The Case For Completeness

Defending the second of the vnm axioms

I’ve previously written an (unnecessarily polemical) article defending transitivity, the first of the vnm axioms. This article will defend completeness, at least in the moral domain. I shall argue that there’s a fact of the matter about, in any given case, whether one action is the best, or whether the actions are equal.

1 It Explains A Lot

Dorr, Nebel, and Zuehl argue “that all gradable expressions in natural language obey a principle that we call Comparability: if x and y are both F to some degree, then either x is at least as F as y or y is at least as F as x.” They claim “that Comparability is needed to explain the goodness of several patterns of inference that seem manifestly valid.”

Some evidence for their conclusion comes from the semantic literature given that “Comparability seems to be taken for granted in most contemporary work by semanticists working on gradable adjectives.” The authors particularly argue that a series of valid inferences are best explained by comparability including the following.

(i) ‘Not as F as’ Our first argument turns on the apparent validity of the following inference:

(13) Max’s room isn’t as tidy as Josh’s. So Josh’s room is tidier than Max’s.

Indeed, they cite the generally seemingly valid inference

x is not as F as y. So, y is F-er than x.

This seems valid in a variety of contexts. Several examples

1. Mark isn’t as tall as Tony. So Tony is taller than Mark.

2. Deontologists are not as wise as utilitarians. So utilitarians are wiser than deontologists.

The authors call this fact “Not As F As.” They additionally note

the converse of Not As F As also seems valid: ‘x is not as F as y’ is intuitively equivalent to ‘y is Fer than x’.

The authors next note

How should these appearances be explained? Here is what we propose. ‘As F as’ is truth-conditionally equivalent to ‘at least as F as’. As a result, given the validity of Exclusive Comparability, Not As F As can be turned into a valid argument-schema by adding an extra premise, ‘x and y are both at least as as F as themselves’. Now, in many instances of Not As F As, this extra premise is an obvious necessary truth. For example, it is presumably necessary that all rooms are as tidy as themselves, and that all composers are at least as good as themselves. So in these cases, the appearance of validity can be accepted at face value. However, this observation doesn’t take us far enough: Not As F As seems like a valid form; and its instances seem valid even when there are no specific grounds for assuming that the relevant objects are F to some degree. (For example, ‘The thing he is thinking about isn’t as expensive as the thing she is thinking about’ seems to entail ‘The thing she is thinking about is more expensive than the thing he is thinking about’.) One could consider explaining this by endorsing the strengthened version of Comparability that drops the ‘to some degree’ proviso. But this is not required: instead, the sense of validity can be explained by positing that comparative sentences like ‘x is as F as y’ carry a presupposition to the effect that x and y are both F to some degree (in our stipulative sense of being at least as F as themselves). Such a presupposition would not be surprising: in uttering ‘x is as F as y’ we are raising the question ‘How F are x and y?’, a question that intuitively presupposes that x and y are F to some degree. It is characteristic of presuppositions to “project through negation”: a sentence and its negation presuppose the same things. So ‘x is not as F as y’ has a false presupposition in the case where one or other of x and y is not F to any degree. Given Comparability, it follows that Not As F As has the following status: whenever its premise neither entails nor presupposes anything false, its conclusion neither entails nor presupposes anything false. Whether or not we want to apply the technical term ‘valid’ to arguments with this status, it seems sufficient to account for the intuitive feeling of validity that arguments like (13) inspire: compare ‘Max doesn’t know that Sue is a spy; so not every spy is known by Max to be a spy’, or ‘Every animal I own is well-trained; so at least one animal I own is well-trained’.

The authors next note that questions are seen to be non-loaded iff they don’t contain objectionable presuppositions. So for example the sentence “Is your Ferrari parked in the garage?” has the totally reasonable answer “Why are you assuming I have a Ferrari?” Similarly, the question “Was the performance by the children’s choir as expensive as the wedding banquet?” merits the totally reasonable answer “Why do you assume they were paid?”

However, in contrast, “(17) Is Max’s room as tidy as Josh’s room?” doesn’t seem to naturally merit the answer “Why do you assume that one of their rooms was at least as tidy as the other?”

They note

Plausibly, every room is tidy to some degree. But if Comparability is false, one would expect it to be pretty easy for a pair of rooms to be such that neither is at least as tidy as the other. So unless we take the questioner in (17) to have some independent evidence about what each of the rooms is like, it should seem like they are unjustifiably ruling out a live scenario. But we have no sense that this is the case: (17) seems a perfectly innocent question.

The point is especially forceful when we turn to sentences involving quantifiers, which should (if ‘as F as’ carries a comparability presupposition but Comparability is false) have strong presuppositions that we can often know to be false. Consider:

(18) Most composers aren’t as good as Beethoven.

