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I need help formally translating the following definition into FOL: "a property F is essential to an object x if and only if x could not have been the object it is without possessing the property F."

So far I have "F is an essential property of x iffdef ∃x(¬F(x) → x ≠ x)"

Would that be correct?

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  • This is a strange sentence to formalise in FOL. It talks about ‘essential’ and ‘could’, both concepts which are more naturally captured in modal logic. It also quantifies over properties – ‘a property’ really means ‘for all properties’ here – and that’s something you can only do in Second-Order Logic. (You can’t formalise ‘F is an essential property’ since you don’t have predicates to apply to F.) You’ll probably need to say something like: For all x, if x is a property, then… - so, have a FO predicate saying ‘…is a property’.
    – MarkOxford
    Commented Jul 4, 2018 at 10:41
  • @MarkOxford - correct; we need properties of properties. Commented Jul 4, 2018 at 10:54

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We need two ingredients : high-order logic, because we have a property of properties.

And, IMO, we need modalities, in order to express the necessary "connection" involved in the concept of essence.

Compare with Gödel's ontological proof.

I would suggest :

Ess(F,x) ↔ ( F(x) & □∀y (~Fy → y ≠ x)).

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