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, 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
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?