Monday, October 25, 2021

Russell on Barriers to Entailment

 Gillian Russell has an interesting paper, How to Prove Hume's Law, which is probably the best thing I've read on the subject in a while. As I've noted before, there is actually no single thing that is Hume's Law, since different people use it to mean different things, all of which are different from what Hume clearly meant. Russell introduces the principle in a common way -- You can't get an Ought from an Is -- but the version that Russell actually considers is that you can't get a normative conclusion from a descriptive premise, which is not the same. In any case, the primary interest is less this than the logical contextualizing of the idea.

Russell takes the principle to be one of a family of principles, barriers to entailment, such as:

You can't deduce a universal conclusion from particular premises.
You can't deduce conclusions about the future from premises about the past or present.
You can't deduce conclusions about how the world must be from premises about how the world is.
You can't deduce indexical conclusions from non-indexical premises.

Russell -- and this is the fundamental value of the paper -- argues for a unified logical account of all of these and then, assuming standard assumptions in logical systems, proves a Limited General Barrier Theorem that handles some commonly proposed counterexamples to the above principles, especially to 'Hume's Law'.  There is a kind of curious twist in the argument -- what Russell actually shows is that all of the above principles are false, and that the major counterexamples (including some Russell does not consider but are analogous to those that are considered) are essentially right, but that the counterexamples each fall under a condition. 

Trying not to get too bogged down in logical technicalities, the essential idea is that we often have cases in which we are dealing with claims that can change truth value depending on context. We find many such cases in the matters with which the above barrier principles are concerned, and in ways that seem closely connected to why people think they are true. For instance, the indexical claim "I am Brandon" is true of me and false of most of you; the context shifts, and the truth value can shift with the context. If "I am Brandon" followed from non-indexical statements, the shift of truth value would not be possible: the truth of the non-indexical statements does not depend on the indexical context, so any conclusions that can be deduced from them will not depend on the indexical context. Thus the indexical shift will block certain kinds of inferences from non-indexicals to indexicals. However, it's obvious that this cannot be an absolute barrier. "All adult Americans have a name" is a non-indexical statement, and from this I can conclude, "If I am an adult American, I have a name", which is indexical. The reason is that, while the latter statement shifts its truth value in different contexts, if we look only at the contexts in which the non-indexical statement is true, the indexical statement has to be true in all of them. This general point can be made more precise with model theory, and it can be generalized to all of the other cases, and, what is more, the point is true in multiple different modal logical systems (Russell looks at S4, B, and S5 in particular), with only minor incidental modifications appropriate to the difference in logical systems.

Of course, it's always possible that there are other logical systems, particularly nonstandard modal logical systems, that plausibly represent some sort of modality we actually use, which will not allow the barrier principle; and deontic reasoning is the form of modal reasoning that has always been hardet to fit with standard moral logical systems. (I also do not think Russell's account of normativity is a very good account of what people generally mean by normativity, even in using the descriptive-normative version of Hume's Law. But it's in the general ballpark.) It's worth remembering that a barrier with exceptions is in fact less a barrier than a gateway, and all of logical reasoning is described by such gateways. As noted above, what Russell actually shows is that all of the above principles are false; despite the title of the paper, what the paper shows is that Hume's Law is wrong. We knew this from the counterexamples -- at least, everyone who wasn't stubbornly holding on to Hume's Law knew it from the counterexamples. Nonetheless, she establishes a general account of how this relates to other principles, and how one can in each case actually make the kind of inference that is forbidden in a way that explains why people would err in thinking that it could not be made.

Apparently Russell is working on a book on Barriers to Entailment; I will definitely have to read it when it comes out.