Two Ways to Pay: The Geach–Kaplan Sentence, the Ancestral, and the Price of Leaving First-Order Logic.

Quine illustrated the power of quantifying over classes with two examples, Frege's ancestral and the Geach–Kaplan sentence "some critics admire only one another". The second became the test case of the debate over plural quantification, a debate that treats ontological commitment as an inventory. We propose to measure a price instead. A sentence has a non-zero price if structures that agree on every first-order fact can disagree on it; the price is ideological or ontological according as the smallest regular logic in which the sentence can be stated keeps or loses the downward Löwenheim–Skolem property. We show that the Geach–Kaplan sentence, in each of its three forms in the literature, generates exactly monadic transitive closure logic, the logic of the ancestral: Quine's two examples are, as logics, one. Its price is ideological: it can be paid in objects or in a logical notion whose logic keeps downward Löwenheim–Skolem, Frege's ancestral. The price of Dedekind completeness is ontological: a new logical notion can state it without new objects, but no logic that states it keeps downward Löwenheim–Skolem. By Barwise's theory of absolute logics, every ideological price we know demands only that an absolute notion such as "finite" be determinate; ontological prices demand that "countable" be determinate and, for sentences with set quantifiers, the power set. The measure concerns what must be determinate, not what can be expressed, and we do not claim that ideological commitment is less substantive than ontological commitment.