THE DOMINANCE ARGUMENT

Huemer's argument comes in two flavours.Footnote ⁴ The first shows that the conjunction of NTB and a plausible principle called ‘Mereological Dominance’ entails that better-than is non-asymmetric. The second shows that the conjunction of NTB and Mereological Dominance entails that better-than is non-irreflexive. Here's Mereological Dominance:

• Mereological Dominance (MD): Given two sets of [value bearers] {A₁, A₂ . . . A _n } and {B₁, B₂ . . . B _n }, if (i) A _i is better than B _i for every i and (ii) there are no evaluative relationships between the A _i and the B _i (e.g., the A _i do not either complement or clash with one another in ways that would generate extra value or disvalue), then [A₁ + . . . + A _n ] is better than [B₁ + . . . + B _n ].Footnote ⁵

This principle is fairly plausible. And of course, better-than is asymmetric and irreflexive. Therefore, the soundness of either version of the argument would show NTB to be false. To be a bit more precise, here's a deductively valid (albeit informal) iteration of the first Dominance Argument:

1. 1. Mereological Dominance

2. 2. (MD & NTB) → ¬ (better-than is Asymmetrical)

3. 3. better-than is Asymmetric

4. 4. ¬ (MD & NTB)     (2, 3)

5. 5. ¬ NTB      (1, 4)

Given premise 1, we could similarly derive conclusions 4 and 5 were we to substitute premise 2 with premise 6 and premise 3 with premise 7:

1. 6. (MD & NTB) → ¬ (better-than is Irreflexive)

2. 7. better-than is Irreflexive

For the advocate of NTB to avoid the conclusion in either case, a premise will need to be given up. In both versions of the argument, premise 4 follows via modus tollens, so it can't be rejected. Premises 3 and 7 each enjoy Moorean certainty – better-than becomes unintelligible if either asymmetry or irreflexivity is given up. So, they too are not candidates for rejection. That leaves premises 1, 2 and 6. But premises 2 and 6 are true.

PREMISES 2 AND 6

Huemer is right about premise 2.Footnote ⁶ That is, if one is committed to Mereological Dominance and NTB, then one is committed to the view that better-than is not asymmetric. To see why, assume that there exists some set of value bearers for which the transitivity of better-than fails to hold: A > B > C > D > E > F > A. Now, split these six value bearers between two subsets:

1. (1a) {A, C, E}

2. (1b) {B, D, F}

Since (A > B), (C > D) and (E > F), according to Mereological Dominance, the mereological sum of 1a is better than the mereological sum of 1b. Now compare 1a with the same set, differently ordered:

1. (1c) {F, B, D}

Since (A < F), (C < B) and (E < D), according to Mereological Dominance, the mereological sum of 1a is worse than the mereological sum of 1c. Thus the mereological sum of 1a is all-things-considered better than the mereological sum of 1b and all-things-considered worse than the mereological sum of 1c. Therefore, better-than fails to be asymmetric in this case and premise two looks fairly well justified.

Premise 6 – that the conjunction of Mereological Dominance and NTB entails that better-than is not irreflexive – can be argued for in a similar fashion to above. Again, for the sake of argument, consider a set containing value bearers A, B and C, which are cyclically ordered (A > B > C > A):

1. (2a) {A, B, C}

2. (2b) {C, A, B}

Since (C > A), (A > B), and (B > C), according to Mereological Dominance, the mereological sum of 2b is better than the mereological sum of 2a. But 2a and 2b are identical. Thus 2b would be all-things-considered better than itself and better-than fails to be irreflexive. As I said at the outset of this section, Huemer is right about premise 2. As just shown, he's also right about premise 6.

PREMISE 1

Mereological Dominance is the only premise left open for criticism. Huemer's defence of the principle involves merely pointing out its prima facie plausibility. To illustrate this, he appeals to an example where driving a Honda is better than driving a Ford and being in California is better than being in Texas. If there are no holistic interactions between cars and places one could live, then surely driving a Honda in California would be better than driving a Ford in Texas.

