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What is the difference between an object K and the thing which satisfies ( is an extension of', 'is an instance of' ) the property P ( or 'universal', 'predicate' , 'qualia', 'concept' ) :

P(x): x is K

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    "is" is ambiguous (as your example shows) between "belongs to" as in : "the object d satisfy the concept/property P" (P(d) or d ∈ P) and "is equal to" as in : "the object d is the same as K". When we say : "Socrates is a philosopher" we assert that the individual Socrates has the "property" of being a philosopher (or it belongs to the set of philosophers). When we say that "Socrates is not Plato" we say that the two names are referring to two individuals. – Mauro ALLEGRANZA Apr 15 '15 at 7:03
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One may formalize this is any formal system with a good concept of equality. One of these is homotopy type theory which knows the identity type (X = K). It is then a theorem that Sum_X (X = K) is a contractible type whose essentially unique inhabitant is K itself.

David Corfield just the other day wrote a note on this for an audience of philosophers interested in structuralism.

  • David Corfield, A note on 'The' and 'The Structure of' in Homotopy Type Theory, 2015 (pdf)

For more along these lines see also the references at nLab:structuralism.

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J. Hintikka has written a classic paper on the problem, which he calls the Fregaean trichotomy: 'is' as predication, identity, existence.

Hintikka J., "Is", Semantical Games, and Semantical Relativity, J. Philosophical Logic, Vol. 8, No. 1 (Jan., 1979), pp. 433-468 (Springer|Jstor)

After discussing these aspects and adding more, he concludes: the trichotomy is probably wrong and notes that "Frege, Russell, Quine, Davidson, Chomsky, Lakoff all were mistaken".

The topic defies surveying but the Ontology site has compiled a bibliography https://www.ontology.co/pdf/existence-predication-biblio.pdf.

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