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In a recent paper Ontological Commitment, Augustin Rayo tries to make the notion of ontological commitment more precise. He specifies:

"Quine’s Criterion (Logical Version)
A first-order sentence ψ carries commitment to Fs just in case "ψ → ∃x P(x)" is a truth of (free) logic for some predicate P expressing F-hood."

He then says that the free logic restriction "is needed to avoid the conclusion that, e.g. an arbitrary sentence carries commitment to every object named by an individual constant in the language."

Question 1: I do not understand why considering a classical logic and not a free one would end up making any arbitrary sentence be committed to an object named by an individual constant.

Furthermore, he considers this criterion not entirely correct in this form:

"The logical version of Quine’s Criterion tells us that ‘∃xWhale(x)’ carries commitment to whales, but not to mammals (since, e.g. ‘∃x (Whale(x)) → ∃x (Mammal(x))’ is not a logical truth). But part of what it is to be a whale is to be a mammal. So if the truth of ‘∃xWhale(x)’ demands of the world that it contain whales, it demands of the world that it contain mammals."

To solve this, he "amends" the biconditional definition in this way:

"A first-order sentence ψ carries commitment to Gs just in case:
(a) "ψ → ∃x P(x)" is a truth of (free) logic for some predicate P expressing F-hood; and
(b) part of what it is to be F is to be G."

Question 2 I do not get why this should solve the problem, since the sentence "∃x(Whale(x)) → ∃x (Mammal(x))" still isn't a logical truth, and therefore one of the two conditions (a) and (b) is not satisfied, even though (b) is satisfied. What am I missing here?

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  • I don't understand your second issue: "one of the two conditions (a) and (b) is not satisfied even though (b) is satisfied." This would mean (a) is not satisfied, right? But (a) is part of the premise, it has to be satisfied to apply this rule. Can you clarify? Aug 21 at 19:40
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    1. Because classical logic requires all individual constants to denote unique objects in the domain of discourse, see Free Logic. 2. "A truth of (free) logic" is a shorthand for "a truth of first order theory based on free logic", not a "logical truth" as in a tautology. And the reason he does not just plaster '∃x (P(x)) → ∃x (Q(x))' there is that he wants something stronger than material implication. It has to be true "in virtue of meanings" or "essentially", as he discusses in footnote 1, but he doesn't want to pick a specific formalization.
    – Conifold
    Aug 21 at 21:20
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Question one: in classical predicate logic, every individual constant denotes an object from the domain of the logic. So just asserting P(a) implies that a is in the domain of the logic, which implies that ∃x P(x).

Question two: I'm not sure what problem you see. Did you note that this rule involves commitment to G's rather than to F's? Another way to put it is that if you have an ontological commitment to F's by the first rule, and if all F's are G's, then you also have an ontological commitment to G's.

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