Search results for: Gemma Robles

Items from 1 to 20 out of 22 results

article

A Note on Gödel-Dummet Logic LC

Gemma Robles, José M. Méndez

Bulletin of the Section of Logic > 2021 > 50 > 3 > 325-335

Let $A_{0},A_{1},...,A_{n}$ be (possibly) distintict wffs, $n$ being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom $(A_{0}\rightarrow A_{1})\vee ...\vee (A_{n-1}\rightarrow A_{n})\vee (A_{n}\rightarrow A_{0})$ is equivalent to Gödel-Dummett logic LC. However, if $n$ is an even number equal to or greater than 2, IPC plus the said axiom is a...

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A 2-set-up Routley-Meyer Semantics for the 4-valued Relevant Logic E4

Gemma Robles, Sandra M. López, José M. Blanco, Marcos M. Recio, more

Bulletin of the Section of Logic > 2016 > 45 > 2

The logic BN4 can be considered as the 4-valued logic of the relevant conditional and the logic E4, as the 4-valued logic of (relevant) entailment. The aim of this paper is to endow E4 with a 2-set-up Routley-Meyer semantics. It is proved that E4 is strongly sound and complete w.r.t. this semantics.

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Relational semantics for the 4-valued relevant logics BN4 and E4

Gemma Robles, José M. Blanco, Sandra M. López, Jesús R. Paradela, more

Logic and Logical Philosophy > 2016 > 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 (relevant) 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.

article

Relational semantics for the 4-valued relevant logics BN4 and E4

Gemma Robles

Logic and Logical Philosophy > 2016 > 25 > 2 > 173–201

article

An Interpretation of Łukasiewicz’s 4-Valued Modal Logic

José M. Méndez, Gemma Robles, Francisco Salto

Journal of Philosophical Logic > 2016 > 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 (rt) and possibilist (pt) theses defined as follows: rt: the value (in a bivalent sense) of modal formulas is equivalent to the value of their...

article

Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

José M. Méndez, Gemma Robles

Journal of Logic, Language and Information > 2016 > 25 > 2 > 163-189

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: (1) the linearly ordered systems Ł3,..., ,..., $$\hbox {L}_{\omega }$$ L ω ; (2) the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type (modal) paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued...

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A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes

José M. Méndez, Gemma Robles

Logica Universalis > 2015 > 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 (modal) 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...

article

A simple Henkin-style completeness proof for Gödel 3-valued logic G3

Gemma Robles

Logic and Logical Philosophy > 2014 > 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 (u-semantics) of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation” (u-interpretation). It is shown that consistent prime theories built upon G3 can be understood as (canonical) u-interpretations...

article

Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

Gemma Robles, Francisco Salto, José M. Méndez

Journal of Philosophical Logic > 2014 > 43 > 2-3 > 303-332

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...

article

Blocking the Routes to Triviality with Depth Relevance

Gemma Robles, José M. Méndez

Journal of Logic, Language and Information > 2014 > 23 > 4 > 493-526

In Rogerson and Restall’s (J Philos Log 36, 2006, p. 435), the “class of implication formulas known to trivialize (NC)” (NC abbreviates “naïve comprehension”) 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.

article

Curry’s Paradox, Generalized Modus Ponens Axiom and Depth Relevance

Gemma Robles, José M. Méndez

Studia Logica > 2014 > 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...

article

A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart

Gemma Robles

Logica Universalis > 2013 > 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...

article

Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem

Gemma Robles

Reports on Mathematical Logic > 2012 > 47 > 147–171

As is stated in its title, in this paper consistency is understood as the absence of the negation of any implicative theorem. Then, a series of logics adequate to this concept of consistency is defined within the context of the ternary relational semantics with a set of designated points, negation being modelled with the Routley operator. Soundness and completeness theorems are provided for each one...

article

Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem

Gemma Robles

Reports on Mathematical Logic > 2012 > 47

article

A modal restriction of R-Mingle with the variable-sharing property

Gemma Robles, José M. Méndez, Francisco Salto

Logic and Logical Philosophy > 2010 > 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.

article

A Routley-Meyer Type Semantics for Relevant Logics Including B_r Plus the Disjunctive Syllogism

Gemma Robles, José M. Méndez

Journal of Philosophical Logic > 2010 > 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 $\vdash A\rightarrow \lnot A\Rightarrow \vdash \lnot A$ 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.

article

Minimal Non-Relevant Logics without the K Axiom II. Negation Introduced as a Primitive Unary Connective

Gemma Robles

Reports on Mathematical Logic > 2010 > 45

In the first part of this paper (RML No. 42) a spectrum of constructive logics without the K axiom is defined. Negation is introduced with a propositional falsity constant. The aim of this second part is to build up logics definitionally equivalent to those displayed in the first part, negation being now introduced as a primitive unary connective. Relational ternary semantics is provided for all logics...

article

The Basic Constructive Logic for Absolute Consistency

José M. Méndez, Gemma Robles

Journal of Logic, Language and Information > 2009 > 18 > 2 > 199-216

In this paper, consistency is understood as absolute consistency (i.e. non-triviality). The basic constructive logic B_Kc6, 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 B_Kc6 up to contractionless intuitionistic logic. All logics defined in this paper...

article

Strong Paraconsistency and the Basic Constructive Logic for an Even Weaker Sense of Consistency

Gemma Robles, José M. Méndez

Journal of Logic, Language and Information > 2009 > 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...

article

The basic constructive logic for a weak sense of consistency

Gemma Robles, José M. Méndez

Journal of Logic, Language and Information > 2008 > 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 B_Kc1 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 B_Kc1 within the spectrum delimited by contractionless...

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Search results for: Gemma Robles

A Note on Gödel-Dummet Logic LC

A 2-set-up Routley-Meyer Semantics for the 4-valued Relevant Logic E4

Relational semantics for the 4-valued relevant logics BN4 and E4

Relational semantics for the 4-valued relevant logics BN4 and E4

An Interpretation of Łukasiewicz’s 4-Valued Modal Logic

Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes

A simple Henkin-style completeness proof for Gödel 3-valued logic G3

Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

Blocking the Routes to Triviality with Depth Relevance

Curry’s Paradox, Generalized Modus Ponens Axiom and Depth Relevance

A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart

Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem

Paraconsistency and Consistency Understood as the Absence of the Negation of any Implicative Theorem

A modal restriction of R-Mingle with the variable-sharing property

A Routley-Meyer Type Semantics for Relevant Logics Including B_r Plus the Disjunctive Syllogism

Minimal Non-Relevant Logics without the K Axiom II. Negation Introduced as a Primitive Unary Connective

The Basic Constructive Logic for Absolute Consistency

Strong Paraconsistency and the Basic Constructive Logic for an Even Weaker Sense of Consistency

The basic constructive logic for a weak sense of consistency

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