Then that minimum does not make a good denominator because it’s always extremely small. It will pick phi to be as powerful as possible to make L small, aka set phi to bottom. (If the denominator before that version is defined at all, bottom is a propositional tautology given A.)
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?
Yup, a monoid, because ϕ∨⊥=ϕ and A∪∅=A, so it acts as an identitity element, and we don’t care about the order. Nice catch.
You’re also correct about what propositional tautology given A means.
Then that minimum does not make a good denominator because it’s always extremely small. It will pick phi to be as powerful as possible to make L small, aka set phi to bottom. (If the denominator before that version is defined at all, bottom is a propositional tautology given A.)
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?