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?
