The Geach-Kaplan sentence reconsidered

The Geach-Kaplan sentence is alleged to be an example of a non-first-orderizable sentence, and the proof of the alleged non-first-orderizability is credited to David Kaplan. However, there is also a widely shared intuition that the Geach-Kaplan sentence is still first-orderizable by invoking sets or other extra non-logical resources. The plausibility of this intuition is particularly crucial for first-orderism, namely, the thesis that all our scientific discourse and reasoning can be adequately formalized by first-order logic. I first argue that the Geach-Kaplan sentence is, in fact, not first-orderizable even by invoking extra non-logical resources, in any sense that is acceptable for first-orderism and adequately corresponds to the sense in which the Geach-Kaplan sentence is deemed to be non-first-orderizable simpliciter via Kaplan's proof. To address this problem on behalf of first-orderism, I then propose an alternative conception of first-orderizability in the sense of which the Geach-Kaplan sentence and any other second-order sentences become first-orderizable by invoking extra non-logical resources; furthermore, in certain circumstances, they are first-orderizable without incurring any extra ontological commitment. My analysis also turns out to yield (as a biproduct) a significant enhancement of the so-called paradox of plurality.