Proof Theoretic Semantics

Peter Schroeder-Heister

University of Tübingen - Germany

Proof-Theoretic Semantics (PTS) is an alternative of model-theoretic (or truth-condition) semantics. It is based on the idea that the central notion in terms of which meanings are assigned to expressions is that of proof rather than truth. In this sense PTS is inferential rather than denotational in spirit. Although the claim that meaning is use has been quite prominent in philosophy for more than half a century, the model-theoretic approach has always dominated formal semantics. This is, as I see it, due to the fact that for denotational semantics very sophisticated formal theories are available, going back to Tarski's definition of truth,  whereas "meaning is use" has often been just a slogan without much formal underpinning. However, within general proof theory several formal approaches to PTS have been developed which promise to provide a 'real' alternative to the model-theoretic approach. They are all based on ideas Gentzen-style proof-theory, which are then turned into logico-philosophical principles.
This tutorial provides in its first part, after some remarks on the historical background (Frege, Hertz, Gentzen, Lorenzen), the basic results of theories of weak and strong normalization (Prawitz, Tait, Martin-Löf, Girard) which are the basic technical tools of PTS.
In its second part it develops and discusses the Dummett-Prawitz approach to PTS and their definition of proof-theoretic validity. It discusses various options of how to define the validity of proofs and relates them to corresponding notions of logical consequence. It puts particular emphasis on the "universal" aspects of these ideas, dealing with general proof structures and arbitrary proof reduction systems as models with respect to which validity is defined.

The third part is devoted to definitional and clausal approaches to PTS, partly developed by the instructor himself jointly with Lars Hallnäs (Gothenburg). This approach puts the validity of rules and inference steps (rather than that ofwhole proofs) first. As compared with the Dummett-Prawitz approach, it is local rather than global, thus not requiring global properties of proofs like normalization or cut elimination to hold in every possible case. Technically it implies a shift from natural deduction to the sequent calculus as the basic model of reasoning. This allows in particular a more general way of dealing with assumptions and negation, including their substructural features. This approach is is not restricted to logical constants but uses clausal definitions in a universal sense as the basis of reasoning, which means that it goes far beyond logic in the narrower sense.  Interesting applications are theories of equality, circular reasoning, universal theories of denial and negation, and extensions logic programming. Whether cut is eliminable in the various theories discussed is always an interesting problem, though not crucial for the approach to be meaningful.

Main reference

R. Kahle & P. Schroeder-Heister (eds.), Proof-Theoretic Semantics, Special Issue of Synthese, 148 (3), 2006, pp. 503-743
P.Schroeder-Heister, Validity concepts in proof-theoretic semantics , ibid., 525-571.
P.Schroeder-Heister, On the notion of 'assumption' in logical systems, Fifth International Congress of the Society for Analytical Philosophy, Bielefeld, 22-26 September 2003, Paderborn: mentis 2004, 27-48
P. Schroeder-Heister, Rules of definitional reflection. In: Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science (Montreal 1993), Los Alamitos 1993, 222-232.