Counterfactual definiteness :: Wolfgang Schwarz

Counterfactual definiteness

Posted on Saturday, 12 Sep 2026. physics conditionals

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):

(1)If this fair coin had been flipped, it would have landed heads/tails.

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.

(2)Alice's measurement would have been the same if Bob had chosen a different axis.

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.

Bennett, Jonathan. 2003. A Philosophical Guide to Conditionals. New York: Oxford University Press.

Cariani, Fabrizio, and Simon Goldstein. 2020. “Conditional Heresies.” Philosophy and Phenomenological Research 101 (2): 251–82. https://doi.org/10.1111/phpr.12565.

Hájek, Alan. 2009. “Fifteen Arguments Against Hypothetical Frequentism.” Erkenntnis 70 (2): 211–35.

Jeffrey, Richard C. 1977. “Mises Redux.” In Basic Problems in Methodology and Linguistics: Part Three of the Proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada-1975, 213–22. Springer.

Joyce, James. 1999. The Foundations of Causal Decision Theory. Cambridge: Cambridge University Press.

Lewis, David. 1981. “Causal Decision Theory.” Australasian Journal of Philosophy 59: 5–30.

Mandelkern, Matthew. 2018. “Talking About Worlds.” Philosophical Perspectives 32 (1): 298–325. https://doi.org/10.1111/phpe.12112.

Marty, Paul, Jacopo Romoli, and Paolo Santorio. 2020. “Counterfactuals and Undefinedness: Homogeneity Vs Supervaluations,” 21.

Moss, Sarah. 2013. “Subjunctive Credences and Semantic Humility.” Philosophy and Phenomenological Research 87 (2): 251–78.

Ramotowska, Sonia, Paul Marty, Jacopo Romoli, and Paolo Santorio. 2025. “Counterfactuals and Quantificational Force: Experimental Evidence for Selectional Semantics.” Semantics and Pragmatics 18: 6:1–43. https://doi.org/10.3765/sp.18.6.

Shaffer, Michael J., and James Beebe. 2019. “Folk Judgments about Conditional Excluded Middle.” In Advances in Experimental Philosophy of Logic and Mathematics, edited by Andrew Aberdein and Matthew Inglis, 251–76. Bloomsbury Academic.

Stalnaker, Robert. 1980. “A Defense of Conditional Excluded Middle.” In Ifs, edited by William Harper, Robert C. Stalnaker, and Glenn Pearce, 87–104. Reidel.

Williams, J. Robert G. 2010. “Defending Conditional Excluded Middle.” Noûs 44 (4): 650–68. https://doi.org/10.1111/j.1468-0068.2010.00766.x.

Comments

# Dmitri Gallow on 13 September 2026, 10:15

I'd be interested to hear how you think about three particle GHZ. Three particles are entangled and sent off to Alice, Bob, and Carol, each of whom can each either measure the X or the Z spin of their particle. QM predicts that, whenever all three measure X spin, an odd number of them will measure 'up', but that, when exactly one measures X spin and the other two measure Z spin, an even number of them will measure 'up'.

Suppose that, in fact, they all measure X spin, and that Alice gets 'up' while Bob and Carol gets 'down'.

Assume the following two principles are not just true but definitely true:

COUNTERFACTUAL LOCALITY: If either Alice, Bob, or Carol had measured Z spin instead, that wouldn't have changed the others' measurements.

NOMIC STABILITY: If either Alice, Bob, or Carol had measured Z spin instead, they wouldn't have proven QM wrong.

Together, these assumptions imply that the following counterfactuals are definitely true:

(3) If Alice had measured Z spin, she'd have gotten 'down'
(4) If Bob had measured Z spin, he's have gotten 'up'
(5) If Carol had measured Z spin, she'd have gotten 'up'

On (3): COUNTERFACTUAL LOCALITY tells us that Bob's and Carol's X-spin down measurements wouldn't have changed. And QM says that, if Alice measures Z-spin while Bob and Carol measure X-spin, then there must be an even number of 'up's. So if they're not going to prove QM wrong, then Alice has to get `down'. So NOMIC STABILITY implies (3). Similar reasoning gets us (4) and (5).

So if we accept both COUNTERFACTUAL LOCALITY and NOMIC STABILITY, then we have to say that each of (3), (4), and (5) are definitely true. Even though measuring Z spin is a paradigmatic chancy process, we would say that there's a definite fact of the matter about how the measurement would have gone, had any of them performed it.

If we reject COUNTERFACTUAL LOCALITY---as I think we should---then there's nothing left of your argument against CEM. If reject NOMIC STABILITY, then the argument against CEM similarly falls apart, since the defender of CEM could likewise endorse COUNTERFACTUAL LOCALITY by insisting that Bell's inequality *would* be violated, were Alice to change her measurement.

