September 25, 2024, 6-7:30 PM (note special time) in 234 Philosophy

Michael Mendler and Luke Burke (Otto-Friedrich University of Bamberg)

The Došen Square under construction: A tale of four modalities

In classical modal logic, necessity []A, possibility <>A, impossibility []~A and non-necessity <>~A form a Square of Oppositions (SO) whose corners are interdefinable using classical negation. The relationship between these modalities in intuitionistic modal logic is a more delicate matter since negation is weaker. Intuitionistic non-necessity [~] and impossibility <~>, first investigated by Došen, have received less attention and — together with their positive counterparts [] and <> — form a square we call the Došen Square. Unfortunately, the core property of constructive logic, the Disjunction Property (DP), fails when the modalities are combined and, interpreted in birelational Kripke structures à la Došen, the Square partially collapses. We introduce the constructive logic CKD, whose four semantically independent modalities [], <>, [~], <~> prevent the Došen Square from collapsing under the effect of intuitionistic negation while preserving DP. The model theory of CKD involves a constructive Kripke frame interpretation of the modalities. A Hilbert deduction system and an equivalent cut-free sequent calculus are presented. Soundness, completeness and finite model property are proven, implying that CKD is decidable. The logics HK[~], HK[], HK<> and HK<~> of Došen and other known theories of intuitionistic modalities are syntactic fragments or axiomatic extensions of CKD.

April 02, 2025, 4-6 PM (note special time) in 234 Philosophy

Harvey Lederman (University of Texas, Austin)

Maximal Social Welfare Relations on Infinite Populations Satisfying Permutation Invariance

We study social welfare relations (SWRs) on an infinite population. Our main result is a characterization of the common core shared by prominent utilitarian SWRs over distributions which realize finitely many welfare levels on this population. We characterize them as the largest SWR (in terms of subset when the weak relation is viewed as a set of pairs) which satisfies Strong Pareto, Permutation Invariance (elsewhere called “Relative Anonymity” and “Isomorphism Invariance”), and a further “Pointwise Independence” axiom. Based on joint work with Jeremy Goodman (Johns Hopkins).