  This paper introduces a novel approach to the axiomatic theory of quadratic
forms. We work internally in a category of certain partially ordered sets,
subject to additional conditions which amount to a strong form of local
presentability. We call such partial orders presentable. It turns out that the
classical notion of the Witt ring of symmetric bilinear forms over a field
makes sense in the context of quadratically presentable fields, that is fields
equipped with a presentable partial order inequationaly compatible with the
algebraic operations. As an application, we show that Witt rings of symmetric
bilinear forms over fields, of both characteristic 2 and not 2, are isomorphic
to Witt rings of suitably built quadratically presentable fields, which
therefore provide a uniform construction of Witt rings for all characteristics.
