# A Misguided Argument for Conditional Excluded Middle

By | February 17, 2023

I have finished reading Kirk MacGregor’s new book defending Molinism. In my previous post I commented on his first 3 chapters and shared why I don’t think his reply to the grounding objection to Molinism is any good. I plan to do another blog post on chapters 4–6 of MacGregor’s book. But right now I want to comment on a paper by J. Robert G. Williams that MacGregor footnotes in his concluding chapter. MacGregor commends the paper as an “outstanding defense of conditional excluded middle” (p. 136).

Williams’ paper is titled “Defending Conditional Excluded Middle”. I’m going to comment on the first section of the paper where he presents a positive argument for conditional excluded middle (CEM). The rest of the paper contains rebuttals to arguments against CEM by David Lewis and Jonathan Bennett. I’m not going to comment on those. But first, I’m going to provide some general background on CEM and conditionals in general.

1. Defining CEM

First of all, let’s define CEM. It’s the thesis that for any given antecedent (A) and any given consequent (C) to a subjunctive conditional, either C would be true given A, or C would not be true given A. If we use ☐→ to represent a would-conditional and ◇→  to represent a might-conditional, then CEM takes the following form:

CEM: (A ☐→ C) ∨ (A ☐→ ~C)

What CEM says, in a nutshell, is that the implications of any hypothetical scenario (A) are always fully determinate, such that for any consequent C it is either determinately the case that C (given A) or determinately the case that ~C (given A).

2. Conditionals in general and what it would take to show that CEM is true

To see what it would take for CEM to be true, let’s take a step back and consider conditionals in general. Every conditional says that the antecedent A is sufficient in some way for the consequent C. I say “in some way” because there are different kinds of sufficiency. For example, there is material or extensional sufficiency. If we suppose that all swans are in fact white—this is not actually the case, but who cares—then being a swan (A) is extensionally sufficient for being white (C) since the extension of A is wholly included in the extension of C. Likewise, A could be logically sufficient for C (e.g., “If x is a red ball then x is red”), conceptually sufficient for C (e.g., “If x is a prime number then x is not a prime minister”), causally/nomologically sufficient for C (e.g., “If you set off that dynamite, then you’ll blow us all to kingdom come”), and so forth. The important point is that the truth of the conditional, whatever type of conditional it is, depends on the truth of a sufficiency claim. Is A actually sufficient for C (in the relevant way)?

The difference between indicative and subjunctive conditionals is simply that, in the latter case, the speaker conveys that he does not accept the antecedent and thus regards it as merely hypothetical, whereas in the former case, the speaker does not signal such doubt and so remains neutral on whether the antecedent is true or not. Consider, for example, the difference between “If you mix those chemicals, you will regret it” (indicative) and “If you were to mix those chemicals, you would regret it” (subjunctive). In many circumstances either conditional could be used to nearly the same effect (warning someone against mixing the chemicals), but the subjunctive doesn’t convey any sense that the addressee’s mixing the chemicals is a likely occurrence—it’s left in the realm of the merely hypothetical—whereas the indicative conveys the idea that the addressee’s mixing the chemicals is at least somewhat likely. CEM is supposed to apply to subjunctive conditionals, or ones where A is regarded as merely hypothetical and thus presumptively contrary-to-fact. That’s why these conditionals are often called “counterfactuals”.

Now, to show that CEM is true one would have to show that, no matter what ideas A and C convey, A is always either sufficient for C (in some way) or sufficient for ~C (in the same way). If A is sufficient for C, then it follows that A ☐→ C, where the “would” connective (☐→) carries the force of whatever kind of sufficiency is in view. And if A is sufficient for ~C, then it follows that A ☐→ ~C, where the “would” connective (☐→) again carries the force of whatever kind of sufficiency is in view. This is a tall order because, again, there seem to be cases where A is clearly not sufficient either for C or for ~C. The Molinist’s CCFs are a case in point. In CCFs the antecedent (A) contains a full causal specification of an indeterministic scenario and the consequent (C) is but one of several causally possible outcomes of that scenario. Because the scenario is explicitly indeterministic, this is a clear-cut situation where A is neither sufficient for C nor sufficient for ~C. Accordingly, both “If I were to toss this fair coin under perfectly indeterministic conditions, it would land heads (C)” and “If I were to toss this fair coin under perfectly indeterministic conditions, it would not land heads (~C)” are intuitively false. What’s true, rather, is that “If I were to toss this fair coin under perfectly indeterministic conditions, it might land heads (C)” and “If I were to toss this fair coin under perfectly indeterministic conditions, it might not land heads (~C)”. We thus arrive at the principle of would/might duality:

