Normal deductions (original) (raw)

Oxford University Press eBooks, 2021

Abstract

The concept of a normal deduction is introduced as a way to capture the notion of a “direct” proof, i.e., one that makes no detours. The latter happens, e.g., when a formula is introduced first by a rule of introduction (say, starting from A and B to introduce the conjunction A ∧ B) and is followed immediately by an elimination of the same connective (yielding, for instance, A). A normalization theorem proves that if one has a proof with detours then there is a proof without detours. Proving such more advanced results in proof theory requires more complex proof methods than simple induction. A first step to more complex proofs is the use of double induction. A first application of double induction is the proof of normalization for a fragment of minimal logic containing conjunction, implication, negation, and the universal quantifier. The result is extended to full intuitionistic logic, followed by a discussion of the structure of normal deductions. Normal deductions have the sub-formula property, and so cannot be proofs of contradictions. This shows that natural deduction is consistent. Normalization can also be used to show that certain formulas cannot be proved, e.g., that the principle of excluded middle cannot be proved in intuitionistic logic. Finally, a full proof of normalization for full classical logic is given.

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