Normalization and excluded middle. I

Natural Deduction and the Isabelle Proof Assistant

“…The first 9 are rules for classical propositional logic and the last 4 are for first-order logic. Intuitionistic logic can be obtained by omitting the rule PBC (proof by contradiction, called "Boole" later) and adding the ⊥-elimination rule (also known as the rule of explosion) [18]. The rules are as follows:…”

A New Normalization Strategy for the Implicational Fragment of Classical Propositional Logic

“…Normal forms in the sense of Seldin have at most one application of Peirce's rule and if this application really occurs, it's the last rule applied in the derivation. The corresponding normalization procedure used by Seldin in [5] depends essentially on the presence of conjunction in the system. The ⊃K⊃Kreductions are responsible for combining two successive applications of Peirce's rule into a single application of Peirce's rule (see [5, p. 199]).…”

“…Reductions →-RI, →-RE m and →-RE M used in this paper can be considered as particular cases of Seldin's ⊃-KR i reductions (see [5, p. 198]); but they are the good particular cases, since they do not duplicate applications of Peirce's rule that already exists in a given derivation. The conceptual apparatus introduced by Seldin in [5] (quasi-chain, chain, regular and irregular chains, connected chains and maximum connected sets) is meant to take care of these duplications produced by ⊃-KR i reductions. Reduction →-RE Mm has no counterpart in [5].…”

“…The conceptual apparatus introduced by Seldin in [5] (quasi-chain, chain, regular and irregular chains, connected chains and maximum connected sets) is meant to take care of these duplications produced by ⊃-KR i reductions. Reduction →-RE Mm has no counterpart in [5]. It is this reduction that takes care of the cases where ⊃-KR i reductions produce bad duplications.…”

Perspectives on Interrogative Models of Inquiry

“…More precisely, thanks to a result due to Seldin (1989), it is possible to show that if A is a theorem of classical logic, then there exists a proof of it using only one occurrence of the CM rule, namely the last one. More precisely, thanks to a result due to Seldin (1989), it is possible to show that if A is a theorem of classical logic, then there exists a proof of it using only one occurrence of the CM rule, namely the last one.…”