Are Proofs Objects or Acts? A Comparison of Prawitz’s and Sundholm’s Semantics

In a paper from 1998, Göran Sundholm has tried to convince Dag Prawitz that a semantic theory of deduction had better employ three notions of proof: proof-object, proof-act and proof-trace. In Prawitz’s semantics of valid arguments, however, the three notions can be said to collapse into each other. In this paper I shall first of all argue that this collapse results in a number of circularity and decidability problems. I shall also argue that it is maybe for getting rid of these problems that Prawitz’s later theory of grounds seems to allow for an objects-acts-traces distinction partly reminiscent of Sundholm’s. But Prawitz’s ground-theoretic picture retains many significant peculiarities. These mainly concern the way objects, acts and traces relate to each other and the epistemic status assigned to proof-objects. The main aim of this paper is to provide an overview and comparison of Prawitz’s and Sundholm’s semantics, and to argue that the divergences between the two stem from a difference in how Prawitz and Sundholm respectively conceive of the notion of assertion. To conclude, I discuss a problem of vacuous validity recently raised by Prawitz and investigate to some extent the possibility of reading it via Sundholm’s (and Martin-Löf’s) approach(es).