Gillian Russell: Barriers to entailment And so to Vrana's objection.
One of the ways of setting out a obligation in first order logic os to simply incorporate an opperator, O. Op is then just "One ought p" This has the advantage of simplicity, Humes rule being the brief remark that there can be no valid inference form φ to Oψ, φ being some statement concerning what is the case, Oψ some statement as to what ought be the case.
Following Russell's strategy, we'd be looking to perhaps show that φ was preserved, while Oψ was fragile, and hence no entailment relation can hold between them. φ → Oψ, then, is mixed, and so in the scheme of things, neither descriptive nor normative.
φ → Oψ is neither preserved nor fragile.
Now the General Barrier Theorem says roughly that no set of satisfiable sentences , each of which is preserved, entails a sentence which is fragile. It is about sentences that are either preserved or fragile.
Vrana's objection is that since φ → Oψ is neither preserved nor fragile, the General Barrier Theorem says nothing about it. So on this account, the Barrier Theorem tells us nothing about Hume's Law... but that's what we wanted!
And the second horn of the dilemma. Suppose we go along with the criticism, and strengthen our barrier to entailment so that no "is" statement can result in φ → Oψ; then we have ~φ ⊨ φ → Oψ; but that is exactly what we do not want! If we strengthen it enough to avoid Vrana's criticism, then it's demonstrably false.