Champollion, Ciardelli, and Zhang on de Morgan's law

Champollion, Ciardelli, and Zhang (2016) argue that truth-conditionally equivalent sentences can make different contributions to the truth-conditions of larger sentences in which they embed. This seems obviously true. 'There are infinitely many primes' and Fermat's Last Theorem are truth-conditionally equivalent, but 'I can prove that there are infinitely many primes' is true, while 'I can prove that there are no integers a, b, c, and n > 2 for which an + bn = cn' is false. Champollion, Ciardelli, and Zhang (henceforth, CCZ) have a more interesting case in mind. They argue that substituting logically equivalent sentences in the antecedent of a subjunctive conditional can make a difference to the conditional's truth-value.

CCZ consider the following scenario. A light is connected to two switches, A and B, each of which is either up or down. The light is on iff both switches are in the same position (both up or both down). In fact, both switches are up and the light is on. CCZ say that (1a) is true in this scenario while (1b) is false:

(1a) If switch A or switch B was down, the light would be off.
(1b) If switch A and switch B were not both up, the light would be off.

They back up these judgements by a poll on Mechanical Turk. Among a few hundred mechanical turkers, 69% judged (1a) true, but only 22% judged (1b) true.

CCZ go on to argue that the difference in meaning between (1a) and (1b) can be explained by combining the inquisitive semantics of Ciardelli, Groenendijk, and Roelofsen (2018) with Alonso-Ovalle (2006)'s model of conditionals.

My hunch is that the difference has a more pragmatic origin.

Note that the data is messy. If (1a) is true, why do only 69% of polled subjects say that it's true? If (1b) is not true, why do 22% say that it's true?

What's more, CCZ report that 21% of their subjects judge that (1c) is true:

(1c) If switch A and switch B were not both up, the light would be on.

These 21% seem to read 'not both up' as 'both not up'. As CCZ mention (p.315), 'not both up' can arguably have this reading if focus is on 'up'. On this reading, it's obvious that (1b) is false, and the antecedents of (1a) and (1b) are not equivalent. If we want to set aside this reading, we have to ignore roughly 21% of subjects, all of which judged (1b) false. Among the remaining subjects, 28% judged (1a) true. So the real spread is: 28% true for (1a) vs 69% for (1b).

What kind of pragmatic effect could explain this noisy difference?

Here's one idea. Perhaps the antecedent of (1a), but not that of (1b), tends to trigger an exclusivity implicature. By comparison, imagine I've checked on the state of the two switches many times over the past few weeks. Now I report either (2a) or (2b).

(2a) Every time I checked, switch A or switch B was down.
(2b) Every time I checked, switch A and switch B were not both up.

The first report indicates that I never found both switches down. The second does not.

If the antecedent of (1a) similarly picks out a set of worlds all or which are such that A or B is down, we might expect to find the same difference.

Here's another idea.

On familiar assumptions about conditionals, the status of (1a) and (1b) depends on whether worlds where both switches are down count as more "remote" than worlds where only one is down. (1a) is only true if they don't. (Otherwise any world where one of A and B is down and the lights are off is equally close as a world where both are down and the lights are on.)

Now the closeness standards are flexible. Context can suggest different resolutions. In particular, the wording of (1a) might suggest a resolution on which both-down worlds are more remote than one-down worlds, while the wording of (1b) might suggest the alternative resolution. That's because (1a) draws attention to the independence of the two switches: one can be down or the other can be down. (1b) instead draws attention to the joint position of the switches: whether they are both up. Varying the joint position gives us both-down just as easily as one-down.

That some such context-dependence is at work is supported by the fact that it's easy to hear (1a) as true and (1b) as false in isolation, but affirming (1a) and denying (1b) in one breath is weird.