(19) Are many contemporary cellists as talented as Pablo Casals was?

(20) I will buy an ice cream for everyone whose room is as tidy as Josh’s.

We should expect such sentences to carry universal presuppositions, just as ‘Most senior managers drive their Ferraris to work’ carries the presupposition that every senior manager has a Ferrari, and ‘Have many programmers stopped using Emacs?’ carries the presupposition that all programmers used to use Emacs. So on the view that equatives carry comparability presuppositions, (18)–(20) should presuppose, respectively, that every composer is comparable in goodness to Beethoven; that every contemporary cellist is comparable in talentedness to Casals; and that everyone’s room is comparable in tidiness to Josh’s. And if Comparability fails, these presuppositions should seem tendentious or even clearly false. But in fact these sentences are all perfectly felicitous for speakers with no special evidence. So, there is an abundance of evidence against the thesis that that ‘x is as F as y’ carries the comparability of x and y as a presupposition which is not entailed by the presupposition that x and y are both F to some degree.

The authors second arguments appeal to the ‘No F-er’ principle, which states “If x and y are both F to some degree, and x is no F-er than y, then y is at least as F as x.” For example “Max’s room is no tidier than Josh’s. So, Josh’s room is at least as tidy as Max’s.” Other examples include

Tim’s name is no longer than Garuk’s, so Garuk’s name is at least as long as Tim’s.

Larry’s hat contains no more live birds than Uraziel’s. So Uraziel’s hat contains at least as many birds as Larry’s.

This is naturally explained by comparability — after all, if comparability is true then there are only three options, more, less, and equal. However, if a room is no tidier than Josh’s that would mean that it can’t be more tidy, so it must be less tidy or equally tidy; in other words, Josh’s room must be at least as tidy as Max’s.

The authors third argument “is based on the apparent validity of the following piece of reasoning: (22) Kara is healthy, and Sam is not healthy. So Kara is healthier than Sam.”

They argue this is best explained by

Strong Monotonicity If x is F and y is not F but y is F to some degree, then x is F-er than y

They note that

Strong Monotonicity is a special case of a more general pattern. The following inference also seems valid: (23) Kara is very healthy, and Sam is not very healthy, so Kara is healthier than Sam.

The impression of validity remains if we replace ‘very’ by other “positive degree modifiers”: ‘extremely’, ‘somewhat’, ‘pretty’, ‘quite’, etc. The case for the validity of Strong Monotonicity thus extends to the following more general schema, where ‘V’ is to be replaced any positive degree modifier: Degree-Modified Strong Monotonicity If x is V F and y is not V F, but y is F to some degree, then x is F-er than y

The authors note

The validity of Degree-Modified Strong Monotonicity is hard to deny. But it makes it very hard to see how there could be any significant incomparability. For, given DegreeModified Strong Monotonicity, items that are divided by a degree-modified, positive form of a gradable adjective—as in (23)—must be comparable. And any gradable adjective can be degree-modified in many different ways. For example, imagine a possibly partial order of people from the not at all funny to the supremely funny: in between there are (in no particular order) people who are barely funny, somewhat funny, pretty funny, moderately funny, slightly more than moderately funny, extremely funny, and so on. All of these degree modifiers would have to impose bottlenecks in the ordering, across which there can be no incomparability. This is implausible. If there is incomparability in Fness, it is best explained by tradeoffs between dimensions of F: for example, two people who are each funny in different respects, where our use doesn’t privilege any particular weighting of these respects in such a way that one of them gets to count as funnier overall. One would expect to be able to find such tradeoffs anywhere in the ordering, not just in between the bottlenecks imposed by degree modifiers.

Put differently, Degree-Modified Strong Monotonicity implies that, if some x is V F, then every y to which x is incomparable is also V F. It follows that positive, degree modified adjectives spread through chains of incomparability. But, if “multidimensional” adjectives give rise to incomparability in the way that has been supposed, one would expect such chains to be ubiquitous. For example, one would expect the following kind of case to be possible. Consider two sequences of careers x1, . . . , xn and y1, . . . , yn, where the subscript represents one’s annual salary in dollars. For example, the x’s might be careers as a philosopher and the y’s might be careers as an artist. For some value of n, xn and yn might both be very good careers. If there is incomparability, we would expect it to be possible for each xi to be neither better nor worse than, nor just as good as, yi . And we would expect a dollar not to break the incomparability: if making i dollars per year as a philosopher is neither better nor worse than, nor just as good as, making i dollars per year as an artist, then making i − 1 dollars per year as a philosopher cannot plausibly be worse than making i dollars per year as an artist. That is the kind of intuition that motivates judgments of incomparability rather than equality in the first place. So suppose that xn—a very good career—is not better than yn, which is not better than xn−1, which is not better than yn−1, . . . , which is not better than x1, which is not better than y1. Because they reject Negative Transitivity, fans of incomparability can say that xn is better than y1. But, given Degree-Modified Strong Monotonicity, they must say that y1 is still a very good career. And that is absurd: no career in which one makes only a dollar a year is very good!

