Home Philosophy Advances in Proof-Theoretic Semantics

## Outlook: Applications and Extensions of Definitional ReflectionWe have not discussed computational issues here. Clausal definitions give rise to computational procedures as investigated and implemented in logic programming, and definitional reflection adds a strong component to such computation (see Hallnäs and Schroeder-Heister [31], Eriksson [17]). This computational aspect is important to proof-theoretic semantics. We should not only be able to give a definition of a semantically correct proof, where 'semantically' is understood in the sense of prooftheoretic semantics, we should also be interested in ways to construct such proofs that proceed according to such principles. Programming languages, theorem provers and proof editors based on inversion principles make important contributions to this task. Devising principles of proof construction that can be used for proof search is itself an issue of proof-theoretic semantics that is a If we want to deal with more advanced mathematical theories, stronger closure and reflection principles are needed. At an elementary level, clauses definition can be used to describe function computation in the form
which is supposed to express that from the arguments
computed. More generally, one might describe functionals tary) clauses the bodies of which describe the evaluation of functions There are several other approaches that deal with the atomic level proof-theoretically, that is, with issues beyond logic in the narrower sense. These approaches include Negri and von Plato's [42] proof analysis, Brotherston and Simpson's [2] infinite derivations, or even derivations concerning subatomic expressions (see Wie˛ckowski [73]), and corresponding linguistic applications, as discussed by Francez and Dyckhoff [19] and Francez et al. [20]. Proof-theoretic semantics beyond logic is a broad field with great potential, the surface of which, thus far, has barely been scratched. |

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