# A∧B-worlds and law of enrichment in Stalnaker semantics

In Stalnaker semantics a conditional A↦B is true in world u if either A is not logically possible (ex absurdo quodlibet holds in the semantics) or A is logically possible and B is true in the nearest world where A is true.

A↦B axiomatically implies the classical logic implication A→B.

I read -in D. Palladino, C. Palladino, Logiche non classiche 'non-classical logics'- that various conditions are assumed in Stalnaker semantics, among them:

1. if the antecedent A of a conditional A↦B is logically possible, then there is at most one possible world where A is true and is nearer to the actual world than any other world (literally quoting the text "when the antecedent of a conditional is logically possible, there is at most one possible world where the antecedent is true and ressembling the actual world more than any other possible world where the antecedent is true (unicity assumption) [i.e. at most one satisfying both the truth of the antecedent and the minimum distance from the actual world]");
2. every world is nearest to itself than any other world;
3. if v is the A∧B-world [i.e., world where A∧B is true, I think] nearest to u, then v is the A-world nearest to u;
4. if the A-world that is nearest to u is a B-world and the B-world that is nearest to u is an A-world, then that A-world and that B-world are the same world.

Then I read that the enrichment law which is stated, in classical logic, as (A→B)→(A∧C→B) does not hold in Stalnaker's system. I think that it means that (A↦B)→(A∧C↦B) does not hold.

The reason why the enrichment law does not hold is, as the book says, that the nearest world where A is true is not necessarily the nearest world where A∧C is true. I think that what is shown here is that A∧C↦B can be false while A↦B is true, which would falsify (A↦B)→(A∧C↦B) and, since we assume axiom (A↦B)→(A→B), would also falsify (A↦B)↦(A∧C↦B). But how can we say that there can be a world w which is the nearest world where A∧C is true, but B is false in w, if condition (3) says that the nearest A∧C-world is also the nearest A-world, where we assume that B is true? I have taken into account that the only possible case that I can see where the nearest A-world and A∧C-world are not the same is given when no A∧C-world exists, but then A∧C↦B would be true by Scotus' quodlibet law, if I am not wrong... Thank you very much for any clarification!

• Is the world indicated in 1. the "Actual World" as in the rigidly designated `@`? Or is it supposed to be the world relative to which we are semantically evaluating the given statement A→B? Mar 6, 2015 at 13:50
• @PaulRoss Forgive me, my text is an introductory manual about non-classical logic and I don't know notation `@`... Anyhow, I would say that the actual world is the world where we are evaluating A↦B. I've also added a literal translation of what the book says for clarity to the OP. Thank you so much for the comment! Mar 6, 2015 at 15:15
• Hmm... It's complicated because in modality there is often some issue as to how the word "Actual" refers - we sometimes use `@` to mean actual in the sense of a distinct privileged world that always refers to this world (the world of the speaker, as it were), even when you're carrying out evaluations relative to some other world. I'll read into the Stalnaker work a bit and see what I can deduce! Mar 6, 2015 at 15:22
• @PaulRoss I'm not sure, but it might well be a privileged, once for all fixed, world. My book doesn't give many details... Thank you again! Mar 6, 2015 at 15:31
• I'm not familiar with this system and I've no access to a specification of this semantics, but I think that the issue is simply : the world where not-C is true can be "nearer" to the actual world than the world where C is : this is why in Stalnaker's theory, A > B does not implies (A & C) > B (see R.Stalnaker, A Theory of Conditionals, (1968), page 106. Mar 6, 2015 at 16:17