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:
- If the right nail failed (B), the left nail would fail because of the right's failure (C).
- If the left nail failed (A), the right nail would fail because of the left's failure (D).
- If the left nail failed (A), the right would fail (B).
- If the right nail failed (B), the left would fail (A).
- If the left nail failed (A), the left nail would fail because of the right's failure (C).
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.)