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?