Bruno Jacinto (LanCog - University of Lisbon): "Ordinals and Logicism"

June 6th, 2025. 3:00 pm.

Venue: Carlos Santamaria Zentroa, room 5.

Abstract:

Ordinal abstraction is the principle according to which any two well-orderings have the same ordinal if and only if they are isomorphic. This principle has been thought to be constitutive of the notion of ordinal. Alas, the Burali-Forti paradox shows that it is inconsistent.

In this talk I will offer some objections to philosophical views of ordinals which, taking a cue from ordinal abstraction, ground their existence and nature in the obtaining of isomorphism relations between well-orderings of individuals.

I will then put forward a novel characterization of ordinal. This characterization will be given in the language of higher-order modal logic. One of its distinctive features is that (in Russellian fashion) it conceives ordinals as properties of well-orderings -- thus taking seriously the idea that ordinals are types of well-orderings. Another distinctive feature is its high impredicativity.

Moreover, I will show that, in the setting of a contingentist-friendly higher-order modal logic, this characterization affords a robust theory of ordinals. I will conclude by arguing that this result paves the way for a defense of an upper logicism (in the sense of (Jacinto 2024)) about ordinals.