So I think that, if you're going to make this argument against CEM, then you're going to have to accept quite a lot of counterfactual definiteness yourself. As a defender of CEM, on the other hand, I get to say that there's some true counterfact about what Alice would have seen, had she measured Z spin, but there's no *definite* counterfact about what she'd have seen. It might have been 'up', and it might have been 'down'.

# Wolfgang Schwarz on 13 September 2026, 11:33

Thanks Dmitri!

I don't follow your derivation of (3). QM doesn't say that if Alice had measured Z-spin while Bob and Carol measure X-spin, then there must be an even number of 'up's. In your setup, X_A*X_B*X_C = +1 and X_A*Z_B*Z_C = Z_A*X_B*Z_C = Z_A*Z_B*X_C = -1. We don't have Z_A*X_B*X_C = -1. In fact, Z_A*X_B*X_C can't have a definite value given that X_A*X_B*X_C has one. So "there's no fact of the matter" about what Alice would have measured.

We could try to fix this problem by changing *two* measurement choices. What if Alice and Bob had both decided to measure Z-spin, while Carol still measures X-spin and gets -1? Since Z_A*Z_B*X_C = -1, we can infer (by COUNTERFACTUAL LOCALITY and NOMIC STABILITY) that Alice and Bob would get the same result. But we can't say which result they would get. There's still no fact of the matter about what Alice would have measured.

Also, about the dialectic. I don't agree that if we reject COUNTERFACTUAL LOCALITY then there's nothing left of my argument. For one thing, my argument draws on the fact that CEM seems to imply *especially peculiar* violations of COUNTERFACTUAL LOCALITY. More importantly, the argument is that these violations are peculiar enough that many competent people who have thought hard about these situations have come to conclusions which (arguably) imply that the relevant instances of CEM are false. The point is not that these instances *are* false. The point is that there are (strange) cases in which a large number of competent speakers make reflective, sincere judgements that clash with the validity of CEM.

# Dmitri Gallow on 13 September 2026, 18:56

Yeah, I agree that I screwed up with the derivation of (3).

On the point about physicists being willing to reject instances of CEM: It's not clear to me that the physicists who deny counterfactual definiteness are really denying instances of CEM. They deny that it's *definite* what would have happened, had Bob's measurement been different. But that's something we defenders of CEM agree with. We will agree that it's definite that there's a fact of the matter about what would have happened with Alice, had Bob measured a different axis. But we'll deny that there's a definite fact of the matter about what would have happened, had Bob measured differently. I'd be surprised if the physicists are distinguishing between "Definitely: ((A>C) or (A>~C))" and "(Definitely: (A>C)) or (Definitely: (A>~C))". And insofar as they need only deny the latter in order to square counterfactual locality with QM, I think CEM does a reasonable job capturing what's going on with the debate about counterfactual definiteness. Is there some reason that they need to deny the former?

# Wolfgang Schwarz on 14 September 2026, 09:33

Right. I'd want to distinguish two question.

You ask whether we *need* to deny CEM to make sense of QM (on any live interpretation of QM). I suspect the answer is no. But holding onto CEM comes at a cost. The details depend on how the CEM-friendly semantics is spelled out, but the basic problem is that every selection function effectively associates a definite outcome with every possible measurement, so relative to every selection function, there are hidden variables of the kind Bell's theorem seem to refute. That is, relative to every selection function, one can derive contradictions with QM if one accepts certain plausible further assumptions X. Given that QM is determinately true, it follows that X is false relative to every selection function; so X is determinately false. That's a cost. But it's only a cost. One can bite the bullet and accept, for example, the non-local counterfactual dependencies that I called "peculiar".

But the blog post is about a different question. It's about whether counterfactual conditionals in English have a semantics that renders CEM valid (e.g. Stalnaker's) or one that renders CEM invalid (e.g. Lewis's or Kratzer's). This is an empirical question that's ultimately decided by observations about how competent speakers use counterfactual conditionals. My claim is that when people think about counterfactuals in EPR-type situations, with Bell's theorem in mind, many competent speakers make judgements that are incompatible with CEM. What's the evidence? 1. This is the most natural interpretation of what they say: the common slogan "unperformed experiments have no results" certainly sounds like "for any unperformed measurement and result x: ¬∃x(M>x)". 2. Many commentators explicitly reject the cost that would come with accepting CEM (see above). 3. Some commentators explicitly endorse the anti-CEM verdict: Vaidman, Griffiths, but also Joyce, Lewis, Jeffrey, myself.

I'm somewhat sympathetic to a third possibility: that the conventions of English don't settle how we should talk about EPR-type situations. Perhaps this is a matter of open texture. In debates about CEM, it often seems to me that there are two dialects of English: some seem to speak a version of English in which CEM is valid, some a version in which it is invalid.

Counterfactual definiteness

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