## Monday, May 30, 2016

### Towards a counterexample to Weak Transitivity for subjunctives

Transitivity for a conditional → says that if A→B and B→C, then A→C. For subjunctive conditionals this rule is generally taken to be invalid. If I ate squash (B), I would be miserable eating squash (C). If I liked squash (A), I'd eat squash (B). But it doesn't follow that if I liked squash, I'd be miserable eating squash.

Weak Transitivity says that if A→B, B→A and A→C, then A→C. The squash counterexample fails, for it's false that if I were eating squash (B), I'd like squash (A).

I don't know whether Weak Transitivity is valid. But here's something that at least might be a counterexample. Suppose a heavy painting hangs on two strong nails. But if one nail were to fail, eventually--maybe several days later--the other would fail. The following seem to be all not unreasonable:

1. If the right nail failed (B), the left nail would fail because of the right's failure (C).
2. If the left nail failed (A), the right nail would fail because of the left's failure (D).
So, by Weakening (if P→Q and Q entails R, then P→R):
1. If the left nail failed (A), the right would fail (B).
2. If the right nail failed (B), the left would fail (A).
If Weak Transitivity holds, then:
1. If the left nail failed (A), the left nail would fail because of the right's failure (C).
But surely (2) and (5) aren't true together.

As I said, I am not sure if Weak Transitivity is valid. If it is, then there is something wrong with (1)-(4), probably with (1) and (2). Maybe there is. But the example should at least give one reason not to be very confident about Weak Transitivity. (There is another reason: Weak Transitivity is incompatible with the non-triviality of the Adams Thesis for subjunctives.)

#### 1 comment:

Alexander R Pruss said...

Alternately, one might deny Weakening.