# I Have a Unique Coffee Mug

I’ll attempt to explore a few of the subtleties surrounding the word unique in English.

I’ve drunk too many of these, and this post is how you can tell.

I’ll begin with an exploration of words like the same and different. Sentences containing these words will prove a fertile initial testing ground.

First, an observation. Consider the sentences:

1. Every coffee mug in my department is brittle.
2. Every coffee mug in my department is the same.

Though these sentences appear structurally similar, their predicates are deceptively different. The predicate is brittle applies individually to each mug, while is the same only makes sense with respect to the group as a whole. (It would make sense to say “The mug is brittle” but not *”The mug is the same.”) I’ll call these sorts of predicates individual and collective, respectively.

Many cultural artifacts play off of this phenomenon. Take, for example, the well-known commandment from Animal Farm:

All animals are equal, but some animals are more equal than others.

This is effectively a pun on the two sorts of predicate suggested above. In the beginning of the sentence, we imagine equal as a collective predicate; later, we’re startled to find it being used as an individual one.

Some predicates are such that the sense in which they should be taken is not clear at all. This has led to hilarity in mathematics. On one public forum, a mathematician remembers:

I gave a homework problem, “Let G1 be the group ___, let G2 be the group ___. Are G1 and G2 isomorphic?” and was astonished to get the response, “G1 is, but G2 isn’t.”

Here, the hapless student mistakes isomorphic for an individual predicate, when actually it’s a collective one.

Different is also a collective predicate, and using it alone seems strange:

1. Every coffee mug in our department is different.
2. *The number of coffee mugs in our department is different.
3. ?My coffee mug is different.

The sentence (5) is plausible, but only because the comparison is implicit: My coffee mug and the others in my department are different. (They’re not.) Indeed, though different is, strictly speaking, a collective predicate, it can, in certain situations, pass as an individual one, where the comparison is implicit.

This will be key in understanding the difficulty surrounding the word unique. Indeed, unique serves as both an individual and collective predicate at once: “syntactically” it’s an individual predicate, while “semantically” it’s a collective predicate. The pool of comparison, moreover, is given implicitly (and often ambiguously!). This will produce a rich array of complications.

Thanks to Richard Teague for enlightening discussion.

### An exploration of unique

To say that an object is unique is to say that it’s the only member of some set. The key question is: Which set? We’ll keep this in mind in what follows.

Things will begin to get complicated now. There seems to be a lexical ambiguity in the word unique. Does it mean “just one”? (This is how mathematicians use it.) Or does it mean “one of a kind”? (This is more colloquial.) I’ll refer to these two respective meanings as sole and special. The latter corresponds more closely to different.

We quickly approach ambiguities in the pool of comparison, which I’ll call the scale. Take the sentence “I have a unique coffee mug.” Immediately two natural pools arise, given respectively by the speaker and the world. I’ll call these the local and global scales, respectively. Fixing the special meaning described above for now, we see these two scales at work:

1. I have a unique coffee mug.
1. special, local. I have a coffee mug that is unlike all my other mugs. (It’s unique in the set of mugs I have that are like it.)
2. special, global. I have a coffee mug that is unlike all other mugs. (It’s unique in the set of mugs that are like it.)

Similar behavior occurs under the sole meaning, with a catch:

1. I have a unique coffee mug.
1. sole, local. I have just one coffee mug. (It’s unique in the set of mugs I have.)
2. sole, global. *I have the sole coffee mug. (It’s unique in the set of mugs.)

The interpretation (6D) of our sentence is infelicitous, but only because it’s excluded after the fact by the use of the English definite article a (the missing meaning would have been given by the). In languages without articles (e.g. Russian), the meaning (6D) would be given too.

Notice that (for either scale) under the special meaning – but not under the sole meaning – it would make sense to say “I have two unique coffee mugs”.

There’s another source of ambiguity I must now confront and will then try to ignore. This is that generated by the type vs. token distinction. Fortunately, the sort of ambiguities so generated find themselves somewhat orthogonal to those I discuss here, and treatable on similar terms. Briefly, even the sentence “I have a unique coffee mug” is ambiguous between I have a unique type of coffee mug and I have a unique token of coffee mug — and, furthermore, independently so for each of the four meanings we’ve studied. (Taking the sole, local case for illustration, the former, but not the latter, would be true if I had multiple copies of the same mug.) Similar sorts of ambiguity can be introduced in each of the sentences I discuss. In every case the reader can disambiguate the sentence by introducing an additional clarifying word as above. In most cases, one meaning will be much more natural, and the other will be all but excluded by intuition. When possible, I’ll indicate this natural meaning. In some cases I’ll explicate both.

