Counterfactual Conditionals

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Semantics

Here, I quickly motivate an account of the semantics of counterfactual conditionals.¹ These are expressions which can be roughly paraphrased as ‘if it were the case that X, then it would be the case that Y’; for instance:

1. You would subscribe if you were cool.

Despite what’s implied by the term ‘counterfactual’, these conditionals need not have false antecedents. For instance, it is true that you’re cool (and so it must be true that you’ll hit the button below).

Counterfactual conditionals are distinguished from indicative conditionals, such as:

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The indicative conditional (‘If X, then Y’) seems to obey two deductive rules. The first is conditional proof: whenever I know that Y is entailed by X and background assumptions ZZ, I may conclude ‘if X, then Y’ (assuming ZZ). The second is modus ponens: whenever I know the conditional ‘if X, then Y’ and the antecedent X, I may conclude the consequent Y. These two deductive rules entail that ‘if X, then Y’ is equivalent to the material conditional (X → Y), which is true just in case the antecedent X is false or the consequent Y is true.

The difference between the counterfactual conditional (1) and the indicative conditional (2) is the ‘would’ (and corresponding ‘were’) in (1). The term ‘would’ can act on its own, as in:

3. She would show up on time.

In particular, (3) doesn’t seem to covertly express a counterfactual conditional; there’s no clear candidate for a hidden antecedent. Rather, (3) seems to express a sort of robustness to her not showing up late; the obvious candidate is that in all relevant (hypothetical) cases, she doesn’t show up late.² It’s clear that the relevant cases depend on context: (3) might be true if we’re talking about classes, or false if we’re talking about parties. We might even be talking about a lecture delivered by a resurrected Hume, such that the real world isn’t among the contextually relevant cases. 

As noted, counterfactual conditionals seem to compose ‘would’ with (X → Y). There are (at least) two distinct ways to compose these operators, with ‘would’ taking either wide scope or narrow scope:

4. would(X → Y)

5. X → would(Y)

But we can see that (4) is more plausible for the counterfactual conditional, by the following contrast:

6. If she were to stay in university, she would still study hard.

7. If she stays in university, she would still study hard.

Suppose that it’s clear from context that we’re discussing what would happen if she won the lottery. Then (6) says that, in the lottery-winning cases where she stays in university, she studies hard. Meanwhile, at least on one reading, (7) says that her actually staying in university indicates that, in the lottery-winning hypothetical cases, she still studies hard. So, (6) is of form (4), while one reading of (7) is of form (5). So, counterfactual conditionals like (6) have ‘would’ taking wide scope over the material conditional. The final picture is that a counterfactual conditional is true just in case, in all contextually relevant cases, the antecedent is false or the consequent is true.

Heuristics

This simple account validates fairly strong logical properties: for instance, □(X → Y) entails □(¬Y → ¬X), so this account validates contraposition; it also entails □((X ∧ Z) → Y), so this account validates strengthening the antecedent. But these properties seems slightly worrying. For instance:

8. If Sydney came over, I’d be happy.

9. If Sydney came over, and she were severely injured, I’d be happy.

This is an instance of strengthening the antecedent. But here, (8) sounds true, whereas (9) sounds false. But don’t we predict that (8) entails (9)? The thing to note is that when one asserts (9), one usually makes salient the abnormal cases where Sydney gets injured on her way here. Thus, although (9) is literally true in contexts where (8) is true, asserting (9) usually forces the context to change, and (9) is false in this new context. This also generates the prediction that after (9) is asserted, it should be hard to assert (8) — it’s easier to expand the range of relevant cases than to contract them. However, we also might imagine one successfully implying that Sydney won’t get injured, but forcefully asserting (8) after (9).

We tend to use the following strategy to evaluate conditionals:

Suppositional Rule: Take the same attitude (e.g., acceptance or rejection) towards ‘If A, then C’ as you take towards C on the supposition that A.

This rule entails principles like the following:

10. Always accept ‘if A, then A’.

On the supposition that A, we should accept A; so, we should always accept ‘If A, then A’. If you don’t do this, it’s not clear that you’re even supposing A.

11. Accept ‘if A, then X’ and ‘if A, then Y’ whenever you accept ‘if A, then X and Y’.

Whenever we accept X and Y on the supposition that A, we accept X and we accept Y on the supposition that A.

12. Accept ‘not(if A, then C)’ whenever you accept ‘if A, then not(C)’

Whenever we would reject C on the supposition that A, we reject ‘if A, then C’.

But (10), (11), and (12) are inconsistent. By (10), we have:

13. If (X and not(X)), then (X and not(X)).

By (11), we have:

14. (If (X and not(X)), then X) and (if (X and not(X)), then not(X)).

Finally, by (12), we have:

15. (If (X and not(X)), then X) and not(if (X and not(X)), then X).

But (15) is an outright contradiction. So, the Suppositional Rule is inconsistent. The culprit is (12): having not(X) in all relevant cases still allows for having X in all relevant cases, so long as there just are no relevant cases! Consider also:

16. If we were to discover a 36-sided Platonic solid, it would have fewer sides than a dodecahedron.

The largest Platonic solid is the icosahedron (known around here as a ‘d20’); mathematically, we know that it’s not possible (in any sense) for there to be a 36-sided Platonic solid — just as we know that logically, it’s not possible (in any sense) for it to be the case that X and not(X). We use the Suppositional Rule as a quick-and-dirty heuristic, and deduce from the fact that 36 is greater than 12:

17. If we were to discover a 36-sided Platonic solid, it would not have fewer sides than a dodecahedron.

Then, we apply the Suppositional Rule — (12) in particular — to get the negation of (16). But this is a mistake; (16) is (vacuously) true. It’s a mistake to use the Suppositional Rule here, since this is exactly where it crashes. Further, if one tries to rescue (17) by positing logically impossible worlds, one dims the guiding light of formal semantics: compositionality. We model the terms and, or, and not (like the terms hits, loves, and subscribes) as functions, such that fixing the values of X should fix the value of not(X). The worlds where not(X) are supposed to be given by the complement of the worlds where X; if some of our worlds are impossible, such that we have both X and not(X), we lose our grip on this simple and powerful tool.

You know what else is simple and powerful? Subscribing to Offhand Quibbles.³

1. See Suppose & Tell (2020); for a quicker version, see part of this paper.

Some people will call these subjunctive conditionals, and reserve the term ‘counterfactual’ for subjunctive conditionals with false antecedents.

2. It’s not automatic that ‘would’ scopes over ‘not’; certainly, ‘she could not lie’ is ambiguous between her not being able to lie, and her being able not to lie (although the first is far more natural).

3. Thanks to Woarna for prompting me to add the second half of this post, and to Mark Young and Amos Wollen for prompting me to fix some pretty bad typos.