Cian pointed out that Conjunction Elimination fails to be valid on Gillies’ favoured account of entailment. On this notion of entailment (‘Entailment v.2.1′, p. 16), P entails Q just in case, for any context s, Q is true in the updated context s[P], that is, just in case:
s[P] = s[P][Q].
But on this account, when T is a tautology, and q an atom, ¬(if T)(q)&q doesn’t entail ¬(if T)(q) (although q&¬(if T)(q) does).
For consider a context s containing both q-worlds and not-q-worlds. The test (if T)(q) has us perform fails, returning the empty context, in s, but passes, yielding the whole updated context, in s[q]. Consequently:
= s[¬(if T)(q)][q]
= (s\s[(if T)(q))[q]
= s[q] (≠ Ø);
s[¬(if T)(q)&q][¬(if T)(q)]
= s[q][¬(if T)(q)]
= s[q]\s[q][(if T)(q)]
= s[q]\s[q] = Ø.
So the entailment fails.
This leaves some doubt about the intended interpretation of material implication on Gillies’ account. Given the semantics for ¬ and &, we cannot rely on the usual paraphrases, such as ¬(P&¬Q) or ¬(¬Q&P), to give us the semantics of material implication; for on this account, these need not be equivalent.
Hopefully, there will soon be a part 2 to this post with our other thoughts.