Generalized definitional reflection and the inversion principle

Logica Universalis 1 (2):355-376 (2007)
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Abstract

. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a principle for reasoning on the basis of rule-based atomic definitions, proposed by Hallnäs and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω- principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself.

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Citations of this work

Proof-Theoretic Semantics.Peter Schroeder-Heister - forthcoming - Stanford Encyclopedia of Philosophy.
Advances in Proof-Theoretic Semantics.Peter Schroeder-Heister & Thomas Piecha (eds.) - 2015 - Cham, Switzerland: Springer Verlag.
Speech Acts, Categoricity, and the Meanings of Logical Connectives.Ole Thomassen Hjortland - 2014 - Notre Dame Journal of Formal Logic 55 (4):445-467.
General-Elimination Stability.Bruno Jacinto & Stephen Read - 2017 - Studia Logica 105 (2):361-405.
On Inversion Principles.Enrico Moriconi & Laura Tesconi - 2008 - History and Philosophy of Logic 29 (2):103-113.

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References found in this work

A natural extension of natural deduction.Peter Schroeder-Heister - 1984 - Journal of Symbolic Logic 49 (4):1284-1300.
Meaning Approached Via Proofs.Dag Prawitz - 2006 - Synthese 148 (3):507-524.
On the idea of a general proof theory.Dag Prawitz - 1974 - Synthese 27 (1-2):63 - 77.
Foundations of Logic Programming.J. W. Lloyd - 1987 - Journal of Symbolic Logic 52 (1):288-289.

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