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Full Idea
There is no condition in a first-order language for a predicate to express identity, rather than indiscernibility within the resources of the language. Leibniz's Law is statable in a second-order language, so identity can be uniquely characterised.
Gist of Idea
Identity can only be characterised in a second-order language
Source
Harold Noonan (Identity [2009], §2)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.4
A Reaction
The point is that first-order languages only refer to all objects, but you need to refer to all properties to include Leibniz's Law. Quine's 'Identity, Ostension and Hypostasis' is the source of this idea.
Related Ideas
Idea 16014 It is controversial whether only 'numerical identity' allows two things to be counted as one [Noonan]
Idea 11095 We should just identify any items which are indiscernible within a given discourse [Quine]
| 16015 | Problems about identity can't even be formulated without the concept of identity [Noonan] |
| 16014 | It is controversial whether only 'numerical identity' allows two things to be counted as one [Noonan] |
| 16017 | Identity is usually defined as the equivalence relation satisfying Leibniz's Law [Noonan] |
| 16016 | Identity definitions (such as self-identity, or the smallest equivalence relation) are usually circular [Noonan] |
| 16020 | Identity can only be characterised in a second-order language [Noonan] |
| 16018 | Indiscernibility is basic to our understanding of identity and distinctness [Noonan] |
| 16019 | Leibniz's Law must be kept separate from the substitutivity principle [Noonan] |
| 16024 | I could have died at five, but the summation of my adult stages could not [Noonan] |
| 16023 | Stage theorists accept four-dimensionalism, but call each stage a whole object [Noonan] |