This paper has a pretty complicated argument and had us all scratching our heads at times. So a disclaimer – we may have missed something obvious in the handout and in the following comments.

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[¬(if T)(q)][q]

= (s\s[(if T)(q))[q]

= s[q] (≠ Ø);

whereas:

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.

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