The Calculus of Higher-Level Rules, Propositional Quantification, and the Foundational Approach to Proof-Theoretic Harmony

Studia Logica 102 (6):1185-1216 (2014)

We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of introduction rules a canonical elimination rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework.
Keywords Proof-theoretic semantics  Assumptions  Higher-level rules  Propositional quantification  Harmony
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DOI 10.1007/s11225-014-9562-3
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References found in this work BETA

The Logical Basis of Metaphysics.Michael DUMMETT - 1991 - Harvard University Press.
General-Elimination Harmony and the Meaning of the Logical Constants.Stephen Read - 2010 - Journal of Philosophical Logic 39 (5):557-576.
A Note on Harmony.Nissim Francez & Roy Dyckhoff - 2012 - Journal of Philosophical Logic 41 (3):613-628.

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Citations of this work BETA

Characterizing Generics Are Material Inference Tickets: A Proof-Theoretic Analysis.Preston Stovall - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy:1-37.

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