Shay Logan

Kansas State University

Varieties of Variable Sharing or: How I Stopped Worrying and Learned to Love Nonuniform Substitutions

Logic and Set Theory Seminar

4th October 2023, 4:00 pm – 5:00 pm
Online, Zoom:, ID: 993 0950 0789

Relevance and variable sharing have gone hand-in-hand since the very start. The relationship in fact predates the seminal texts of the movement, as one can see by comparing the publication dates of [1] and [2].

But both both parties to this marriage have changed over the decades, and the marriage looks quite different now than it did when it first began. In particular, variable sharing results no longer play at being quite so hard to get. This is the result of a novel way to prove strong variable sharing results. The key bit (introduced in [3]; [4]) is the use of nonuniform substitutions. It turns out that this key bit is key in more than one way: not only does it unlock easy proofs of strong variable sharing results, it also opens a door behind which hide a plethora of novel and quite unanticipated forms of variable sharing as well. For each of these forms of variable sharing, a proof that is not interestingly different from the proof of the main result in [3] shows that weak-enough logics exhibit that form of variable sharing.

Given all this, the goal of this talk is twofold. First, I’ll survey the state of the art in order to show you how to use nonuniform substitutions to achieve profit and fame. After that, I’ll try to convince you that you shouldn’t feel bad about doing so.

[1] Nuel D. Belnap, Entailment and relevance, Journal of Symbolic Logic,
vol. 25 (1960), no. 2, pp. 144–146.
[2] Alan Ross Anderson and Nuel D. Belnap, Entailment: The Logic of Relevance
and Neccessity, Vol. I, Princeton, Princeton University Press (1975).,
[3] Shay Allen Logan, Depth Relevance and Hyperformalism, Journal of Philosophical
Logic, vol. 51 (2022), no. 4, pp. 721–737.
[4] Shay Allen Logan, Correction to: Depth Relevance and Hyperformalism,
Journal of Philosophical Logic, vol. 52 (2023), no. 4, p. 1235.

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