Although induction is omnipresent, inductive theorem proving in the form of descente infinie has not yet been integrated into full first-order deductive calculi. We present such an integration for (possibly even higher-order) classical logic. This integration is based on lemma and induction hypothesis application for free variable sequent and tableau calculi. We discuss the appropriateness of these types of calculi for this integration. The deductive part of this integration requires the first combination of raising, explicit variable dependency representation, the liberalized -rule, and preservation of solutions.
Paul Bernays and David Hilbert carefully avoided overspecification of Hilbert's ε-operator and axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the ε-operator underspecified. In the meanwhile, there have been several suggestions for semantics of the ε as a choice operator. After reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as rightuniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the ε supports proof search optimally and is natural in the sense that it does not only mirror some cases of referential interpretation of indefinite articles in natural language, but may also contribute to philosophy of language. Finally, we ask the question whether our ε within our free-variable framework can serve as a paradigm useful in the specification and computation of semantics of discourses in natural language.
We present a combination of raising, explicit variable dependency representation, the liberalized δrule, and preservation of solutions for first-order deductive theorem proving. Our main motivation is to provide the foundation for our work on inductive theorem proving.
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