In flowchart languages, predicates play an interesting double role. In the
textual representation, they are often presented as conditions, i.e.,
expressions which are easily combined with other conditions (often via Boolean
combinators) to form new conditions, though they only play a supporting role in
aiding branching statements choose a branch to follow. On the other hand, in
the graphical representation they are typically presented as decisions,
intrinsically capable of directing control flow yet mostly oblivious to Boolean
combination. While categorical treatments of flowchart languages are abundant,
none of them provide a treatment of this dual nature of predicates. In the
present paper, we argue that extensive restriction categories are precisely
categories that capture such a condition/decision duality, by means of
morphisms which, coincidentally, are also called decisions. Further, we show
that having these categorical decisions amounts to having an internal logic:
Analogous to how subobjects of an object in a topos form a Heyting algebra, we
show that decisions on an object in an extensive restriction category form a De
Morgan quasilattice, the algebraic structure associated with the (three-valued)
weak Kleene logic K3w. Full classical propositional logic can be
recovered by restricting to total decisions, yielding extensive categories in
the usual sense, and confirming (from a different direction) a result from
effectus theory that predicates on objects in extensive categories form Boolean
algebras. As an application, since (categorical) decisions are partial
isomorphisms, this approach provides naturally reversible models of classical
propositional logic and weak Kleene logic.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX
