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Hi, sorry I'm not directly familiar with that Dewey work. As far as classical American pragmatism goes, I can only point to Brandom's endorsement of Cheryl Misak's transformative new way of looking at it in Lecture 4 of this course, which might be helpful for drawing this connection.
Thanks for the comment! Probability theory is a natural thing to reach for in order to both recover defeasibility while still upholding "logicism about reasons" (seeing the inferences as underwritten by a logical formalism). And of course it's a very useful calculational tool (as is classical logic)! But I don't think it can play the role that material inference plays in this theory. I think a whole post would be needed to make this point clearly[1], but I will try in a comment.
Probability theory is still a formal calculus, which is completely invariant to substituting nonlogical vocabulary for other nonlogical vocabulary (this is good - exactly what we want for a formal calculus). However, the semantics for such nonlogical vocabulary must always always presupposed before one can apply such formal reasoning. The sentences to which credences are attached must be taken to be already conceptually contentful, in advance of playing the role in reasoning that Bayesians reconstruct. This is problematic as a foundation for inferentialists who are concerned with the inferences that are constitutive of what 'cat' means within sentences like "Conditioning on it being a cat, it has four legs".
Footnote 8 applies to trying to understand what we really mean to say as covert probabilistic claims. Laws of probability theory still impose a structure on relations between material concepts (there are still forms of monotonicity and transitivity), whereas the logical-expressivist order of explanation argues that the theoretician isn't entitled to a priori impose such a structure on all material concepts: rather, their job is to describe them.
Logical expressivism isn't committed to assuming inferences and beliefs are binary! There are plenty of friendly amendments to make - it's just that even in the binary, propositional case there's a lot of richness, insight, and work to be done.
- ^
Some of this is downstream of deeper conflicts between pragmatism and representationalism, so I don't see myself as making the kinds of arguments now that could cause that kind of paradigm shift.