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In the Wikipedia article theory of descriptions, there is a traditional example of translating statements with descriptions into one that does not use descriptions, from Russell.

Before translation: The current Emperor of Kentucky is gray. 

1. There is an x such that x is an emperor of Kentucky
2. For every x and every y, if both x and y are emperors of Kentucky, then y is x (i.e. there is at most one emperor of Kentucky).
3. Anything that is an emperor of Kentucky is gray.

However, for some objects the second statement may not held. A good example is the well known mathematics object i (the root of -1). There is no mathematical properties can distinguish i and -i, unless you have already arbitrarily specified i to be one of them. This fact was mentioned by Conway's famous book "On Numbers and Games".

In this case, the uniqueness of description simple not held. What we want to translate the equation e^(pi * i) = -1 would somehow looks like (although e and pi are also descriptions to keep it simple only i being translated):

1. There is at least one x such that x^2=-1
2. If i satisfy x^2=-1, then e^(pi * i) = -1

In a longer detail, a translation may looks like

1. There is at least one x such that XXX
2. If x satisfy XXX then 
    2.1 (statements about x)
    2.2 (other statements about same x)
...

In other words, we need to fix the meaning of x in following discussions.

I am just asking, is this translation without uniqueness part of theroy of descriptions? Are there any references talking about this?

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    Welcome to SE! This is a bit puzzling. There is nothing in the wikipedia link you've given about translation - you might want to give a bit more detail about your concerns. – Mozibur Ullah Dec 23 '13 at 9:30
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This is a really interesting question. Let me see if I can make a little progress on it.

Are all of the following true?

  1. i and -i are indistinguishable--there is no property that i has that -i lacks and vice versa.
  2. Nevertheless i and -i are distinct (i.e. it is false that i = -i, i and -i are distinct locations along the number line).

It seems to me that the uniqueness of i simply has follow from i's distinctness from -i. i has to be the unique object picked out by the definite description "the imaginary unit" rather than not -i.

But, what you have here looks to be a neat argument for what metaphysicians call haecceities (>Latin "thisness"). The idea is that everything must have something unique to it which is not some normal qualitative property, but rather a bare individual non-qualitative property.

I was going to suggest writing this argument up, but it looks somebody's beat you to the punch already. Cf.

http://philmat.oxfordjournals.org/content/16/3/285.full.pdf+html

and

http://analysis.oxfordjournals.org/content/66/4/303.extract

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  • i and -i, mathematicians call a "dual". Same with logical conjunction + truth/logical disjunction + falsehood, carthesian product and disjoint union and so forth. It is a quite important concept. – Karl Damgaard Asmussen Apr 23 '14 at 8:17
  • I thought it was operators that were duals. Your examples are all operators. But i isn't an operator, but a value no? – user5172 Apr 23 '14 at 9:19
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    Duals are roughly anything where you can swap two sets of symbols and still have true statements. If you take any statement about complex numbers and swap i for -i and vice versa; you get true statements. There are also other kinds of duals, but the symbolic swap thing is commonly notices outside of category theory. – Karl Damgaard Asmussen Apr 23 '14 at 9:50
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The analysis of Russell regarding description aims to explain what are the truth-conditions of statemets involving descriptions.

His famous example is :

The King of France is bald.

What happens if there is no King ruling France ?

His transaltion (better: analysis) shows that the apparently "simple" statement must be "unpacked" into a conjunction of three statements; so (with the rules for propositional conectives) if one of the conjuncts is false the complex statement will also be false.

Your example, if I understant it well, concerns the root of -1 , which is not unique.

So, part 1) (the existence clause, i.e. there is an x such that x*x = -1) is true

but part 2) (the uniqueness clause, i.e. if x and y are both roots of -1, then x = y) is false.

According to Russell's analysis, a statement like : "The root of -1 has the property so and so" is false.

In order to apply Russell's translation, you must restore uniqueness. Is it possible to use, for example : "The positive root of -1 ..." ?

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  • "In order to apply Russell's translation, you must restore uniqueness. Is it possible to use, for example : "The positive root of -1 ..."" -- No, this point was made earlier. It's not possible to define positive/negative in a way compatible with the algebraic structure of the complex numbers. The distinction between i and -i is purely arbitrary. If you reversed them, nothing would change. This is much different than the distinction between 1 and -1 in the real numbers. – user4894 Feb 22 '14 at 1:44
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I think what you're proposing is that for some explanations of descriptive phrases in common use, the only correct form of translation to a first order logical form would be an indefinite description, because there might be ambiguity involved in trying to specify a singular referent of the phrase. This certainly isn't alien to Russell's project, though others have taken and picked at the notion of ambiguity involved in indefinite descriptions in their own ways; see, for example, the SEP article on Descriptions.

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