The Logic and Metaphysics Workshop will meet on November 7th from 4:15-6:15 (NY time) in-person at the Graduate Center (Room 7314) for a talk by Victoria Gitman (CUNY).
Title: Set theory without the powerset axiom
Abstract: Many natural and useful set-theoretic structures fail to satisfy the Powerset axiom. For example, the universe of sets can be decomposed into the H_alpha-hierarchy, indexed by cardinals alpha, where each H_alpha consists of all sets whose transitive closure has size less than alpha. If alpha is a regular cardinal, then H_alpha satisfies all axioms of ZFC except, maybe, the Powerset axiom (it will only satisfy Powerset if alpha is inaccessible). Class forcing extensions of models of ZFC will often fail to satisfy ZFC, but if the class forcing is nice enough, then it will preserve all the axioms of ZFC except, maybe, the Powerset axiom. Finally, a strong second-order set theory, extending Kelley-Morse by adding a choice principle for classes (Choice Scheme), is bi-interpretable with a strong first-order set theory without the Powerset axiom. Thus working in a strong enough second-order set theory can be reinterpreted as working in a strong first-order set theory in which the Powerset axiom fails. It turns out that simply taking the axioms of ZFC and removing the Powerset axiom does not yield a robust set theory. I will discuss robust (and strong) axiomatizations of set theory without Powerset and how much of the standard set theoretic machinery is still effective even in the strongest theories in the absence of Powerset. Because of the bi-interpretability of a strong set theory without Powerset with Kelley-Morse plus Choice Scheme, these results will have consequences for which set theoretic machinery continues to work in set theories with classes. Time permitting, I will also talk about some unexpectedly strange models of set theory without Powerset.
The Logic and Metaphysics Workshop will meet on October 31st from 4:15-6:15 (NY time) via Zoom for a talk by Friederike Moltmann (CNRS, Côte d’Azur).
Title: The semantics of special quantification: Higher-order metaphysics and nominalization approaches
Abstract: Prior’s problem consists in the impossibility of replacing clausal complements of most attitude verbs by ‘ordinary’ NPs; only ‘special quantifiers’ that is, quantifiers like something permit a replacement, preserving grammaticality or the same reading of the verb;
(1) a. John claims that he won.
b. ??? John claims a proposition / some thing.
c. John claims something.
In my 2013 book Abstract Objects and the Semantics of Natural Language, I have shown how this generalizes to nonreferential complements of various other intensional predicates and argued for a Nominalization Theory of special quantifiers. In this talk, I will review and extend the range of linguistic generalizations that motivate the Nominalization Theory and show that they pose serious problems for a simple higher-order semantics of special quantifiers. I will outline a new version of the Nominalization Theory for special quantifiers with attitude verbs and address the question whether there can be a unified semantics of special quantifiers for the various contexts in which they display a nominalizing force.
The Logic and Metaphysics Workshop will meet on October 24th from 4:15-6:15 (NY time) in-person at the Graduate Center (Room 7314) for a talk by Rohit Parikh (CUNY).
Title: A measure of group coherence
Abstract: The Stanford Encyclopedia of Philosophy has an article on Social Epistemology and also one on group rights. Wikipedia has an article on group coherence. Clearly, groups are important and that importance is acknowledged. But what is missing is a measure of group coherence or as I shall say, groupiness. The Democratic party is a group but the Squad is a more coherent subgroup and works more closely with each other. The bees in a beehive work coherently with each other but it is not clear if this coherence is buttressed by common beliefs. The purpose of this talk, and of this project is to propose a measure of groupiness, investigate its properties, ask about the extent to which it enables group action, and about the extent to which it comports with epistemic logic and with the theory of information.