Their fourth argument is as follows

(iv) ‘The F-est things’. Strong Monotonicity has an analogue for superlatives that seems similarly compelling: Superlatives If x is one of the F-est Ks, and y is a K and is F to some degree but is not one of the F-est Ks, x is F-er than y. 25

Here K is schematic for a noun phrase, perhaps complex. The claim that Superlatives is valid can be supported by appeal to the intuitive validity of various specific arguments: for example ‘Parmesan is one of the most beloved cheeses, and Grana Padano is not one of the most beloved cheeses; so, Parmesan is more beloved than Grana Padano’.26

Indeed, it is rather tempting to think that whenever x is F-er than y and both are Ks, we can find some candidate denotation for ‘the F-est Ks’ that includes x but not y. If so, then given Superlatives, it would follow that any such x and y are separated by a bottleneck, so that Negative Transitivity is true restricted to the Ks. We could sharpen this hunch by turning to plural definite descriptions of the form ‘the n F-est Ks’, which do not seem to be subject to the peculiar context-sensitivity of ‘the F-est Ks’ (although they of course inherit any context sensitivity there might be in ‘F’ and ‘K’). The following principle concerning such cardinality-specific plurals seems rather natural:

Cardinal Superlatives If x is F-er than y, x and y are both Ks, and there are at most n Ks, then either x is the Fst K, or x is one of the two F-est Ks and y is not, or x is one of the three F-est Ks and y is not, . . . , or x is one of the n − 1 F-est Ks and y is not

But given the obvious extension of Superlatives to cardinality-specific definites, Cardinal Superlatives entails that Negative Transitivity holds restricted to the Ks whenever there are finitely many Ks. It follows that Negative Transitivity holds in full generality: just take K to be ‘thing identical to x or y or z’ for any given trio. And as we saw in section 1, there is no interesting way of accepting Negative Transitivity while rejecting Comparability.

Their fifth argument is as follows.

(v) ‘Much Fer than’. Our final argument is in the spirit of Degree-Modified Strong Monotonicity but does not require any appeal to the positive or superlative forms of gradable expressions. Consider the following inferences:

(24) Ann is much more creative than Bob. Cat is more creative, but not much more creative, than Bob. So Ann is more creative than Cat.

(25) Io had a lot more fun than Jim. Kyle had more fun, but not a lot more fun, than Jim. So Io had more fun than Kyle.

(26) Xan swam way faster than Yair. Zev swam faster, but not way faster, than Yair. So Xan swam faster than Yair. These seemingly impeccable inferences suggest the following generalization (where V now stands for positive degree modifiers of the same sort as ‘much’, ‘a lot’, and ‘way’): Much F-er Than If x is V F-er than z and y is F-er than z but not V F-er than z, then x is F-er than y. 2

Later, they say

Not As Much F-er As If x and y are both F-er than z, but y is not as much F-er than z as x is, then x is F-er than y.

The validity of this schema explains the badness of speeches like the following:

(27) The tuna and the sable were both tastier than the whitefish. But the sable wasn’t as much tastier as the tuna. However, the tuna was not tastier than the sable.

All of these considerations provide a very strong case for comparability. Much like Lewis invoking modal realism to explain properties, modality, and more, the authors argue that we get to parsimoniously explain the validity of many rules of inference by positing one fairly meager rule.

2 Money Pumping

Part 1 of the money pump: My version

It is always possible to reject some plausible premise to escape an otherwise tricky conclusion — this argument is certainly no exception. However, in their attempt to avoid money pumping, the person with incomplete preferences would seem to have to accept some strange things.

Let’s start by defining the choice worthiness relation as follows

X and Y are equally choice-worthy iff one has, all things considered, no decisive reason to choose either X or Y over the other. X is more choiceworthy than Y iff one has, all things considered, decisive reason to choose X over Y, and Y is more choiceworthy than X iff one has, all things considered, decisive reason to choose Y over X.

Suppose X and Y are equally choiceworthy. Then there’s a following principle which is as follows

Bribery: If any two things are equally choiceworthy, it is rational to accept payment to exchange one for the other.

