Upcoming Events

Logic Colloquium

November 05, 2021, 4:10 PM

https://berkeley.zoom.us/j/94041701380?pwd=dVBobHNqSGNjT2ttdVpEMWxmcG93dz09
Registration with Zoom is required for access.

Snow Zhang
Bersoff Faculty Fellow in Philosophy, New York University

Conglomerability, Countable Additivity and the Continuum Hypothesis

According to standard Bayesianism, rational agents should have degrees of belief (i.e. credences) that satisfy the Kolmogorov axioms of probability. In particular, rational credences should be countably additive. One of the strongest arguments for countable additivity is the conglomerability argument. Roughly, the idea is that rational agents should have credences that are countably additive because otherwise they would be disposed to change their credences in a predictable way. One objection against the conglomerability argument is that it begs the question: given the coherent theory of conditional probability, countable additivity is sufficient for conglomerability in countable partitions, but does not guarantee conglomerability in partitions of higher cardinality. But why should rational credences be conglomerable only in countable partitions, and not in uncountable partitions?

This talk explores whether the conglomerability argument succeeds if one adopts Kolmogorov’s theory of conditional probability instead. I argue that the answer is not obvious. In particular, under plausible assumptions, a natural generalization of the conglomerability argument fails because it entails both the continuum hypothesis and its negation.

Logic Colloquium

November 19, 2021, 4:10 PM

https://berkeley.zoom.us/j/94041701380?pwd=dVBobHNqSGNjT2ttdVpEMWxmcG93dz09
Registration with Zoom is required for access.

Joel (Ronnie) Nagloo
Associate Professor of Mathematics, University of Illinois Chicago

TBA

TBA

Logic Colloquium

December 03, 2021, 4:10 PM

https://berkeley.zoom.us/j/94041701380?pwd=dVBobHNqSGNjT2ttdVpEMWxmcG93dz09
Registration with Zoom is required for access.

Thomas Icard
Associate Professor of Philosophy, Stanford University

Interleaving Logic and Counting

Reasoning with quantifiers in natural language combines logical and arithmetical features, transcending divides between qualitative and quantitative. This practice blends with inference patterns in “grassroots mathematics” such as pigeon-hole principles. Our topic is this cooperation of logic and counting, studied with small systems and gradually moving upward. We start with monadic first-order logic with counting. We provide normal forms that allow for axiomatization, determine which arithmetical notions are definable, and conversely, discuss which logical notions can be defined out of arithmetical ones, and what sort of (non-)classical logics are induced. Next we study a series of strengthenings in the same style, including second-order versions, systems with multiple counting, and a new modal logic with counting. As a complement to our fragment approach, we also discuss another way of controlling complexity: changing the semantics of counting to reason about “mass” or other aggregating notions than cardinalities. Finally, we return to natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary, modules such as monotonicity reasoning, and procedural semantics via semantic automata. We conclude with some thoughts on further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on empirical aspects of our findings. Joint work with Johan van Benthem.