Would/Might Duality (WMD):
would (A ☐→ C) = not-might-not ~(A ◇→ ~C)
might (A ◇→ C) = not-would-not ~(A ☐→ ~C)

If WMD is true, then CEM is false because (A ☐→ C) ∨ (A ☐→ ~C) is not a proper instance of LEM since there are three possible ways in which A can be related to C:

• A is sufficient for C: (A ☐→ C) (would)
• A is sufficient for ~C: (A ☐→ ~C) (would not)
• A is neither sufficient for C nor for ~C: ((A ◇→ C) ∧ (A ◇→ ~C)) (might and might not)

Again, to show that CEM is true one would have to show that this third scenario cannot obtain for any A–C pair. If it so much as possibly obtains, then CEM is not the conceptual truth that it purports to be. Obviously this is a tall order. To exclude the might and might not case, one would have to show that would not (A ☐→ ~C) is semantically equivalent to—has the same meaning as—not would (~(A ☐→ C)). Since the former entails the latter (i.e., (A ☐→ ~C) entails ~(A ☐→ C)), one would have to show that the converse entailment also holds. That is, one would have to show that not would (~(A ☐→ C)) entails would not (A ☐→ ~C) for any arbitrary A–C pair.

Incidentally, my derivation of WMD does not in any way depend on endorsing Stalnaker–Lewis “possible world” semantics. So it can’t be undermined simply by attacking that semantics. My case for WMD rests only on (a) the sufficiency requirement, i.e., that in a true conditional the antecedent must be sufficient, in some relevant way, for the consequent and on (b) the possibility that that requirement not be met for some A–C pairs.

3. Williams’ argument for CEM

Let’s now turn to Williams’ argument for CEM and see whether he satisfies the burden of proof that I just articulated. His argument is clever and subtle. It depends on a series of (alleged) semantic equivalences and attempts to show by a series of steps that ~(A ☐→ C) is semantically equivalent to A ☐→ ~C. If that’s true, then CEM follows by noting that (A ☐→ C) ∨ ~(A ☐→ C) is an instance of the law of excluded middle (LEM) and then substituting A ☐→ ~C for the second disjunct.

Here’s Williams’ argument:

Premise 1: A and B are equivalent:

A. No student would have passed if they had goofed off.
B. Every student would have failed to pass if they had goofed off.

Premise 2: (A) and (B) are equivalent to the following:

A*. [No x: student x](x goofs off ☐→ x passes).
B*. [Every x: student x](x goofs off ☐→ ~(x passes)).

Premise 3: For any F, “[No x: Fx]Gx” is equivalent to “[Every x: Fx]~Gx”.

The argument now proceeds as follows: By Premise 3, (A*) is equivalent to

C. [Every x: student x] ~(x goofs off ☐→ x passes).

We can now construct a chain of equivalences to show that (C) is equivalent to (B*). Thus, (C) is equivalent to (A*) (Premise 3), which is equivalent to (A) (Premise 2), which is equivalent to (B) (Premise 1), which is equivalent to (B*) (Premise 2). Thus, the following are equivalent:

C. [Every x: student x]~(x goofs off ☐→ x passes).
B*. [Every x: student x](x goofs off ☐→ ~(x passes)).

Williams’ last step is to generalize this result. Thus, we can generalize A and B to “No F would G if they H” and “Every F would fail to G if they H”, respectively, and then apply the same chain of equivalences as above to conclude that

D. [Every x: Fx]~(Hx ☐→ Gx) is equivalent to [Every x: Fx](Hx ☐→ ~Gx).

From there, it seems a short step to show that ~(A ☐→ C) is equivalent to A ☐→ ~C, which yields CEM. Williams only claims, however, that (D) is an extensional equivalence (p. 652), which is not enough to establish an intensional or semantic equivalence between ~(A ☐→ C) and A ☐→ ~C. Since the latter sort of equivalence is what CEM requires, the argument doesn’t get us all the way to CEM, but it does point suggestively in that direction.

4. Assessing Williams’ argument

There are four possible places where Williams’ argument could conceivably fail: the three premises and the generalization step. Let’s examine each in turn.

