Enjoyable Blog of Possibilities

Lengrand & Miquel (2008). Greco-Roman Fω, orthogonality and symmetrical candidates. Annals of Pure and Put on Logic 153:3-20.

We portray a version of system Fω, bade Fω^C, in which the layer of type
constructors is fundamentally the traditional one of Fω, whereas provability
of types is Graeco-Roman. The proof-term calculus accounting for the classic
reasoning is a variant of Barbanera and Berardi’s symmetrical λ-calculus.
We show that the hale calculus is powerfully normalising. For the
layer of type constructors, we apply Tait and Girard’s reducibility method
combined with orthogonality techniques. For the (authoritative) layer of terms,
we expend Barbanera and Berardi’s method based on a symmetrical notion of
reducibility candidate. We test that orthogonality does not catch the
fixpoint construction of symmetrical candidates.

We plant the consistency of Fω^C, and bear on the calculus to the
traditional system Fω, too when the latter is extended with axioms for
Greco-Roman logic.