## Kocurek on chance and would

A lot of rather technical papers on conditionals have come out in recent years. Let's have a look at one of them: Kocurek (2022).

The paper investigates Al Hajek's argument (e.g. in Hájek (2021)) that "chance undermines would". It begins with a neat observation.

One way of putting Hajek's argument goes like this. Imagine a chancy coin. We all agree that (1) is true.

(1) If the coin were flipped, there would be a chance of tails.

From this, one might be tempted to infer (2):

(2) If the coin were flipped, it could land tails.

And arguably, (2) is incompatible with (3):

(3) If the coin were flipped, it would land heads.

Kocurek notes that this line of reasoning leads to trouble if we consider a stronger antecedent:

(1') If the coin were flipped and then landed heads, there would be a chance of tails.

This seems true. By the same reasoning that leads from (1) to (2) – from 'there would be a chance of tails' to 'the coin could land tails' –, we could infer (2'):

(2') If the coin were flipped and then landed heads, it could land tails.

If 'could tails' is incompatible with 'would heads', we could infer that (3') is false.

(3') If the coin were flipped and then landed heads, it would land heads.

But (3') is obviously true!

To me, this settles that 'there would be a chance of A' is compatible with 'would not A'. If that's what the "chance undermines would" thesis denies, then the thesis has been refuted.

But I still feel the pull of "chance undermines would". If there's a chance of tails, it's not true that the coin would land heads!

Hajek should say that (1) isn't the right premise about chance. What undermines 'would' is not that there would be a chance. What undermines 'would' is the actual chanciness of the coin. There is a chance of tails, conditional on flip. There is no chance of tails conditional on flip ∧ heads.

The right premise (1) in Hajek's argument should be something like (1H):

(1H) There is a chance that the coin lands tails if flipped.

Intuitively, this entails (2). (At least if we add the premise that the coin isn't flipped.) And (2) seems incompatible with (3). If we replace 'flipped' with 'flipped and heads' then the first premise becomes obviously false.

So that's a good lesson about how to spell out the "chance undermines would" argument.

Kocurek doesn't consider revising premise (1). Instead, he immediately descends into the rabbit hole of formal semantics.

He also conjectures that the intuition behind "chance undermines would" is a more general intuition that 'there is a chance of A' conflicts with 'not A', as witnessed by the oddness of (9).

(9) ??The coin will land heads and there's a chance it won't.

Kocurek spells out a semantics that explains this oddness and makes 'there would be a chance of A' incompatible with 'would not A', despite the apparent counterexample from (1')-(3').

The semantics is modeled on information-sensitive accounts of epistemic modals. Truth is defined relative to a world and a chance function. Two notions of logical consequence are distinguished. I won't go through the details, mainly because I don't see a good motivation for the exercise.

In light of the counterexample from (1')-(3'), any semantics that makes 'there would be a chance of A' incompatible with 'would not A' is going to have dubious consequences. Even Hajek should reject the incompatibility. When it comes to (9), we don't need a special semantics to explain the oddness. If you know that there's a chance that the coin will land tails then you arguably can't know that the coin will land heads (unless you have very unusual information). So (9) is unknowable and hence unassertable.

I also don't agree that the "chance undermines would" intuition is based on a general intuition about the incompatibility between chanciness and truth. For example, there's nothing odd at all about a past version of (9):

(9') The coin landed heads and there was a chance it wouldn't.

Kocurek ends up rejecting the original semantics he develops in favour of an alternative, "indeterminist semantics" that doesn't make 'there would be a chance of A' incompatible with 'would not A'. The indeterminist semantics still defines truth relative to a world and a chance function, and it still has two notions of entailment. I'm not sure exactly how it works because Mind has messed up a lot of the symbols.

An important new element of the new semantics is the assumption of "counterfacts": for every world w and (non-empty) proposition A, there is a unique world w' that would be the case if A were the case. The chance parameter in the definition of truth is only really used to interpret statements like (1): 'if A then the chance of B would have been x' is true relative to a world w and a chance function f iff f(B//A) = x, where f(*//A) results from f by moving the probability of every non-A world w to the unique world w' that would be the case at w if A were the case.

Little explanation is given for all these choices. What is the chance parameter supposed to represent? Do we have to believe in counterfacts? Why the imaging operation on the chance function? Do we have to believe that the physical chances are defined under these operations?

Besides, the semantics seems to render 'if the chance of heads were 0.9 then the chance of heads would be 0.9' contingent.

I think of proposals like Kocurek's as "proof-theoretic semantics".

The aim of proof-theoretic semantics is to predict certain inferential patterns. To this end, the semantics introduces models and semantic values relative to points of evaluation, but all this machinery is assessed merely by whether it determines a notion of entailment (or several such notions) that match the relevant inferential patterns.

Real semantics, by contrast, would try to spell out what has to be the case for a sentence to be true. Perhaps Kocurek's semantics can be understood as real semantics. But I don't know how this would work. The paper doesn't explain.

Hájek, Alan. 2021. “Counterfactual Scepticism and Antecedent-Contextualism.” Synthese 199 (1): 637–59. doi.org/10.1007/s11229-020-02686-0.
Kocurek, Alexander W. 2022. “Does Chance Undermine Would?” Mind 131 (523): 747–85. doi.org/10.1093/mind/fzab055.