Oh, I see what the issue is. Propositional tautology given A means A⊢pcϕ, not A⊢ϕ. So yeah, when A is a boolean that is equivalent to ⊥ via boolean logic alone, we can’t use that A for the exact reason you said, but if A isn’t equivalent to ⊥ via boolean logic alone (although it may be possible to infer ⊥ by other means), then the denominator isn’t necessarily small.
So the valuation of any propositional consequence of A is going to be at least 1, with equality reached when it does as much of the work of proving bottom as it is possible to do in propositional calculus. Letting valuations go above 1 doesn’t seem like what you want?
Oh, I see what the issue is. Propositional tautology given A means A⊢pcϕ, not A⊢ϕ. So yeah, when A is a boolean that is equivalent to ⊥ via boolean logic alone, we can’t use that A for the exact reason you said, but if A isn’t equivalent to ⊥ via boolean logic alone (although it may be possible to infer ⊥ by other means), then the denominator isn’t necessarily small.
So the valuation of any propositional consequence of A is going to be at least 1, with equality reached when it does as much of the work of proving bottom as it is possible to do in propositional calculus. Letting valuations go above 1 doesn’t seem like what you want?