Logic

My one distinguishing feature is that I don't have a distinguishing feature - paradox?

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Fun question.

Let's say that a characteristic or property or whatnot is intrinsic if we can tell whether someone has it without needing information about other people/things. The fact that I have blue eyes is an intrinsic feature in that sense. My eye color doesn't depend on your eye color. But to know that I'm the shortest person in the room, you have to know things about the other people in the room as well as things about me (namely, our heights.) Being the shortest person in the room isn't an intrinsic property/quality/characteristic. Note that we're using "property", "characteristic", "quality" so as to include abstract things, and things that depend in possibly quite recondite ways on how an individual is related to other individuals, sets of individuals...

We don't tend to use the word feature so abstractly. Your features are the things we'd talk about to describe you yourself. Some of them, like height, may not be purely intrinsic, but to make things simple, we'll set those aside.

If we think of a feature in this way, as an intrinsic quality like eye color, shape of nose..., i.e., as a description of the person him/herself, and if by "distinguishing feature" we mean a feature that no one else has, then things seem clear. Suppose there are 5 people and 4 relevant features: height, hair color, eye color, weight. The people fit these descriptions:

1: dark-skinned; brown hair; brown eyes; attached earlobe.
2: dark-skinned; brown hair; blue eyes; detached earlobe.
3: dark-skinned; black hair; brown eyes; detached earlobe.
4: light-skinned; brown hair; brown eyes; detached earlobe.
5: dark-skinned; brown hair; brown eyes; detached earlobe.

The first person's earlobe configuration is unique; the second person's eye color is unique; the third person's hair color is unique; the fourth person's skin-tone is unique. Each of these people has a distinguishing feature. But none of the fifth person's features are unique. So no feature as we're using that word distinguishes this person. However, s/he is distinguished by an abstract, relational characteristic. S/he is the only person with no unique features.

Is it true that the person's "one distinguishing feature is that s/he has no distinguishing features?" Not if we use "feature" only for intrinsic characteristics. To make the sentence come out true, we'd need equivocate on the word "feature," using it broadly in the first part of the sentence and narrowly in the second. To clean it up, we'd need to reword it as something like "My one distinguishing property is that I don't have a distinguishing feature." In that version, of course, there's no paradox as long as we keep in mind that properties can be non-intrinsic but features can't—at least, they aren't on one way of understanding the word. The air of paradox comes from the fact that we can et away with the equivocation: we don't have a settled usage for "feature." We often use it only for intrinsic, visible characteristics, but sometimes use it more broadly.

Notice, by the way, that having no distinguishing features doesn't entail having a distinguishing property. Suppose we add a sixth person to our list above, and the description of person 6 is exactly like the description of person 5. Then neither 5 nor 6 has a distinguishing feature. And neither has a distinguishing property either.

From here we could go on to put the question in the context of Leibniz's principle of the Identity of Indiscernibles. If that principe is right, then everything has at least one distinguishing property, though not necessarily a distinguishing feature But I'll leave sorting that out as an exercise for the reader. ;-)

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