3.Gemma Robles - 2014 - Logic and Logical Philosophy 23 (4):371-390.
    A simple Henkin-style completeness proof for Gödel 3-valued propositional logic G3 is provided. The idea is to endow G3 with an under-determined semantics of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation”. It is shown that consistent prime theories built upon G3 can be understood as u-interpretations. In order to prove this fact we follow Brady by defining G3 as an extension of Anderson and Belnap’s positive fragment of First Degree Entailment Logic.
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  14.  14
    The Logic B and the Reductio Axioms.Gemma Robles & José M. Méndez - 2004 - Bulletin of the Section of Logic 33 (2):87-94.
  15.  21
    Relevance Logics, Paradoxes of Consistency and the K Rule II. A Non-Constructive Negation.José M. Méndez & Gemma Robles - 2007 - Logic and Logical Philosophy 15 (3):175-191.
    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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  16. Dual Equivalent Two-Valued Under-Determined and Over-Determined Interpretations for Łukasiewicz’s 3-Valued Logic Ł3.Gemma Robles, Francisco Salto & José M. Méndez - 2014 - Journal of Philosophical Logic 43 (2-3):303-332.
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  17.  18
    Relational Ternary Semantics for a Logic Equivalent to Involutive Monoidal T-Norm Based Logic IMTL.Gemma Robles & José M. Méndez - 2005 - Bulletin of the Section of Logic 34 (2):101-116.
  18.  12
    Dual Equivalent Two-Valued Under-Determined and Over-Determined Interpretations for Łukasiewicz's 3-Valued Logic Ł3.Gemma Robles, Francisco Salto & José M. Méndez - 2013 - Journal of Philosophical Logic (2-3):1-30.
    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 (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley (...)
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  19.  8
    Admissibility of Ackermann's Rule Δ in Relevant Logics.Gemma Robles - 2013 - Logic and Logical Philosophy 22 (4):411-427.
    It is proved that Ackermann’s rule δ is admissible in a wide spectrum of relevant logics satisfying certain syntactical properties.
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  20.  67
    Two Versions of Minimal Intuitionism with the Cap. A Note.Gemma Robles & José M. Méndez - 2005 - Theoria 20 (2):183-190.
    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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  21.  15
    A General Characterization of the Variable-Sharing Property by Means of Logical Matrices.Gemma Robles & José M. Méndez - 2012 - Notre Dame Journal of Formal Logic 53 (2):223-244.
    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 (...)
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  22.  20
    Axiomatizing S4+ and J+ Without the Suffixing, Prefixing and Self-Distribution of the Conditional Axioms.Gemma Robles & José M. Méndez - 2010 - Bulletin of the Section of Logic 39 (1/2):79-91.
  23.  5
    Ticket Entailment Plus the Mingle Axiom has the Variable-Sharing Property.José M. Méndez, Gemma Robles & Francisco Salto - 2012 - Logic Journal of the IGPL 20 (1):355-364.
    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.
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  24.  22
    The Basic Constructive Logic for Negation-Consistency Defined with a Propositional Falsity Constant.José M. Méndez, Gemma Robles & Francisco Salto - 2007 - Bulletin of the Section of Logic 36 (1-2):45-58.
  25.  17
    Relevance Logics and Intuitionistic Negation.José M. Méndez & Gemma Robles - 2008 - Journal of Applied Non-Classical Logics 18 (1):49-65.
    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.
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  26.  33
    Strong Paraconsistency and the Basic Constructive Logic for an Even Weaker Sense of Consistency.Gemma Robles & José M. Méndez - 2009 - Journal of Logic, Language and Information 18 (3):357-402.
    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 define (...)
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  27.  27
    The Basic Constructive Logic for Absolute Consistency.José M. Méndez & Gemma Robles - 2009 - Journal of Logic, Language and Information 18 (2):199-216.
    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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  28.  1
    Belnap-Dunn Semantics for Natural Implicative Expansions of Kleene's Strong Three-Valued Matrix with Two Designated Values.Gemma Robles & José M. Méndez - forthcoming - Journal of Applied Non-Classical Logics:1-27.
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  29.  8
    Converse Ackermann Property and Constructive Negation Defined with a Negation Connective.Gemma Robles & José M. Méndez - 2006 - Logic and Logical Philosophy 15 (2):113-130.
    The Converse Ackermann Property is the unprovability of formulas of the form (A -> B) -> C when C does contain neither -> nor ¬. Intuitively, the CAP amounts to rule out the derivability of pure non-necessitive propositions from non-necessitive ones. A constructive negation of the sort historically defined by, e.g., Johansson is added to positive logics with the CAP in the spectrum delimited by Ticket Entailment and Dummett’s logic LC.
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  30.  1
    The Basic Constructive Logic for Absolute Consistency Defined with a Propositional Falsity Constant.Gemma Robles - 2008 - Logic Journal of the IGPL 16 (3):275-291.
    The logic BKc6 is the basic constructive logic in the ternary relational semantics adequate to consistency understood as absolute consistency, i.e., non-triviality. Negation is introduced in BKc6 with a negation connective. The aim of this paper is to define the logic BKc6F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc6 and BKc6F are definitionally equivalent. Then, we show how to extend BKc6F within the spectrum of logics delimited by contractionless intuitionistic logic. All logics (...)