This is very plausible and has lots of explanatory power. For example, five dollars are just as choiceworthy as 20 quarters — so it would be rational to accept payment to give up five dollars in exchange for 20 quarters. If there are two houses that you value exactly equally, you should opt for the cheaper one. Likewise with colleges.

Well now consider X+ which is slightly better than X, but both X and X+ are incommensurable with Y. Suppose that one is indifferent between X+ and X plus a dollar. If this is true, you can be money pumped.

If a person starts with X+, they’d accept ten cents to trade for Y, and ten cents to trade for X, but then a dollar to trade back for X+, so they’re down 80 cents and are back where they started. Doing this infinitely puts the person with incomplete preferences infinitely far in the hole — a bad place to be.

They’d no doubt reject bribery, but this is implausible. If two things are equally choice worthy, getting some benefit from exchanging them is surely good.

To get out of this, they will plausibly accept

Reasons Range Vagueness: There are often no precise facts about how much better one thing is than another. However, these imprecise facts can often hold over a range — so X and Y can be incommensurable, while X plus 5 dollars and Y are also incommensurable.

One problem with this view is that it is self-evidently false. I am generally reluctant to declare things self evidently false — I don’t believe I’ve ever done so on this blog. However, this claim is just plainly absurd. If you have no greater reason to choose A over B or B over A, and then A gets better, you would obviously have decisive reason to choose A. Those who deny this are just confused about what reasons are — or so it seems to me. There are more forceful money pumps coming up.

Part 2: The second most forceful version

Johan E. Gustafsson appeals to diachronic dominance

Diachronic Dominance It is rationally required that one does not make a sequence of choices to which an alternative sequence of choices is preferred.

Gustafsson points out that in a basic way, this is violated by deniers of completeness. Suppose that being a priest and being a lawyer are incommensurable. Call lawyer+ a career that’s like being a lawyer but a little bit better. Well, on this view, moving from lawyer+ to priest isn’t irrational, and then moving to lawyer from there isn’t irrational, but two choices together which weren’t irrational combined to be irrational.

This, however, isn’t that forceful. The third one is the most forceful. It’s also the most complex. Strap in!

Part 3: Gustaffson’s OP money pump

Gustaffson appeals to the following principles to derive a money pump

2 Symmetry of Souring Sensitivity If (i) 𝑋 − is a souring of 𝑋 and (ii) 𝑋 ≻ 𝑋− ∥ 𝑌 ∥ 𝑋, then there is a prospect 𝑌 − such that (i) 𝑌 − is a souring of 𝑌 and (ii) 𝑌 ≻ 𝑌− ∥ 𝑋.

4 Decision-Tree Separability The rational status of the options at a choice node does not depend on other parts of the decision tree than those that can be reached from that node

5 The Principle of Unexploitability seems like a basic requirement of rationality 

 If 

(i) 𝑋 − is a souring of 𝑋, 

(ii) 𝑋 ≻ 𝑋− , 

(iii), at node 𝑛, it holds that 𝑃 and 𝑃 − are two available plans such that 𝑃 is the only available plan that amounts to walking away from all offers by an exploiter and the prospect of following 𝑃 is 𝑋 and the prospect of following 𝑃 − is 𝑋 − , and 

(iv) one knows what decision problem one faces at 𝑛, then one does not follow 𝑃 − from 𝑛.

6 The Principle of Rational Decomposition If an agent, whose credences and preferences are not rationally prohibited, makes a sequence of choices which violates a requirement of rationality, then some of those choices are rationally prohibited.

8 The Principle of Future-Choice Independence The rational status of an option at a choice node and the rational status of the agent’s credences and preferences at that node do not depend on what would in fact be chosen at later choice nodes.

10 The Irrationality of Single Sourings If 

(i) 𝑋 − is a souring of 𝑋, 

(ii) 𝑋 ≻ 𝑋− ,

 (iii) node 𝑛 is a choice between node 𝑛 ∗ and 𝑋, 

(iv) node 𝑛 ∗ is a choice between 𝑋 − and 𝑌, and 

(v) one knows at node 𝑛 what decision problem one faces, then the sequence of choices consisting in choosing node 𝑛 ∗ at node 𝑛 and 𝑋 − at node 𝑛 ∗ violates a requirement of rationality

11 According to precautionary backward induction, it is irrational to choose an option 𝑋 over an option 𝑌 if there is a rationally allowed outcome of 𝑋 (that is, a prospect of an available plan consisting in choosing 𝑋 followed by choices that are not irrational) that is less preferred than some rationally allowed outcome of 𝑌 and there is no rationally allowed outcome of 𝑌 that is less preferred than some rationally allowed outcome of 𝑋.

You can read about the money pump in the paper more — it’s extremely compelling. This, in my mind, pretty much settles the issue.