Argumentation is a proof theoretic paradigm for reasoning under uncertainty. Whereas a ‘proof’ establishes its conclusion outright, an ‘argument’ can only lend a measure of support. Thus, the process of argumentation consists of identifying all the arguments for a particular hypothesis φ, and then calculating the support for φ from the weight attached to these individual arguments. Argumentation has been incorporated as the inference mechanism of a large scale medical expert system, the ‘Oxford System of Medicine’ (OSM), and it is therefore important to demonstrate that the approach is theoretically justified. This paper provides a formal semantics for the notion of argument embodied in the OSM. We present a categorical account in which arguments are the arrows of a semilattice enriched category. The axioms of a cartesian closed category are modified to give the notion of an ‘evidential closed category’, and we show that this provides the correct enriched setting in which to model the connectives of conjunction (&) and implication (⇒).Finally, we develop a theory of ‘confidence measures’ over such categories, and relate this to the Dempster-Shafer theory of evidence.