These phenomena surrounding the word unique interact in interesting ways with scope and quantification. Consider the sentence “Every person in our department has a unique coffee mug.” Here, we start to see some ambiguities in the implicit pool of quantification. Notably, we now observe, in addition to the local and global scales, a third, intermediate scale, corresponding to the group over which quantification occurs (in this case the members of the department). I’ll call this scale middle. (This intermediate scale will occur in any sentence featuring restricted quantification.) We’ve also, of course, introduced a scope interaction between the determiners every and a. Let’s study this sentence’s meanings.

First I’ll study those meanings available under the sentence’s surface reading. Many familiar behaviors will recur. I’ll indicate whether I take type or token to be more natural; as before, in each case both interpretations could be taken.

1. Every person in our department has a unique coffee mug.
1. surface, special, local (token). Every person in our department is such that that person has a coffee mug that is unlike all of that person’s other mugs. (It’s unique in the set of mugs that that person has that are like it.)
2. surface, special, middle (token). Every person in our department is such that that person has a mug that is unlike all those had by others in the department. (It’s unique in the set of mugs in the department that are like it.)
3. surface, special, global (token). Every person in our department is such that that person has a mug that is unlike all other mugs. (It’s unique in the set of mugs that are like it.)
4. surface, sole, local (token). Every person in our department is such that that person has just one coffee mug. (It’s unique in the set of mugs that that person has.)

Though these have come out just fine, it’s not clear how we should make sense of sole at levels larger than that of the individual. Indeed, doing so demands deciding how to generalize sole‘s local (6C) and global (6D) cases treated above; I offer an attempt below.

On the other hand, the larger-scale renditions of sole invite us to investigate the inverse scope reading of our sentence, in which a is quantified above every. Indeed, when sole is used at a level higher of uniqueness than the local one, we effectively force a meaning like that given by the inverse scope, and we might as well quantify the object first; in this way we achieve a meaning identical to that which would have been given by surface under the larger scale. In other words, the meanings given under surface and inverse overlap, namely when sole is used at the middle and global scales. In fact, one result of taking the inverse reading is to exclude the local scale. We’ll see that the missing sole readings are given below, together with new ones:

1. Every person in our department has a unique coffee mug.
1. (surface or) inverse, sole, middle (type). There is a coffee mug that every person in our department has, and no other mug has the property that every person in our department has it. (It’s unique in the set of mugs that every person in our department has.)
2. (surface or) inverse, sole, global (type). *Every person in our department has the sole coffee mug. (It’s unique in the set of mugs.) Again, this sentence is excluded by the English use of a. It would be permitted in Russian – this sentence could have come in handy during the days of communist rule.
3. inverse, special, middle (type/token). There is a coffee mug that every person in our department has, and it’s unlike every other mug having the property that every person in our department has it. (It’s unique in the set of mugs every person in our department has that are like it.)
4. inverse, special, global (type/token). There is a coffee mug that every person in our department has, and it’s unlike all other coffee mugs. (It’s unique in the set of mugs that are like it.)

Constructing the generalization (7E) of (6C) and (6D) required some discretion. Note that it could still happen that certain individuals have other mugs. Indeed, there are other mugs, just not other mugs that ever person in our department has. This subtlety makes sole at the middle level more complicated than at the local or global levels.

I could have instead required additionally that each person had no other mugs. In any case, the latter meaning can be achieved by “combining” or “superimposing” this meaning with the surface, sole, local meaning (7D) given above. In fact, any of the meanings discussed here – which, after all, are all given by the same single sentence – can be easily combined, provided that they don’t contradict each other. (I’ve tried to identify the fundamental building blocks.) These combinations give further readings.

The final two sentences (7G) and (7H) raise interesting subtleties. For both, I’ve refrained from specifying type or token; though in each case both meanings are perfectly coherent, these meanings might seem strange. In short, the inverse scope reading seems to preclude a token interpretation, while the special sense of uniqueness seems to preclude a type interpretation.

