Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star)

Restall, Greg

doi:10.1007/978-94-015-9309-0_3

Cited by 96 publications

(50 citation statements)

References 8 publications

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“…Relevant logicians such as Dunn, Restall, and Mares [4,10,15] responded that the Routley star can be made sense of through the conception of negation as modality of impossibility [3]. According to it, negation as impossibility is understood in terms of a binary relation C on frames:…”

Section: Ternary Relation Routley-star and Negation As Modalitymentioning

confidence: 99%

Understanding Negation Implicationally in the Relevant Logic R

Onishi

2016

Stud Logica

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A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.The defining feature of display calculus is the display property that every substructure of a sequent can be "displayed" through display rules. Cut elimination theorem can be proved in a generic way based on this property. Definition 4.3. (Antecedent/succedent parts) In X Y , we say X is an antecedent part (AP) and Y is a succedent part (SP). We define:• If W ; Z is an AP (SP) in X Y , so are W and Z, and • In any substructure W > Z of X Y , W (Z) is an AP (SP) in X Y .Definition 4.4. We say two sequents are display equivalent if they are derivable from each other using only display rules.

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Section: Ternary Relation Routley-star and Negation As Modalitymentioning

confidence: 99%

Understanding Negation Implicationally in the Relevant Logic R

Onishi

2016

Stud Logica

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“…He descries some sort of negation rolling around in FDE. (Restall, 1997) 21 For more, see Gabbay and Woods (2003b). what he takes to be defining conditions on ideal rationality, and his theorems give them a binding force on the idealized agents of his model. Consider Super-Draw, the ideally rational agent who draws every consequence of anything he assents to.…”

Section: Concerning Question (2)mentioning

confidence: 99%

Advice on the Logic of Argument

Woods

2013

RHV

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ResumenDesde su creación moderna a principios de la década de los 70, la lógica informal ha puesto un especial énfasis en el análisis de las falacias y los esquemas de diálogo argumentativo. Desarrollos simultáneos en los círculos que se ocupan de los actos de comunicación de habla exhiben una concentración en el carácter dialéctico de la discusión. PALABRAS CLAVE: Lógica informal, argumento, diálogos AbstractSince its modern inception in the early 1970s, informal logic has placed a special emphasis on the analysis of fallacies and argumentative dialogue schemes. Concurrent developments in speech communication circles exhibit a like concentration on the dialectical character of argument.

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“…The condition for negation then utilises the compatibility relation: I have used Dunn's compatibility account of negation to give a philosophically respectable account of the semantics of negation in relevant logics [5]. The point here, however, is a formal one.…”

Section: Definition 12 a Relation Between Points In P And Atomic Promentioning

confidence: 99%

Defining double negation elimination

Restall

2000

Logic Journal of IGPL

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In his paper "Generalised Ortho Negation" [2] J. Michael Dunn mentions a claim of mine to the effect that there is no condition on 'perp frames' equivalent to the holding of double negation elimination ∼∼A A. That claim is wrong. In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses. 1 Compatibility FramesDunn's work on general models for negation has been a significant advance in our understanding of negation in non-classical logics [1,2]. These models generalise Kripke models for intuitionistic logic and Routley-Meyer models for relevant implication. I will recount the essential details of these models for negation here before we look at the behaviour of inference patterns such as double negation elimination. Definition 1.1 A frame is a triple P, C, consisting of a set P of points, and two binary relations C and on P such that• is a partial order on P . That is, is reflexive, transitive and antisymmetric on P .• C is antitone in both places. That is, for any x and y in P , if xCy, x x and y y then x Cy .The relation between points may be interpreted as one of information inclusion. A point x is informationally included in y if everything warranted by x (or encoded by x or made true by x or included in x or however else information is thought to be related to points) is also warranted by y. This is a transitive, reflexive relation on points. If the information content of a point is all you care about (as it is here) then antisymmetry is also an appropriate condition for inclusion.C is the relation of compatibility between points. 2 A point x is compatible with y if and only if all claims excluded by x are not warranted by y. It is clear that this merits the antitonicity condition -if x is compatible with y, then x (excluding no more than x) must be compatible with y (warranting no more than y).It is important to realise what "excluding" amounts to here. A point x excludes a piece of information when it rules it out. A point might neither warrant nor exclude 1 Thanks to anonymous referees of the journal for comments which helped improve the presentation of this paper. 2 I use 'C' for a relation of compatibility instead of '⊥' for the relation of incompatibility used by Dunn, simply because I use '⊥' as the false proposition which entails all others. The difference is merely notational. 853L.

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Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star)

Cited by 96 publications

References 8 publications

Understanding Negation Implicationally in the Relevant Logic R

Understanding Negation Implicationally in the Relevant Logic R

Advice on the Logic of Argument

Defining double negation elimination

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