To begin with, we can exclude Premise 1. It’s not essential to the argument provided that (A*) and (B*) are, in fact, equivalent. If they are, then we can shorten the chain of equivalences to go from (A*) to (C) to (B*).

Next, I don’t think the problem lies in Premise 2. (A*) says “No student is such that, if that student were to goof off, then that student would pass” and (B) says “Every student is such that, if that student were to goof off, then that student would not pass”. In both cases we’re told that goofing off is sufficient for not passing. Notice, however, that the equivalence between (A*) and (B*) is supposed to be an intensional equivalence. They aren’t merely saying that goofing off and not passing apply, as it so happens, to the same individual students, such that there are no actual students who both pass and goof off. That would be an extensional equivalence such as could be expressed by an indicative material conditional: (∀x)((Sx ∧ Gx) ⊃ ~Px)). Because these are subjunctive conditionals, however, they apply to merely hypothetical students, and they say that (given the test conditions) not goofing is a necessary condition for passing. In other words, it’s not merely that there is no extensional overlap between students who pass and students who goof off, but that (given the test conditions) there couldn’t possibly be any such overlap.

Next, I do discern a problem with Premise 3 or, more precisely, it’s application to (A*) and (B*). The problem is that the equivalence in Premise 3 is an extensional equivalence. Applying it uncritically to the intensional equivalence between (A*) and (B*) has the effect of suppressing relevant information, and that’s where the argument breaks down. To see that Premise 3 is extensionally correct it suffices to draw a simple Venn diagram with two overlapping circles. Label one circle “F” and the other “G”. What Premise 3 says is that “No Fs are Gs” is equivalent to “All Fs are non-Gs”. These are extensionally equivalent because they both say that the class of Fs and the class of Gs are disjoint, i.e., the overlap region is empty. Now, when we try to apply Premise 3 to (A*) and (B*), the class of Fs becomes (hypothetical) students and the class of Gs becomes something like persons who would pass after having goofed off. Applying Premise 3 to (A*) therefore yields:

E. All (hypothetical) students are non-(persons who would pass after having goofed off).

The problem is that (E) contains a scope ambiguity in that it is not clear how the negation is supposed to interact with the class of Gs. Of particular significance is that “non-” can interact with would to give either a narrow-scope or wide-scope reading:

• Narrow-scope: All students are persons who would not pass after having goofed off.
• Wide-scope: It is not the case that (all students are persons who would pass after having goofed off).

Williams’ argument masks this scope ambiguity. In fact, his case for CEM depends on there being no scope ambiguity between would not and not would. But they are clearly distinct. The narrow-scope (would not) reading says essentially what (A*) and (B*) say, namely, that goofing off is sufficient for not passing. The wide-scope reading, however, is compatible with two different scenarios.

• Scenario 1: Narrow-scope reading: Goofing off is sufficient for not passing.
• Scenario 2: Failure of sufficiency: Goofing off is neither sufficient for passing nor for not passing. Some (hypothetical) students might pass after having goofed off and some (hypothetical) students might not pass after having goofed off.

Scenario 2 is precisely the might-and-might-not case that grounds would–might duality and falsifies CEM. By applying Premise 3, an extensional equivalence principle, to the intensional equivalences of (A*) and (B*) without reflecting on how negation interacts with a complex would-predicate, Williams in effect suppresses the scope ambiguity between would not and not would. It’s only by doing so that he gets anywhere close to CEM. For this reason alone, his argument fails.

Finally, I have no problems with the generalization step of the argument. One consequence of his using an extensional principle like Premise 3, however, is that his argument is ill-suited to show that ~(A ☐→ C) and A ☐→ ~C are intensionally or semantically equivalent. So even if there were no problems with scope ambiguity, Williams’ argument couldn’t establish CEM, strictly speaking. The most his argument could do is point suggestively toward CEM. But, again, it can’t even accomplish that much because the scope ambiguity involved in applying Premise 3 to (A*) invalidates the chain of equivalences that he wants to draw. This is not surprising because, far from being an intuitive principle, CEM is highly counterintuitive. Why would anyone think it remotely plausible that, for every A–C pair, is it conceptually necessary that A is either sufficient for C or sufficient for ~C? Anyone who thinks indeterminism is true, or even could be true, should definitely reject CEM. (And that includes all you Molinists!)