Abstract
In this paper, we examine a number of relevant logics’ variable sharing properties from the perspective of theories of topic or subject-matter. We take cues from Franz Berto’s recent work on topic to show an alignment between families of variable sharing properties and responses to the topic transparency of relevant implication and negation. We then introduce and defend novel variable sharing properties stronger than strong depth relevance—which we call cn-relevance and lossless cn-relevance—showing that the properties are satisfied by the weak relevant logics \({\textbf{B}}\) and \({{\textbf{B}}}{{\textbf{M}}}\), respectively. We argue that such properties address a sort of semantic lossiness of strong depth relevance.
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Notes
The applicability of two-component approaches to topic in mainstream relevant logics has recently seen an extraordinarily compelling defense and model theory in Tedder (2023).
It is worth mentioning how Meyer, Dunn, and Leblanc aptly refer to the variable sharing property as the “crudest but most memorable result” (Meyer et al., 1974) concerning relevance with respect to the logic \({\textbf{R}}\).
The theory of topic is still in its relative infancy especially with respect to its ontology. We try to remain somewhat agnostic about the particular model of topic—favoring instead less formal intuitions—but note that many, if not all, of the historical approaches reviewed in e.g. Yablo (2014) cohere with these informal intuitions.
To be sure, we do not offer the foregoing as, say, an exegesis of Brady’s goals but rather as a possible explanation. We should note, however, that there is undoubtedly a topic-theoretic reading of Brady’s interpretation of his model theory of contents.
Note that this characterization coincides with the definition of antecedent and consequent parts of a sentence, i.e., that B appears positively (resp., negatively) in A corresponds to its being an antecedent part (resp., consequent part) of A.
For a clear discussion of the distinctions between these two expansions of \({\textbf{R}}\), see Mares and Goldblatt’s (2006).
This is closely related to a problem in Meyer’s relevant arithmetic \({\textbf{R}}^{\sharp }\), described by Brady as the that in \({\textbf{R}}^{\sharp }\), “\(m=n\rightarrow m' = n'\) [is a theorem] which has the natural numbers m and n in common, leads to \(0=0\rightarrow 100=100\), where the two numbers involved can be as far apart as you like.” (Brady, 2006, p. 11) That this theorem “smacks of irrelevance” (as Dunn says in Dunn (1987)) involves a similar acknowledgement that the overlap of terms—in this case two instances of 0 followed by sequences of \('\)—need not ensure relevance. Estrada-González and Tapia-Navarro’s (2021) takes up this matter in more detail.
For a broader range of examples, one can consult the discussion of intensionality and subject-matter in Ferguson (2023).
This priority is not universal—Dov Gabbay considers several subtle means through which consequents can exert a similar role in Gabbay (1972)—but the priority is extremely entrenched.
It’s worthwhile to note that there is in the literature another way in which depth relevance has been ‘expanded’; namely in the exploration of depth relevant logics that aren’t typically included in the class of relevant logics. This project has been taken up in e.g. Robles and Méndez (2014).
As a reviewer points out, one can extract from this proof a proof of the claim that weak-enough relevant logics are depth relevant in the sense of Brady (1984). In fact, the proof in Logan (2022) (which was corrected in Logan (2023)) is of exactly this form, though for a slightly broader class of logics.
We thank both reviewers of this paper for drawing our attention to this assumption.
It’s interesting that the same modification of \(\sigma \)—that is, the change from \(\sigma \) to \(\sigma ^{c/nc}\)—does the job in both cases.
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Acknowledgements
The input of two reviewers for this journal was incredibly useful; we appreciate their remarks. Thomas Ferguson’s contributions to this paper were written with the support of the MetaMuSo project (Czech Science Foundation project GA22-01137 S).
Funding
This work has received funding by Czech Science Foundation, grant no. 22-01137 S. There are no competing interests.
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Ferguson, T.M., Logan, S.A. Topic Transparency and Variable Sharing in Weak Relevant Logics. Erkenn (2023). https://doi.org/10.1007/s10670-023-00748-6
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DOI: https://doi.org/10.1007/s10670-023-00748-6