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Lengrand & Miquel (2008). Hellenic 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 basically the traditional one of Fω, whereas provability
of types is Greco-Roman. The proof-term calculus accounting for the Hellenic
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 expend Tait and Girard’s reducibility method
combined with orthogonality techniques. For the (classic) layer of terms,
we expend Barbanera and Berardi’s method based on a symmetrical notion of
reducibility candidate. We show that orthogonality does not catch the
fixpoint construction of symmetrical candidates.

We show the consistency of Fω^C, and link the calculus to the
traditional system Fω, as well when the latter is extended with axioms for
Hellenic logic.

Related Posts:
Fω^C: a symmetrically definitive variant of System Fω
Fω^C: a symmetrically definitive variant of System Fω