That’s an interesting theory, thanks for this post!
However… We arguably already have a theory of material inference. Probability theory! Why do we need another one? Unlike Brandom’s material logic, probability theory is arguably a) relatively simple in its rules (though the Kolmogorov axioms aren’t explicitly phrased in terms of inference), and b) it supports degrees of belief and degrees of confirmation. Brandom’s theory assumes all beliefs and inferences are binary.
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.
But if p is “It’s a cat” and q is “It has four legs”, and P describes our beliefs (or more precisely, say, my beliefs at 5 pm UTC October 20, 2024), then P(q|p)>P(q). Which surely means p is a materially good reason for q. But p⊬q, so the inference from p to q is still logically bad. So we don’t have logicism about reasons in probability theory. Moreover, probability expressions are not invariant under substituting non-logical vocabulary. For example, if r is “It has two legs”, and we substitute q with r, then P(r|p)<P(r). Which can only mean the inference from p to r is materially bad.
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.
I think the axioms of probability can be thought of as being relative to material conceptual relations. Specifically, the additivity axiom says that the probabilities of “mutually exclusive” statements can be added together to yield the probability of their disjunction. What does “mutually exclusive” mean? Logically inconsistent? Not necessarily. It could simply mean materially inconsistent. For example, “Bob is married” and “Bob is a bachelor” are (materially, though not logically) mutually exclusive. So their probabilities can be added to form the disjunction. (This arguably also solves the problem of logical omniscience, see here).
That’s an interesting theory, thanks for this post!
However… We arguably already have a theory of material inference. Probability theory! Why do we need another one? Unlike Brandom’s material logic, probability theory is arguably a) relatively simple in its rules (though the Kolmogorov axioms aren’t explicitly phrased in terms of inference), and b) it supports degrees of belief and degrees of confirmation. Brandom’s theory assumes all beliefs and inferences are binary.
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.
But if p is “It’s a cat” and q is “It has four legs”, and P describes our beliefs (or more precisely, say, my beliefs at 5 pm UTC October 20, 2024), then P(q|p)>P(q). Which surely means p is a materially good reason for q. But p⊬q, so the inference from p to q is still logically bad. So we don’t have logicism about reasons in probability theory. Moreover, probability expressions are not invariant under substituting non-logical vocabulary. For example, if r is “It has two legs”, and we substitute q with r, then P(r|p)<P(r). Which can only mean the inference from p to r is materially bad.
I think the axioms of probability can be thought of as being relative to material conceptual relations. Specifically, the additivity axiom says that the probabilities of “mutually exclusive” statements can be added together to yield the probability of their disjunction. What does “mutually exclusive” mean? Logically inconsistent? Not necessarily. It could simply mean materially inconsistent. For example, “Bob is married” and “Bob is a bachelor” are (materially, though not logically) mutually exclusive. So their probabilities can be added to form the disjunction. (This arguably also solves the problem of logical omniscience, see here).