*uniform substitution*: it is not true that

for all substitutionsFor example, the formulaf(i.e., functions from atoms to formulas) and for all formulas φ: if φ is N-valid thenf(φ) is N-valid.

*p*→¬¬

*p*is N-valid, but if we plug in

*q*∧

*q*for

*p*we get (

*q*∧

*q*)→¬¬(

*q*∧

*q*), which turns out to be N-invalid. Although N does not validate uniform substitution in its full generality, N is closed under

*some*classes of substitutions:

- renamings of atoms
- double negating atoms

- substituting N-validities for atoms,

*is N*More precisely, the question can be put this way. Let N′ be the closure of N under uniform substitution (i.e., N′ is the smallest set of formulas extending N that is closed under uniform substitution). We then ask:

**consistent**with uniform substitution?- Is N′ consistent? (We can vary the notion of consistency here, thereby precisifying the question.)
- Does N′ have a suitably straightforward dialogical characterization?

*not*the answer to question 2 above that we seek!

We, or at least I, also think that replacing atoms with their negations is also validity-preserving.

ReplyDeleteYes, that does seem to be valid, but a proof escapes me at the moment. Unlike the proofs of the two cases where we do have positive answers (renaming and double negating atoms), the approach used there (working directly with the game tree) doesn't immediately seem to work in the case of simply negating atoms, because the stance of the two players switches in a way that, at the moment, I'm not sure how to handle. For example, when dealing with double negating atoms, one can work directly with the dialogue tree and argue that, since Opponent gave up an atom in one game tree, he will assert the double negation, which Proponent can then attack, and he (may) give up the atom again. But if Opponent now does

ReplyDeletenotgive up an atompbut only ¬p, Proponent is in a different situation.Still, yes, the data does seem to suggest that this is another class of substitutions under which N is closed. I'm just kvetching about not knowing how to prove that.

I also think there is a generalization of the double-negation substitution; I think not only is replacing atoms with their double-negations validity-preserving, but I suspect that replacing any arbitrary subformula with its double-negation is validity-preserving, and that the proof would be similar for the case of double-negating atoms.

ReplyDelete