University of Tübingen - Germany |
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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. |
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.
R. Kahle & P. Schroeder-Heister (eds.), Proof-Theoretic Semantics, Special Issue of Synthese, 148 (3), 2006, pp. 503-743 |
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