Identity and Indiscernibility
There are two principles here that must be distinguished (equivalent versions of each are given in the language of the predicate calculus). Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic.
- The indiscernibility of identicals
- For any x and y, if x is identical to y, then x and y have all the same properties.
- For any x and y, if x is identical to y, then x and y have all the same properties.
- The identity of indiscernibles
- For any x and y, if x and y have all the same properties, then x is identical to y.
- For any x and y, if x and y have all the same properties, then x is identical to y.
Principle 1 is taken to be a logical truth and (for the most part) uncontroversial. Principle 2 is controversial. Max Black famously argued against 2. (see Critique, below).
The above formulations are not satisfactory, however: the second principle should be read as having an implicit side-condition excluding any predicates which are equivalent (in some sense) to any of the following:
- "is identical to x"
- "is identical to y"
- "is not identical to x"
- "is not identical to y"
If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology: if x is non-identical to y, then there will always be a putative "property" which distinguishes them, namely "being identical to x".
On the other hand, it is incorrect to exclude all predicates which are materially equivalent (i.e. contingently equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two non-identical objects, because all distinguishing predicates are materially equivalent to at least one of the four given above (in fact, they are each materially equivalent to two of them), the two non-identical objects are identical – which is a contradiction.
Read more about this topic: Identity Of Indiscernibles
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