Another paper by Gillian Russell that I think worth some consideration. This one is of direct relevance, amongst other things, to discussions of induction, the is/ought barrier, temporal logic and modality.

Hume’s Law and other Barriers to Entailment

This is the 2013 paper that led to her book, Barriers to Entailment: Hume's Law and other Limits on Logical Consequence, 2023. I’ll stick to the paper here, since it is openly available.

A barrier to entailment occurs when a set of premises that might on the face of it appear to be good reasons for some conclusion, can’t be made to formally perform as required. So, for instance, it can be argued that if every time one finds a poppy, it is purple, one might feel entitled to conclude that all poppies are purple. This is the familiar problem of induction - no sequence of individual statements, “Here is a purple poppy”, “There is a purple poppy”, can deductively justify the move to “All poppies are purple”.

A couple of other such barriers have been mooted. Of particular interest are that no list of things that are the case can lead one to a conclusion as to what ought to be the case. And no list of things that have been the case can lead to a conclusion as to what will be the case in the future. And no sentence about what might be the case can lead to a conclusion as to what must be the case. There are others, but these will do for now.

Now, each of these carries its own baggage of some considerable literature, and each has its arguments and answers. Part of what is special about the present paper is that it attempts to analyse their common logical structure, formally. And it attempts to do so using the machinery of first-order logic.

Russell sets out her case quite formally, but finishes with a discussion of natural language examples. Informally, she identifies a group of satisfiable sentences that do not change their truth value if new objects are introduced into the conversation, and another set that can be made false by introducing new objects. So “That is a purple poppy” will remain true if we introduce more poppies, either purple or red; but “This is the only purple poppy” will change its truth value if another purple poppy is introduced. Russell calls sentences that do not change their truth value “particular” and those that must, “Universal”.

It should be noted that the term “Universal” is not a reference to the quantifier. These are not, for instance, just those sentences beginning with a universal quantifier. It is also worthy of note that the categories “particular” and “universal” do not cover all sentences - there are sentences that are neither.

We can form sets of particular sentences, such as {this is a purple poppy, that is a purple poppy}. These sets will themselves remain particular, in that adding more objects to the discussion - more purple poppies, or a white swan - will not change the particularity of the set.

And here’s the point. From sets of particular sentences alone, we cannot derive a universal sentence. There is a barrier to entailment between particular and universal sentences such that the first cannot imply the second.

That account is of course a mere outline, the devil is in the detail of the article. The relation to induction should, I hope, be apparent. Russell offers ways of understanding Hume’s Law, Temporality, and other instances of barriers to entailment using this frame. There are other approaches, notably some involving relevance logic, that seek a similar result. Russell’s approach seems to be more intuitive.

Perhaps some will find this approach interesting.