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  31. Extensions of the Basic Constructive Logic for Negation-Consistency BKc4.Gemma Robles - 2008 - Logique Et Analyse 51.
  32.  3
    Reduced Routley–Meyer Semantics for the Logics Characterized by Natural Implicative Expansions of Kleene’s Strong 3-Valued Matrix.Gemma Robles - forthcoming - Logic Journal of the IGPL.
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  33.  39
    Curry's Paradox, Generalized Modus Ponens Axiom and Depth Relevance.Gemma Robles & José M. Méndez - 2014 - Studia Logica 102 (1):185-217.
    “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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  34. Intutionistic Propositional Logic with the Converse Ackerman Poperty.Gemma Robles - 2003 - Teorema: International Journal of Philosophy 22 (1-2):46-54.
  35.  23
    A Constructive Negation for Logics Including TW+.Gemma Robles & José M. Méndez - 2005 - Journal of Applied Non-Classical Logics 15 (4):389-404.
    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 paper.
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  36.  22
    A Weak Logic with the Axiom Mingle Lacking the Variable-Sharing Property.Gemma Robles, Francisco Salto & José M. Méndez - 2011 - Bulletin of the Section of Logic 40 (3/4):195-202.
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  37.  21
    The Basic Constructive Logic for Weak Consistency and the Reductio Axioms.Gemma Robles & José M. Méndez - 2009 - Bulletin of the Section of Logic 38 (1/2):61-76.
  38.  26
    El Sistema Bp+ : Una Lógica Positiva Mínima Para la Negación Mínima (the System Bp+: A Minimal Positive Logic for Minimal Negation).José M. Méndez, Francisco Salto & Gemma Robles - 2007 - Theoria 22 (1):81-91.
    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, some of (...)
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  39.  21
    A Semantical Proof of the Admissibility of the Rule Assertion in Some Relevant and Modal Logics.Gemma Robles - 2012 - Bulletin of the Section of Logic 41 (1/2):51-60.
  40.  15
    A Note on the Non-Involutive Routley Star.Gemma Robles - 2008 - Bulletin of the Section of Logic 37 (1):19-27.
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  41.  26
    Paraconsistent Logics Included in Lewis’ S4.Gemma Robles & José M. Méndez - 2010 - Review of Symbolic Logic 3 (3):442-466.
    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 (...)
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  42.  16
    Restricting the Contraction Axiom in Dummett's LC: A Sublogic of LC with the Converse Ackermann Property, the Logic LCo.Francisco Salto, José M. Méndez & Gemma Robles - 2001 - Bulletin of the Section of Logic 30 (3):139-146.
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  43.  12
    Exhaustively Axiomatizing Rmo→ with a Select List of Representative Theses Including Restricted Mingle Principles.Francisco Salto, Gemma Robles & José M. Méndez - 1999 - Bulletin of the Section of Logic 28 (4):195-206.
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  44.  7
    A Companion to Brady's 4-Valued Relevant Logic BN4: The 4-Valued Logic of Entailment E4.Gemma Robles & José M. Méndez - 2016 - Logic Journal of the IGPL 24 (5).
  45.  11
    The Non-Relevant De Morgan Minimal Logic in Routley-Meyer Semantics with No Designated Points.Gemma Robles & José M. Méndez - 2014 - Journal of Applied Non-Classical Logics 24 (4):321-332.
    Sylvan and Plumwood’s is the relevant De Morgan minimal logic in the Routley-Meyer semantics with a set of designated points. The aim of this paper is to define the logic and some of its extensions. The logic is the non-relevant De Morgan minimal logic in the Routley-Meyer semantics without a set of designated points.
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  46.  9
    Exhaustively Axiomatizing S3°→ and S4°→.Gemma Robles, Francisco Salto & José M. Méndez - 2008 - Teorema: International Journal of Philosophy 27 (2):79-89.
  47.  12
    A Constructive Negation Defined with a Negation Connective for Logics Including Bp+.Gemma Robles, Francisco Salto & José M. Méndez - 2005 - Bulletin of the Section of Logic 34 (3):177-190.
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  48. Two Versions of Minimal Intuitionism with the CAP. A Note.José Manuel Méndez Rodríguez & Gemma Robles - 2005 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 20 (2):183-190.
  49.  4
    El sistema Bp+ : una lógica positiva mínima para la negación mínima.José M. Méndez, Francisco Salto & Gemma Robles - 2007 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 22 (1):81-91.
    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, some of (...)
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  50.  7
    Negation Introduced with the Unary Connective.Gemma Robles - 2009 - Journal of Applied Non-Classical Logics 19 (3):371-388.
    In the first part of this paper a minimal and an intuitionistic negation is introduced in a wide spectrum of relevance logics extending Routley and Meyer's basic positive logic B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule. Negation is introduced with a propositional falsity constant. The aim of this paper is to build up logics definitionally equivalent to those in the aforementioned paper, negation being now introduced with the (...)
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