# Definite descriptions and essentially indistinguishable participants

In the analysis of definite descriptions there is a problem called "The Problem of Indistinguishable Participants", exemplified by the so-called bishop sentences:

If a bishop meets a bishop, the bishop blesses the bishop.

These sentences provide a specific example of incomplete definite descriptions. In the typical case incomplete definite descriptions are taken to have some sort of implicit content that "completes" them. For example, take the sentence from Strawson:

The table is covered with books.

Since there is more than one table in existence "the table" cannot pick out a unique table. A variety of implicit contents could be proposed, but generically we could say the complete description is something like "the contextually salient table". What renders a particular table contextually salient could vary, e.g., maybe it's the table I'm pointing at, but the crucial point is that there is some feature that distinguishes the relevant table from other tables -- the participants are only accidentally indistinguishable.

My question is about a situation in which the participants are essentially indistinguishable, and so could not be distinguished by some implicit content. For example, consider the following sentence:

If the square root of -1 is added to the other square root of -1, their sum is 0.

The issue is that "the square root of -1" could refer to either i or -i. If we start with the field of real numbers and take their algebraic closure, the result is unique up to isomorphism (in a second-order context, at least), and has the complex numbers as its intended modal. The problem is that there is no formula in the language of complex analysis that distinguishes i from -i (more generally, no formula exists to distinguish a+bi from a-bi for any value of a or b). Put another way, there exists a non-trivial automorphism mapping i to -i and so i and -i are indiscernible -- they have all of the same properties that can be expressed in the language.

Clearly we could avoid the problem by enriching the language or by embedding the complex field within a richer structure. But that doesn't seem like it should be needed. Either i or -i could be the referent of "the square root of -1", and it doesn't much matter which one is singled out since they have all of the same properties. But since they have all of the same properties we can't "complete" the description by positing some implicit content that distinguishes i and -i.

A quick thought is that "square root of -1" has {i, -i} as its reference set and the definite article could serve to introduce a choice function (or Hilbert's epsilon operator) that operates on the reference set to select just one of the two candidates. Suppose it selects i. Then "the other square root of -1" can be interpreted as a complement anaphor with the semantic value {i, -i} \ {i} = -i.

The problem I'm having is that while I've seen choice functions used in the analysis of indefinites (particularly, with specific indefinites), I'm not familiar with them being used in the analysis of definites. Furthermore, many of the conditions I've seen put forward to distinguish definites from indefinites seem not to apply here. For instance, it is traditionally help that definites must refer to a familiar (i.e., previously introduced) entity, while indefinites introduce a novel referent. Absent further context it seems as though "the square root of -1" is introducing a novel referent and so is semantically functioning more as an indefinite despite seeming to be definite. While this would legitimate the use of choice functions, it does seem a bit odd that the semantic analysis should be divorced from syntactic form. (This sort of thing is pervasive in mathematics. In most cases "the intended model" of a theory is only unique up to isomorphism and so there will be a similar plurality of indistinguishable candidates for its referent.)

Has there been work on "essentially indistinguishable participants"? Alternatively, have there been analyses of definites that utilize choice functions or something similar to select a unique referent from a number of indistinguishable candidates? Or, finally, is there precedent for analyzing a class of seemingly definite expressions as actually being disguised specific indefinites?

I think it solves most of the natural language problem to realize that a definite description conveys the uniqueness of its referent (rather than assuming it). And the rest is solved when you realize that there is a variety of circumstances that make utterances difficult for hearers to interpret. The possibility that the speaker apparently conveys something that is false (as here, the uniqueness of the referent of a description), is just one example from a vast array of potential communication problems.

