We are a Working Group devoted to the discussion of historical and philosophical issues in symbolic logic, mathematics, and science. We meet on occasional Wednesday evenings for a talk and a lively discussion. The group is funded by the Doreen B. Townsend Center for the Humanities and the Department of Philosophy.
All members of the academic community are welcome to attend. We have regular participants in many different fields, including philosophy, mathematics, history of science, and psychology.
The group organizers are Wesley Holliday, John MacFarlane, Paolo Mancosu, Seth Yalcin, and Xueyin (Snow) Zhang.
Our next event
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.