## Thursday, 10 March 2016

### Transitivity and Conditionals

Followup:

Let us assume that a conditional A > C is true iff in all relevant scenarios the corresponding material conditional is true.

Let's leave it completely open here what makes a scenario relevant for a conditional. Let's also leave it open what scenarios are like.

(That something like the above is true for counterfactual or subjunctive conditionals seems more widely accepted than that something like it is true for indicatives, so the following will be most widely acceptable as an observation about the logic of counterfactuals. I think it probably applies to indicatives too. That it holds on the assumption of the above schematic semantics seems to me to be almost beyond dispute.)

In their 2008 paper 'Counterfactuals and Context', Brogaard and Salerno attempt to block a famous counterexample to transitivity for counterfactuals (cf. Lewis , p. 33) with the proposal that to have conditionals for which different scenarios are relevant figuring in the same argument is illicit.

But an inference from A > B and B > C to A > C will be truth-preserving as long as the set of relevant scenarios for the second is a subset of that for the first, and the set of relevant scenarios for the third is a subset of that for the second. (Note I don't say 'proper subset': they could all be the same set, but that's a special case.)

Illustration: If I had spoken to a cat then I would have spoken to an animal. If I had spoken to an animal I would have been happy. Therefore, if I had spoken to a cat then I would have been happy. (It is natural to think of the set of relevant scenarios for the first sentence as larger than that for the second. This could be further brought out by adding something like 'no matter what' to the first sentence.)

(This post builds on this.)

References

Brogaard, Berit & Salerno, Joe (2008). Counterfactuals and context. Analysis 68 (297):39–46.

Lewis, David K. (1973). Counterfactuals. Blackwell Publishers.

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