• Am I right to understand you as advancing the view that the uniqueness requirement for definites is a (defeasible) pragmatic implication of some sort? I'm familiar with views on which there is no semantic distinction between definite and indefinite descriptions -- only differing pragmatic implications and felicity conditions. Would the idea, then, be to treat this case as akin to a specific indefinite? Nov 30, 2019 at 16:12
• I suppose another option that just occurred to me is to treat "the square root of..." as a generic definite description. Nov 30, 2019 at 17:46
• @Dennis, Or to suppose that "the square root of" means the same as "√", since "√-1" refers only to the positive root. Nov 30, 2019 at 19:06
• @Dennis, no, I'm saying that in human conversations, "the" conveys the information that there is a unique referent. If this happens not to be true, it may not matter, but even if it does matter, this means only that the speaker's expression was slightly defective, not that the sentence is nonsense. Nov 30, 2019 at 19:22
• I guess my question is what you mean by “conveys” such that information is conveyed that is not part of the meaning. Nov 30, 2019 at 19:26

I didn't read the question to end. Just one example

The table is covered with books

it's not a full clause if it's taken out of context. It's missung the relative pronoun. Usually we assume that there's an implicite one at the beginning of a top level clause. While I don't want to say that's a matter of established fact. While I'm not sure. While it is attractive, nevertheless, that inserting "that" into your example yields a mere Noun Phrase with embedded structure:

The table that is covered with books

You might as well say there would be no referent to

the table

that's obvious and does not need any explanation. Most if not all referents do need an explanation. Vice versa, it's child's play to keep askin. Which books? What press. What content? Why? Why? Why? Why do you abuse "essential", which in my understanding means vitaly neccesary? Although my understanding might be biased by the biologic meanining of "essential", it's essentially indistinguishable: If we have to eat, we have to stop, or at least pause asking for specifics and deal with the unknown. Can I eat that? Only one way to find out.

The bishop example is different though because it's ambiguous, but not unspecific: Each bishop blesses all bishops (that) they meet, if they get the chance.

Now, what was the question. I'm not sure you are correct about -i in analysis. As far as I know, -i represents a 270 degrees arc on the unit circle. Of curse that's symmetric to i, the imaginary number, representing 0 degrees. You are correct only insofar i = -(-i), because the choice of the symbol is arbitrary. But i is very specific, innthe sense that you cannot exchange i for -i in e.g. Euler's identity. You can try, but can't change it in all instances of i, so why would you try.

After al: You are just falling for a very basic mathematical mistake, because you ars not aware that sqrt(x)^2 isn't always sqrt(x^2). I can't explain that.

• I think you're confused about something. I wasn't saying that i = -i, only that i and -i possess all of the same algebraic properties. Although the complex field is unique up to isomorphism, it isn't unique up to unique isomorphism -- see here. What you are doing in your last paragraphs is moving to a different structure, 2D Euclidean vector space, and then utilizing the linear order on the reals to distinguish i and -i. Importantly, there is no natural order on the complex numbers. Nov 30, 2019 at 2:20
• well, i posesses the property \not \equal -i so you are evidently wrong, unless you mean definitory properties, which but depend on the approach. PS: you seem to know what you are talking about though Nov 30, 2019 at 2:22
• yes, there is no property (formula with one free variable) definable in the language of the complex field that holds of i but not -i. In any case, this is getting a bit far from the question. For the sake of the question it would probably be best if you just granted me this point (that they’re indistinguishable) or substituted an example you prefer, e.g., the indistinguishability of points in the Euclidean plane. Nov 30, 2019 at 2:49
• This answer seems fundamentally confused in a lot of ways. Surely if a theory of English syntax says "the table is covered with books" isn't a valid sentence, that theory is flawed? I'd be very surprised if any native-speaker said that was ungrammatical. $i$ represents a rotation of 90 degrees, not 0. And you absolutely can swap $i$ for $-i$ in Euler's identity: $e^{-i\pi}=-1$ is perfectly true.
– Draconis
Nov 30, 2019 at 4:15
• @vectory You're still not making any sense. Most linguists are not, in fact, "making [it] up as [we] go", and are working based on established theories and hard data. And the fact that -i×pi and i×-pi are completely and utterly equivalent (not just isomorphic) is a fundamental property of multiplication over the complex numbers.
– Draconis
Nov 30, 2019 at 17:13