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A little clarification is probably required. By "non-monotonic logic", I have in mind the various formal treatments of commonsense entailment that have popped up in AI over the years (e.g. Default logic, autoepistemic logic, circumscription, etc.)

Now, any substructural logic lacking weakening is going to be non-monotonic. But the motivations for dropping weakening (in, say, the Lambek Calculus) are quite different than what's going on when we're worrying about defeasible argument. And substructural logics all tend to be subclassical while nonmonotonic ones are all supraclassical. Still, the surface formal similarity may suggest that there may be some interesting way to tie them together (perhaps by introducing a new connective to serve as a defeasible conditional in a suitable substructural logic?).

Anyone have any references for exploring this connection?

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    Any modern psicology attempt applied to marketing? Sarcasm aside, the fields are young and all I've seen around is foundational. Someone can even argue that there is no point in drawing "strong" theoretical connections between such "weak logic" fields but sooner or later some will attempt it nonetheless: just not yet as far as I know – Lorenzo Sep 15 '16 at 06:31
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    Linear logic is a substructural logic that admits weakening but is monotonic. So the first sentence of your second paragraph is incorrect. – Rob Arthan Sep 15 '16 at 21:47
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    Not sure I understand this. Linear Logic doesn't admit uncontrolled weakening, and isn't monotonic as far as I know (any failure of the weakening schema also giving a failure of monotonicity...) –  Sep 15 '16 at 22:41
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    Apologies, I meant "affine logic" not "linear logic". (Affine logic is linear logic plus weakening.) How are you justifying your claim that weakening leads to a failure of non-monotonicity? – Rob Arthan Sep 16 '16 at 23:26
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    Maybe I'm being naive, but if the logic doesn't admit weakening, there must be some sequent with $\Gamma \vdash \Delta$ but not $\Gamma, X\vdash \Delta$. If you think of turnstile as characterizing the entailment relation, that just gives you a counterinstance to monotonicity. –  Sep 17 '16 at 01:53
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    A logic is non-monotonic if adding a hypothesis can make a deduction invalid. How doe the weakening rule conflict with this? – Rob Arthan Sep 19 '16 at 21:30
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    If weakening is admissible as an inference rule, the logic will automatically be monotonic (at least finitely). From a deduction $\Gamma\vdash\Delta $, weakening states that you can "add" the information "$\Pi$" and have a valid deduction $\Gamma,\Pi\vdash\Delta$". The inadmissibility of weakening is thus the statement that this format of argument is not, in general, valid (i.e. the consequence relation is nonmonotonic). I might be being naive, but this seems clear to me... –  Sep 20 '16 at 17:02

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I am also presently trying to figure out the exact relations between the two fields. Maybe the following paper is of help:

Aucher, Guillaume. (2015). When Conditional Logic and Belief Revision Meet Substructural Logics. CEUR Workshop Proceedings. 1423.

Rappatoni
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    While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review – Jan Mar 11 '20 at 15:22
  • Definitely, I will add to the answer once I have read and understood this (and potentially other) papers on the topic better. – Rappatoni Mar 12 '20 at 16:47