(3) ? If switch A or switch B was down then the light would be off, but I don't think that if switch A and switch B were not both up then the light would be off.
(4) ?? The light would be off if switch A or switch B was down, but not if switch A and switch B were not both up.
Alonso-Ovalle, Luis. 2006. “Disjunction in Alternative Semantics.” PhD thesis.
Champollion, Lucas, Ivano Ciardelli, and Linmin Zhang. 2016. “Breaking de Morgan’s Law in Counterfactual Antecedents.” In Semantics and Linguistic Theory, 26:304–24.
Ciardelli, Ivano, Jeroen Groenendijk, and Floris Roelofsen. 2018. Inquisitive Semantics. Oxford University Press.


# on 22 April 2023, 17:58

Nice to see you discussing the latest iterations of 'logical equivalents in antecedents'.

I believe Romoli, Santorio and Wittenberg gave a semantic explanation for these cases in terms of questions under discussion and interpretation of negation. Perhaps yours is a pragmatic version of that:

I explicitly argued against the exclusivity implicature in antecedents in a separate paper by manipulating cases similar to CZC:;

# on 24 April 2023, 15:14

Interesting, thanks! I need to have a closer look at your paper in particular, as I'm currently thinking about SDA. I wonder why you don't consider the possibility of applying exhaustification twice to the entire conditionals, as in: exh(exh(A > C)).

# on 24 April 2023, 15:56

Thanks, Wo! I did not consider applying double exhaustification, because as far as I remember they were not helping in the cases I was interested in. My particular interest was in antecedents of the form (Av(A&B)) and I only considered embedded exhaustification to the weaker disjunct A [exh(A)v(A&B)]. Perhaps I am missing something but double exhaustification in this case did not really help to obtain the right results.

Could you tell more how you thought double exhaustification would help?

# on 24 April 2023, 18:35

If I did the calculations right then exh(exh(Av(A&B) > C)) entails A&B > C. So if (3a) is double-exhaustified, we can see why it's false. The same is true for (4b). But not for (4a).

# on 24 April 2023, 20:50

Interesting! Can you share your derivation of exh[exh[(Av(A&B))>C]], since I feel like I am making a mistake.

Still assuming your calculation is correct, I have a query. Take scenario 1 in the paper (Both children are on the right and the seesaw is balanced) and evaluate the following:

(3a') If Blue or both of them were on the left, then the seesaw would be unbalanced.

In that scenario I feel like (3a') is false, because if Blue was on the left, the seesaw would be balanced! But if exh(exh(Av(A&B)>C)) is equivalent to A&B>C, then we should expect (3a') to be true, since if both children were on the left, it would indeed be unbalanced.

Perhaps this is also relevant. In the paper, exhaustifying Av(A&B) to obtain (A&-B)v(A&B) was forcing a dilemma on the friends of the exhaustification. They get the (3a)-(4a)-(4b) trio right, but then they could not get the (9a)-(10a)-(10b) trio right. I cannot tell whether double exhaustification is helping with that or not. Is it?

Also thank you so much for the discussion!

# on 25 April 2023, 07:18

Here's a derivation that exh(exh(AvB > C)) entails A > C:

Assume Alt(exh(AvB > C)) = { exh(A > C), exh(B > C) } = { (A > C) & ~(B > C), (B > C) & ~(A > C) }.

Then exh(exh(AvB > C)) = (AvB > C) & ~[(A > C) & ~(B > C)] & ~[(B > C) & ~(A > C)].

~[(A > C) & ~(B > C)] & ~[(B > C) & ~(A > C)] is equivalent to (A > C) <-> (B > C).

Assuming that AvB > C entails (A > C) v (B > C), it follows that exh(exh(AvB > C)) entails (but is not equivalent to) A > C and B > C.

In your examples, one or both of A and B in the antecedent is complex, which might introduce some more alternatives, but I'd expect the above entailment to remain.

As far as I can tell, this also gets (9)/(10) right.

Add a comment

Please leave these fields blank (spam trap):

No HTML please.
You can edit this comment until 30 minutes after posting.