Though Mereological Dominance may be sufficient to capture such intuitions, it's not necessary. To see why, consider a logically weaker principle, which does the job just as well:

• Mereological Dominance*: Given two sets of value bearers {A₁, A₂ . . . A _n } and {B₁, B₂ . . . B _n }, if (i) for some bijection between the two sets, A _i is better than B _i for every i, (i*) there's no bijection between the two sets such that A _i is worse than B _i for every i and (ii) there are no evaluative relationships between the A _i and the B _i , then [A₁ + . . . + A _n ] is better than [B₁ + . . . + B _n ].

(i) of this principle is just a different way of stating (i) in Huemer's formulation. (i*) is new. If the cars and states are not cyclically ordered, (i*) will be satisfied, so Mereological Dominance* captures just the same intuitions as Huemer's original principle.

More interestingly, the Dominance Argument fails if we accept this weaker principle in lieu of Huemer's stronger principle. Reconsider sets 1a and 1b. The mereological sum of 1a satisfies (i) of my revised principle for being better than the mereological sum of 1b. Now, consider sets 1a and 1c. The mereological sum of 1a fails to satisfy (i*) for being better than the mereological sum of 1c. Since 1b and 1c are identical, 1a fails to meet this sufficient condition for its mereological sum to be all-things-considered better than the mereological sum of 1b/1c.

Therefore, it's not the case that the conjunction of NTB and Mereological Dominance* entails that better-than is non-asymmetrical. We can state this point more generally to show that Mereological Dominance* doesn't introduce any symmetries into better-than: if (i) is satisfied for [A₁ + . . . + A _n ] being better than [B₁ + . . . + B _n ], (i*) could not be satisfied for [B₁ + . . . + B _n ] being better than [A₁ + . . . + A _n ].

Mereological Dominance* and NTB also don't entail that better-than is non-irreflexive. Reconsider 2a and 2b. 2b satisfies the first conjunct of the revised sufficient condition for 2b's elements jointly being all-things-considered better than the joint elements of 2a. But since 2a just is a rearrangement of 2b for which A _i is worse than B _i for every i, i* is not fulfilled. Therefore, the conjunction of NTB and Mereological Dominance* does not entail that better-than is non-irreflexive.

Therefore, the validity of the Dominance Arguments crucially relies on Huemer's particular statement of the Mereological Dominance principle.

HAS THE DOMINANCE ARGUMENT BEEN UNDERMINED?

Huemer gives no defence of his formulation of Mereological Dominance beyond its prima facie plausibility. Both his principle and my logically weaker one are in a stalemate on this count. That is, the two principles deliver the same verdict in nearly all cases. Furthermore, they only depart on some – but as we will see in a moment, not all – cases of non-transitive orderings.

Moreover, on transitive orderings, if a set of value bearers satisfies (i), necessarily, it satisfies (i*). So, Mereological Dominance gets any of the plausibility points that fulfilling (i*) would add for free on the assumption that better-than always generates acyclic orderings. It's worth stressing again: if better-than is transitive, (i*) holds anyway – and therefore, Mereological Dominance entails Mereological Dominance*. Of course, assuming transitivity in this discussion would beg the question. But discounting the plausibility of Huemer's Mereological Dominance in the case of cyclic orderings because it would entail that NTB is false would also be question-begging.

Since Mereological Dominance entails the weaker Mereological Dominance*, Huemer has to argue that though Mereological Dominance* is necessary, it fails to be sufficient to capture all of the intuitions in non-question begging cases. If there is no non-question begging case which Mereological Dominance* fails to capture, the Dominance Argument fails to be convincing.

Why does it fail to be convincing? Well, the defender of NTB can hold each premise of the Dominance Argument besides premise 1 to be true and accept a principle that is similarly favoured by all of the same evidence as premise 1. Given this, we might think that premise one lacks the justification needed for the Dominance Arguments to be cogent. The defender of NTB thus has her response: Huemer has no reason to prefer the stronger Mereological Dominance to the weaker Mereological Dominance* taken by itself. And it isn't clear what such a reason could be. Therefore, there's little reason to grant the stronger principle as a premise. But to be clear: this doesn't mean Mereological Dominance* is more intuitively plausible.