The former, token, is strange because it forces us to imagine a whole department’s worth of people clinging onto a single (token of a) mug. This isn’t completely implausible, though; in Russia (why do these examples keep coming back to Russia?) our math building had a supply of dirty ragtag communal mugs, among which I’m quite sure that at least one was unique (special).

Taking the type interpretation, on the other hand, is strange because statements of uniqueness seem to become trivially true. If two types did have completely identical attributes, we would simply combine them and consider them as one. All “unique” (in the special sense) means, meanwhile, is different from the others, and so every type — after, of course, we identify pairs of facsimiles to one — should become trivially unique in this way.

The assertion of a type’s uniqueness can be made non-tautological if we strengthen our definition of uniqueness – to serve to deny not just the property of being identical to but even the property of being similar to any others. (In effect, we make the equivalence relation associated with the word unique strictly coarser than that which unites members of a type.) In this way two types which are different — and which, thus, can remain separate as types — might still fail to be unique. This makes uniqueness of a type non-automatic. Imagine, for example, that everyone in the department has an identical collection of exactly three mugs; two of these are identically white except for the fact that one has an (a) written on it while the other has a (b) written on it, while the third is multi-colored. The sentences above would then be true, but in a non-trivial way; the (a) and (b) types of mug would be considered (loosely) identical while only the third would be unique (special).

### Multiple quantifiers

I’ll briefly sketch the possible extension of this analysis to situations featuring more complex quantification. Take the following sentence (assume that the prepositional phrase in every department scopes over the subject):

1. Every person in every department at our university has a unique coffee mug.

Here, we quantify over multiple groups, namely all those departments populating the university.

The complexity quickly mounts. We now have two intermediate stages of scale, corresponding respectively to each department as well as to the university, in addition to the usual local and global scales. Thus the scale will, in general, vary over four levels. Furthermore, even if we insist that every department outscopes every person, the direct object a unique mug might still take, in addition to the narrow and wide scopes, an intermediate scope between the subject’s two determiners. This gives three scope readings to consider. Still further, after choosing a scope reading we must take care to consider only those scales of uniqueness which reside at a level higher than the scope taken by the object.

Thus, for example, if mug scoped between every person and every department, then the scales available to us would be the higher three of the four, namely departmentuniversity, and global.

We can take this further, proceeding with a sentence as at below (assume that, syntactically, in every university attaches to in every department and not directly to every person, and that each prepositional phrase scopes over the phrase it modifies):

1. Every person in every department at every university in this country has a unique coffee mug.

In addition to the local and global scales, we now can imagine three intermediate scales, corresponding to the departments, the universities, and the country.

In general, it appears that in each such sentence we produce scales corresponding to: local; each sub-pool over which quantification occurs (in order of growing size), of which there may be many; the overall pool represented by the quantification’s restriction, if it exists; and finally global. Furthermore, I imagine, the direct object will scope below, above, or anywhere within the range of values represented by the above scales. A choice of scope reading will exclude all scales below it. Finally, for each choice of scope and scale we have, as before, special and sole meanings together with the issues presented by type and token.

We soon encounter, of course, the perennial problem whereby linguistic intuitions begin to grow sparse. In any case, they can be elicited even here, with some work, I contend, and, with luck, they’ll continue to support the analysis I’ve proposed.

### Thoughts on validity

I’ll end with a word on philosophical issues in the above study. For the claims I make to be linguistically valid, in the philosophical sense — for them to correspond to reality — something like following should be true:

• that the meanings I suggest are indeed meanings of the English words,
• that they’re distinct form each other,
• that the ambiguities so produced hinge not on arbitrary contextual factors but rather on intrinsic structural properties of the sentences in question, and finally
• that these structural properties generate in a regular, predictable, and testable way the ambiguities I describe.

I could have introduced arbitrary and ridiculous scales. I could have proposed, for example, that “I have a unique coffee mug” actually asserts the uniqueness of my mug among the set of those mugs whose geographical longitudinal coordinate has ones digit a 6. Given an appropriately absurd pragmatic context, furthermore, such an assertion could have been warranted.

The scales I introduce are not like this, I contend, because they are given as natural meanings of the sentences by their linguistic features, and not by arbitrary extralinguistic content such as that above. Finally, the ambiguities I describe reflect in a highly regular way the structures of the underlying sentences.

Want to verify what I’ve claimed? Try to understand the sentences (8) and (9) and see how things match up. Try coming up with a sentence (10).