Counterfactual definiteness
A peculiar question that has divided philosophers since the Middle Ages (and possibly earlier) concerns the status of Conditional Excluded Middle. CEM says that for any propositions A and C, either C would be the case if A were the case, or ¬C would be the case if A were the case. Some accept CEM as valid, others reject it. In the last few decades, the CEM supporters have been gaining ground, especially in the field of semantics. See, for example, Williams 2010, Moss 2013, Mandelkern 2018, Shaffer and Beebe 2019, Cariani and Goldstein 2020, Marty, Romoli, and Santorio 2020, and Ramotowska et al. 2025.
I belong to the other camp: I believe that CEM is invalid. My reasons aren't particularly original; they are essentially the ones given in Jeffrey 1977, 193, Hájek 2009, 217, Joyce 1999, 172f., Lewis 1981, 331, and Bennett 2003, 191, among others.
Most supporters of CEM don't seem to understand these reasons.
A principle is invalid if it has at least one counterexample. The opponents of CEM that I've cited think that the principle is false in certain cases involving indeterministic chance events: if a certain type of chance process A has both C and ¬C as possible outcomes, and A doesn't occur, then neither C would have occurred nor ¬C would have occurred if A had occurred.
The usual toy example is (1):
But – here's the misunderstanding: our claim is not that statements like (1) are false. Our claim is that they are false on the assumption that the antecedent describes a particular kind of physical process. Whether any processes of this kind exist in worlds like ours is an open question. Bohmian mechanics, for example, implies that there are none.
I want to tear my hair out when I read papers like Marty, Romoli, and Santorio 2020 and Ramotowska et al. 2025 in which ordinary people are asked to make truth-value judgements about sentences like (1). That's not what our dispute is about!
The anti-CEM hypothesis is that CEM fails in a particular class of cases. What needs to be checked is how people are disposed to talk about these cases, not about ordinary coin flips.
To appreciate the point about chance, you have to know a bit of physics.
Suppose we prepare two electrons in an entangled state, sending one to Alice and the other to Bob. Alice has a measuring device with which she can measure the spin of her electron along one of two axes, depending on the setting of a switch. Bob also has such a device for his electron, with axes that are different from Alice's. Alice and Bob both choose an axis, measure, and register the outcome ("up" or "down").
Suppose this process is repeated many times, with Alice and Bob choosing different axes at different times. Quantum mechanics predicts that the recorded measurements will display certain statistical correlations, which have been verified experimentally.
Now assume that there's always an answer to what would have happened if Alice or Bob (or both) had chosen different axes. For example, we can ask whether, on a particular run, Alice would have obtained the same result if Bob had chosen a different axis. Intuitively, the answer is "yes". We can assume that Alice and Bob are really far apart, so that there's no causal connection between Bob's choice and Alice's measurement. So (2) seems plausible.
But this assumption of counterfactual locality, together with the assumption of counterfactual definiteness (that there is always an answer to what would have been measured if Alice and Bob had chosen such-and-such axes), contradicts the verified predictions of quantum mechanics. This is a version of Bell's theorem.
(Obviously, deriving the contradiction requires a little more precision. The derivation also requires a further "no conspiracy" assumption that is widely, but not universally accepted.)
So the friend of CEM has to deny (2). More precisely, they have to say that as the experiment is repeated over and over, there is an increasingly precise number of runs (around 41%) in which Alice's measurement would have been different if Bob had chosen a different axis. They have to posit a mysterious counterfactual dependence between spacelike-separated events.
Bohmian mechanics endorses this dependence, and suggests a mechanism of how it might come about. But most physicists think the mechanism doesn't work. They reject the dependence. So they deny the other ingredient in the derivation of the contradiction: counterfactual definiteness. On this view, it is neither true that Alice would have measured "up" nor that she would have measured "down" (or anything else) if she had chosen a different axis. There is "no fact of the matter" about what outcome she would have obtained.
To me, this looks like a reasonable response. The Wikipedia page on counterfactual definiteness even suggests that the failure of counterfactual definiteness is implied by various interpretations of quantum mechanics. I don't think that's quite correct. But the important point is that physicists who have supported these interpretations have explicitly rejected counterfactual definiteness.
If CEM were valid, rejecting counterfactual definiteness would not even be an option. All these physicists – and all the philosophers I've cited above – would misunderstand their own language, promoting theories that are analytically false.
Worse, if CEM were valid, the grammar of English, together with findings from physics, would force us to believe in mysterious patterns of counterfactual dependence between spacelike-separated events. This is giving way too much power to grammar.
I've assumed that supporters of CEM would have to endorse the "counterfactual definiteness" that figures in Bell's theorem. Perhaps this could be denied. Some friends of CEM admit that there may be "no fact of the matter" about what would have happened if a certain antecedent had been true. Stalnaker 1980 is the classic source. On Stalnaker's account, counterfactuals are evaluated relative to a selection function f, but the context of utterance often doesn't supply a unique f. For a counterfactual to be true in a context ("supertrue"), it has to be true relative to all selection functions f that are compatible with the context.
But does this help? It's not enough to somehow vindicate the judgements that there's "no fact of the matter". We need to get around the trouble raised by Bell's theorem. As far as I can see, counterfactual definiteness remains true relative to each selection function f. Granting the other ingredients of Bell's theorem, and holding fixed f, we can still derive violations of counterfactual locality: we can derive that in a certain number of runs, Alice's measurement would have been different if Bob had chosen a different axis. So this comes out supertrue.