Calling all modal logicians, I need help with a problem.

[Schwarzwald]

Hey all, I'm looking for someone familiar with Mares' "impossible world" constructs, specifically from the 1997 paper, "Who's afraid of impossible worlds?" which can be found here:https://projecteuclid.org/euclid.ndjfl/1039540767

However, an unfamiliarity with Mares' work is not necessary if one has sufficient knowledge of David Lewis' work in Counterfactuals and S.

My question regards Mares' outlining of the truth condition for propositions in impossible worlds:

i ╞ A □→ B ≡

ƎS (S ∈ $(i) & Ǝj ∈ S(j ╞ A) & ∀k ∈ S(k ╡ A ∨ k ╞ B))

∨ ¬ƎSƎj(S ∈ $(i) & Ǝj ∈ S & j ╞ A)


My first question is, is S being used here as a generic formulae, or as the function where S is a ternary relation regarding some selection frame K, or SaXb ≡ b ∈ f(a, X)?

My second question is, in this impossible world talk, what is the relationship of formulae: A, B, to variables: j, k, and what do they refer to, that is, what order are we in? When I state j holds at A, am I stating that some fact substantiates the truth value of some proposition in some world, or some proposition substantiates some formula described by some index <R, a-1...a-n, π>?

My third question is, why must k fail to hold at A in order for A □→ B?

I'll take any help you all have to offer. General explanations of Mares' project are welcome too, :]

[Schwarzwald]

:[ Pretty please?

[Chrysippus'Doge]

So I'll try my hand here, though it's been a wee bit since I've looked at counterfactual semantics. I think S is being used to represent a particular sphere around the index i... Which is just all those indexes which fall within a given "distance" from i. So I don't think it's a function or a generic formulae but rather a set of indices, which would explain why we see it used like 'Ǝj ∈ S'. This just says: "There exists an index in S s.t." yada yada yada. Which answers the second question: I think j,k, etc. are just additional variables for indices. Thus j makes true or false propositions represented by variable A,B etc., which is abbreviated to j ╞ A, j  ╡B respectively. 

So I guess then as far as #3 is concerned, did you mean why must A be made false at k for A □→ B to hold? Or am I missing the point...

[Schwarzwald]

4 hours ago, Chrysippus'Doge said:

So I'll try my hand here, though it's been a wee bit since I've looked at counterfactual semantics. I think S is being used to represent a particular sphere around the index i... Which is just all those indexes which fall within a given "distance" from i. So I don't think it's a function or a generic formulae but rather a set of indices, which would explain why we see it used like 'Ǝj ∈ S'. This just says: "There exists an index in S s.t." yada yada yada. Which answers the second question: I think j,k, etc. are just additional variables for indices. Thus j makes true or false propositions represented by variable A,B etc., which is abbreviated to j ╞ A, j  ╡B respectively. 

So I guess then as far as #3 is concerned, did you mean why must A be made false at k for A □→ B to hold? Or am I missing the point...

Thanks for your response! I agree with your interpretation over mine, I'd initially thought as much, but then I was thrown off by reading one of Lewis' papers on the function/relation S.

So, what Mares' is saying is that...

a counterfactual implication is true at an index if and only if there exists some sphere of indices, part of the set of spheres of that index, and there exists some index j, part of set of spheres, S, such that the antecedent A is true at j, and for all indices, k, part of the set of spheres, S, the antecedent A is false at k, or, the consequent, B, is true at k, OR...there is not some sphere of worlds, S, where there exists some index, j, inwhich S is part of the set of spheres of the relevant index and there exists some index part of set of spheres, S, where the antecedent, A, is true at j

Is this correct?

My problem is trying to wrap my head around the intuition of this truth condition.

We're trying to  alleviate counterfactual implication of a sort of mathematical triviality, right? We want the truth values of counterfactuals to not be in any sense vacuous. So, we treat ersatz or possible indices/worlds with as much reality as we can. However...

Mares' is saying that a counterfactual conditional can be true when just the antecedent is true in some index, and false in all other indices within the sphere, or, the consequent is true in all other indices within that sphere, OR, when there is no sphere with an index that supports A to begin with.

How does this solve the problem of triviality? One of the purposes of relevant logics is to do away with assigning the value of true to conditionals with false antecedents automatically? However, Mares' rule states 1.) that the antecedent can be true at j and false at k and still be considered true, or, the antecedent can be entirely vacuous (as in the last line of the truth condition) due to no spheres related to i supporting it. This seems pretty trivial? So I assume I'm misunderstanding Mares and Lewis.

This is why I need a bit more information on the relation of j to k. Simplistically speaking, Lewis (with the limit assumption in mind) wanted counterfactuals to be assigned values based on the facts of the worlds they took place in. As such, a counterfactual condition would be true just in case the consequent was true for every world that departed the least from our world to make the antecedent true. So with Mares' tools, we set up these relational constructs of states of affairs trying to make the antecedent true, (I assume we have to make all of them for this to work?) and then we rate them on their similarity to create the set of spheres, $(i), and only if the consequent holds across the sphere most similar to i, S, do we state the counterfactual implication is true. This would seem in line with Lewis (which is who Mares states he gets the condition from), but ironically, this is counterfactual to what I've written above about Mares' truth condition. Soooo...I'm lost?

[Schwarzwald]

On second thought, Mares seems to be stating that this model is a classical variant and that the problems of triviality can only be fully resolved in a relevant model that he'll propose at a later paper. In the above paper, classical logic's triviality is somewhat combated through the $ function, Humean supervenience, Lewis' delineation between ersatz and possible worlds, and Mares' well-formed formulae rule for validity. So with that stated, unless I'm missing something, thanks Chryssipus, you really set me straight, I was starting down a wrong route misinterpretating S that would've had me confused for days. You're a life saver.