Topos Theory as a Framework for Partial Truth

Butterfield, Jeremy (2000) Topos Theory as a Framework for Partial Truth.

Abstract

This paper develops some ideas from previous work (coauthored, mostly
with C.J.Isham). In that work, the main proposal is to assign as the
value of a physical quantity in quantum theory (or classical physics),
not a real number, but a certain kind of set (a sieve) of quantities
that are functions of the given quantity. The motivation was in part
physical---such a valuation illuminates the Kochen-Specker theorem; in
part mathematical---the valuations arise naturally in the theory of
presheaves; and in part conceptual---the valuations arise from
applying to propositions about the values of physical quantities some
general axioms governing partial truth for any kind of proposition.

In this paper, I give another
conceptual motivation for the proposal. I develop (in Sections 2
and 3) the notion of a topos (of which presheaves give just one kind of
example); and explain how this notion gives a satisfactory
general framework for making sense of the idea of partial
truth. Then I review (in Section 4) how our proposal applies
this framework to the case of physical theories.