The defender of NTB could try to get a bit more out of this line of thought. Given the stalemate in justification between the two principles, she might think that there is some extra bit of justification in holding Mereological Dominance* in lieu of Mereological Dominance on coherentist grounds. So if their arguments for NTB are sufficiently compelling and Mereological Dominance is incompatible with such arguments, then she may be at least weakly justified in rejecting the stronger Mereological Dominance Principle.

The methodological principle is not that the arguments for NTB directly undercut Mereological Dominance. It's that given two principles with equal non-coherentist justification, one may be justified in believing one principle in lieu of another on coherentist grounds. This is importantly different from claiming that the defender of NTB can reject Mereological Dominance out of hand because of their arguments for NTB. But even if this coherentist argument would fail, that wouldn't undermine the line of thought two paragraphs ago.

But, one might object, the lines of argument offered thus far would similarly justify the following:

• Mereological Dominance**: Given two sets of value bearers {A₁, A₂ . . . A _n } and {B₁, B₂ . . . B _n }, if (i) A _i is better than B _i for every i, (i**) all value bearers in the union of both sets can be acyclically ordered and (ii) there are no evaluative relationships between the A _i and the B _i , then [A₁ + . . . + A _n ] is better than [B₁ + . . . + B _n ].

This is clearly ad hoc. It states, roughly, that mereological dominance principles only hold in cases where all of the value bearers can be acyclically ordered. Obviously, such a principle couldn't be used to show NYB is false. This may seem troubling because it would deliver many of the same verdicts as Mereological Dominance* and the lines of argument just sketched might appear to apply equally to it. Furthermore, the underlying methodological point would seem generalizable to many other philosophical debates: one can simply add a clause to a premise that says the premise holds except in cases of contradiction, which looks pretty bad.

But of course, the reason such a revision looks bad is that there doesn't appear to be any independent reason to accept it other than that it avoids the contradiction. So if the defender of NTB is to say something about why (i*) isn't objectionable in the way (i**) is, she will need to provide independent grounds for accepting (i*) that don't apply to (i**).

A couple of quick notes: one may think that the construction of (i*) is less objectionable than (i**) simply on the grounds that (i*) makes no reference to the issue in question – namely, whether better-than always generates transitive orderings. Additionally, the general structure of ‘in one way better and in no way worse’ – which is structurally similar to the conjunction of (i) and (i*) in my revised principle – is fairly common among dominance principles (i.e. pareto optimality), while a clause akin to (i**) is not. So given that there are other dominance principles with a similar structure, Mereological Dominance* certainly doesn't have the same flavour of ad hocness as Mereological Dominance**.

But more can be said. Consider two sets in which better-than fails to be transitive but (i*) still holds:

1. (3a) {A, C, E}

2. (3b) {B, D, F}

For the sake of argument, assume the value bearers are intransitively ranked as follows: A > B > C > D > E > F > C and A > F. The condition common to each revision holds: (i). But there's no bijection from one set to the other for which (i*) fails to hold and ex hypothesi, (i**) does fail to hold. Therefore in this case, Mereological Dominance and Mereological Dominance* both deliver the same verdict: the mereological sum of 3a is better than the mereological sum of 3b. This is the intuitive verdict. But Mereological Dominance** fails to do so since it claims that (i) and (ii) are only sufficient when there is a transitive ordering.

Therefore, we might conclude that Mereological Dominance** and other similarly ad hoc principles do not obviously capture the same virtues that Mereological Dominance* does either in their predictions or in the plausibility of their formulations. Moreover, Huemer has no principled reason to have accepted Mereological Dominance over Mereological Dominance* in his arguments other than that it provided him with an objection to NTB. But such a justification isn't allowable in this case.

I've argued that the defender of NTB ought not be moved by Huemer's Dominance Arguments. Because the logically weaker Mereological Dominance* principle has the same implications as Huemer's Mereological Dominance principle in all non-question-begging cases, the first premise of Huemer's argument fails to be compelling. Therefore, I think, the Dominance Arguments will not settle the question of whether better-than always generates transitive orderings.Footnote ⁷