# When are Counterfactuals transitive?

Analysis: For brevity, define 'gone' = 'gone to the party'.
In this scenario, while it is true that had P gone, then A would have gone,
and it is true that if A would have gone, then M would have gone,
it is NOT true that had P gone, then M would have gone.
If A had gone to the party, P still would not have gone, but M would have gone (because he heard about P's arrest). The first ... and ... second premise[s are] true. This exceptional case proves that this form of argument is invalid, because it overlooks the possibility that
even if P had gone, M would still not have gone...
Notice, however, we may avoid the fallacy if we could assume that if A would have gone, then P would have gone. Sometimes by adding another premise we can rule out all cases where transitivity fails. But in this scenario, we need not make that assumption.

Conclusion: Transitivity does not always fail for counterfactual arguments, but since it does sometimes, hypothetical syllogisms are unreliable and thus invalid.

1. Please explain how the bolded would 'avoid the fallacy'?

2. In general, what 'premise(s)' can be added, or are required. to ensure transitivity?

To precisify the elusive issues orbiting conditionals it's best to work within some formal framework. I'll adopt (a simplified version of) Stalnaker's conditional logic, which extends the modal logic T. Fix a language, L, for propositional modal logic in the usual way and add the two-place conditional connective >.

A model , M, is a triple (W, R, f), where W is a non-empty set of possible worlds, R is a two-place reflexive relation on W and f: W x L → W.

Intuitively, f takes some world w and some formula A and returns the unique world, where A is true and which otherwise minimally differs from w. Truth in a world from some model is defined in the usual way, save for the conditional connective, which is defined as follows:

Let M be a model and w ∈ W. A > B is true with respect to M, w iff B is true wrt M, f(w, A).

Let logical consequence (⊨) be truth preservation in all worlds from all models. That is, for some set of formulas F, let F ⊨ A, if for all models M and all w ∈ W: If B is true wrt M, w, for all B ∈ F, then A is true wrt M, w. Finally, let tautologies be the logical consequences of the empty set.

Now, concerning your first question it is easy to show that

A > B, B > A, B > C ⊨ A > C

To see this note that the following is a tautology

(B > A & A > B) → ((B > C) → (A > C))

So, if A > B, B > A, B > C are true wrt M, w it trivially follows that A > C is true wrt M, w.

Your second question is much more difficult to answer. Part of the difficulty is that a general answer would require a general framework for the various conditional logics, and I don't know if anything of this sort has already been developed.
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