Showing papers by "Jean-Yves Girard published in 2006"

Geometry of interaction IV: the feedback equation

Abstract: The first three papers on Geometry of Interaction [9, 10, 11] did establish the universality of the feedback equation as an explanation of logic ; this equation corresponds to the fundamental operation of logic, namely cut-elimination, i.e., logical consequence ; this is also the oldest approach to logic, syllogistics ! But the equation was essentially studied for those Hilbert space operators coming from actual logical proofs. In this paper, we take the opposite viewpoint, on the arguable basis that operator algebra is more primitive than logic : we study the general feedback equation of Geometry of Interaction, h(x⊕y) = x′⊕σ(y), where h, σ are hermitian, ‖h‖ ≤ 1, and σ is a partial symmetry, σ3 = σ. We show that the normal form which yields the solution σJhK(x) = x′ in the invertible case can be extended in a unique way to the general case, by various techniques, basically order-continuity and associativity. From this we expect a definite break with essentialism à la Tarski : an interpretation of logic which does not presuppose logic !