Formal theory of SOGATs with phase distinction

Develop the missing technical details needed to give a rigorous formal account of second-order generalized algebraic theories equipped with a proposition and its associated open and closed modalities.

Background

The paper proposes extending the theory of signatures for second-order generalized algebraic theories with a proposition and associated open and closed modalities. This construction is intended to provide the formal framework underlying synthetic phase distinctions and the paper’s treatment of systems such as the simply typed lambda calculus and System F.

The definition is explicitly described as conjectural because significant technical work remains to fill in the details. The paper notes that related work is being pursued in greater generality, but does not supply the complete formal development itself.

References

The next definition is ``conjectural'' in the sense that there is significant work to be done to fill in the technical details.

— Hippogriff: a semantic approach to uniting core and modules  (2608.19728 - Lynch et al., 20 Aug 2026) in Section 5.1, immediately before Definition “SOGAT with phase distinction”