On the Ternary Relation and Conditionality

Beall, Jc; Brady, Ross T.; Dunn, J. Michael; Hazen, Allen; Mares, Edwin D.; Meyer, Robert K.; Priest, Graham; Restall, Greg; Ripley, David; Slaney, John; Sylvan, Richard

doi:10.1007/s10992-011-9191-5

Cited by 66 publications

(46 citation statements)

References 17 publications

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“…One message we have for logicians is that it can have more clarity and impact when the model theory is grounded in concrete engineering or scientific problems; that is, where the elements of the model have a clear reading or interpretation apart from their role in defining the semantics of sentences. For example, relevant logicians have admitted to have struggles in interpreting the meaning of the elements in their formal semantics based on ternary relations (Beall et al 2012). Their semantics enjoys completeness theorems with respect to their proof theories, but the subject matter of the models themselves is not evident.…”

Section: Resultsmentioning

confidence: 99%

Why Separation Logic Works

2018

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One might poetically muse that computers have the essence both of logic and machines. Through the case of the history of Separation Logic, we explore how this assertion is more than idle poetry. Separation Logic works because it merges the software engineer's conceptual model of a program's manipulation of computer memory with the logical model that interprets what sentences in the logic are true, and because it has a proof theory which aids in the crucial problem of scaling the reasoning task. Scalability is a central problem, and some would even say the central problem, in applications of logic in computer science. Separation Logic is an interesting case because of its widespread success in verification tools. For these two senses of model-the engineering/conceptual and the logical-to merge in a genuine sense, each must maintain their norms of use from their home disciplines. When this occurs, both the logic and engineering benefit greatly. Seeking this intersection of two different senses of model provides a strategy for how computer scientists and logicians may be successful. Furthermore, the history of Separation Logic for analysing programs provides a novel case for philosophers of science of how software engineers and computer scientists develop models and the components of such models. We provide three contributions: an exploration of the extent of models merging that is necessary for success in computer science; an introduction to the technical details of Separation Logic, which can be used for reasoning about other exhaustible resources; and an introduction to (a subset of) the problems, process, and results of computer scientists for those outside the field.

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Section: Resultsmentioning

confidence: 99%

Why Separation Logic Works

2018

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“…n+1-ary) relation on worlds. Subsequently, a number of philosophical interpretations of this ternary relation have been proposed and we will briefly recall some of them at the end of this section (see [18,58,50] for more details). However, one has to admit that providing a non-circular and conceptually grounded interpretation of this relation remains problematic.…”

Section: Updates As Ternary Relationsmentioning

confidence: 99%

“…Moreover, like the ternary relation corresponding to a product update (Equation (10)), this ternary relation is 'two-sorted': the antecedent of a conditional takes value in a set of worlds (instead of a single world). 2 So, the dynamic reading of the ternary semantics is consistent with the dynamic reading of conditionals proposed by Ramsey. This dynamic reading was not really considered and investigated by substructural logicians when they connected the substructural ternary semantics with conditional logic [18]. On the other hand, the dynamic reading of inferences has been stressed to a large extent by van Benthem [27,30] (we will come back to this point in Section 4.2), and also by Baltag and Smets who distinguished dynamic belief revision from static (standard) belief revision [12,13,14].…”

Section: Updates As Ternary Relationsmentioning

confidence: 99%

“…In this interpretation, the ternary relation R represents the informational links in situations: if there is an informational link in situation x that says that an infon σ carries the information that the infon π also holds, then if Rxyz holds and y contains the infon σ , then z contains the infon π. Other interpretations of the ternary relation have been proposed in [18], with a particular focus on their relation to conditionality.…”

Section: Other Interpretations Of the Ternary Relationmentioning

confidence: 99%

“…The wellknown semantics for substructural logics is based on a ternary relation introduced by Routley & Meyer for relevance logic in the 1970's [60,61,59,62]. However, the introduction of this ternary relation was originally motivated by technical reasons, and it turns out that providing a non-circular and conceptually grounded interpretation of this relation remains problematic [18]. As we shall see, the ternary semantics of DEL provides a conceptual foundation for Routley & Meyer's semantics.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamic Epistemic Logic as a Substructural Logic

Aucher

2014

Outstanding Contributions to Logic

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Dynamic Epistemic Logic (DEL) is an influential logical framework for reasoning about the dynamics of beliefs and knowledge. It has been related to older and more established logical frameworks. Despite these connections, DEL remains, arguably, a rather isolated logic in the vast realm of non-classical logics and modal logics. This is problematic if logic is to be viewed ultimately as a unified and unifying field and if we want to avoid that DEL goes on "riding off madly in all directions" (a metaphor used by van Benthem about logic in general). In this article, we show that DEL can be redefined naturally and meaningfully as a two-sorted substructural logic. In fact, it is even one of the most primitive substructural logics since it does not preserve any of the structural rules. Moreover, the ternary semantics of DEL and its dynamic interpretation provides a conceptual foundation for the Routley & Meyer's semantics of substructural logics.

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Bad Worlds

Girard

Weber

2015

Thought: A Journal of Philosophy

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The idea of relevant logic—that irrelevant inferences are invalid—is appealing. But the standard semantics for relevant logics involve baroque metaphysics: a three‐place accessibility relation, a star operator, and ‘bad’ (impossible/non‐normal) worlds. In this article we propose that these oddities express a mismatch between non‐classical object theory and classical metatheory. A uniformly relevant semantics for relevant logic is a better fit.

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On the Ternary Relation and Conditionality

Cited by 66 publications

References 17 publications

Why Separation Logic Works

Why Separation Logic Works

Dynamic Epistemic Logic as a Substructural Logic

Bad Worlds

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