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. Apr 15 '15 at 7:03

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.


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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