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In The Oxford Companion to Philosophy, the entry on existence includes this paragraph.

It is often held that ‘exist’ is not a firstlevel predicate. What this means is that ‘exist’ does not express a property of objects, as verbs like ‘shine’ and ‘fall’ do. According to Frege and Russell, ‘exist’ is a second-level predicate, expressing a property of properties. Thus ‘God exists’ does not have the same logical form as ‘Sirius shines’, predicating a property of a particular object. Rather, it is equivalent to ‘Godhood is instantiated’, asserting that the property of being divine has at least one instance, or that there is at least one thing possessing that property.

We may notate the proposition 'Sirius shines' thus: $ \exists s(Ss) $

However, I don't know how to notate the proposition 'the property of Godhood is instantiated'. I presume it involves second order logic.

How do logicians notate a proposition that posits the instantiation of a property?

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  • $\begingroup$ Actually, given that "Sirius" is a proper name, I think it'd be more adequate to simply write $Ss$, with $s$ a constant denoting Sirius. As for the "the property of Godhood is instantiated", what's wrong with $\exists x Gx$, where $G$ is a predicate variable whose intended interpretation is "Godhood"? $\endgroup$ – Nagase Apr 12 '15 at 19:21
  • $\begingroup$ @Nagase I thought about $\exists x (Gx)$. However, it seems to express the proposition 'God exists'. The proposition 'something instantiates the property Godhood' entails that God exists, but I don't think that it is identical to the proposition 'God exists'; although, perhaps it is. $\endgroup$ – Hal Apr 12 '15 at 19:23
  • $\begingroup$ Actually, $\exists x Gx$ literally says that there is an $x$ such that $x$ satisfies $G$, which seems pretty much equivalent to $G$ is instantiated. $\endgroup$ – Nagase Apr 12 '15 at 19:25
  • $\begingroup$ You wrote $ \exists Gx$, I've never seen a predicate and a subject quantified together. Did you mean $\exists x (Gx)$? $\endgroup$ – Hal Apr 12 '15 at 19:27
  • $\begingroup$ I'm omitting the outer parenthesis, since there's no risk of ambiguity here, if that's what you mean. $\endgroup$ – Nagase Apr 12 '15 at 19:28
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From the SEP article on existence, it seems that Nagase and Mauro ALLEGRANZA are correct.

To say that foxes exist is to say that there are some things that are foxes; that is, the property of being a fox is instantiated. This is reflected in the standard regimentation of the sentences ‘Foxes exist’ and ‘There are foxes’ in first-order quantificational logic as ∃xFx, where Fx is the translation for the predicate ‘is a fox’.

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