A cut-elimination proof in intuitionistic predicate logic

Gentzen’s “Cut Rule” and Quantum Measurement in Terms of Hilbert Arithmetic. Metaphor and Understanding Modeled Formally

“…6 The same two stages can be illustrated also by a relevant proof in intuitionistic predicate logic (e.g. as in: Borisavljević 1999) since that logic postulates a gap between finiteness and infinity due to the suspension of the "excluded middle" as to infinite sets. 7 As to the relevant counterpart for the arithmetization of the Gödel-like incompleteness in paradoxes of set theory, the paper of Dean (2019) suggests a detailed outlook, and Wiegand (2000), a generalized phenomenological-semantic viewpoint.…”

Gentzen’s ‘Cut Rule’ and Quantum Measurement in Terms of Hilbert Arithmetic. Metaphor and Understanding Modeled Formally

“…The same two stages can be illustrated also by a relevant proof in intuitionistic predicate logic (e.g. as in:Borisavljević 1999) since that logic postulates a gap between finiteness and infinity due to the suspension of the "excluded middle" as to infinite sets.…”

On permuting cut with contraction

“…To demonstrate that the figure of ( * * ) is somehow simpler, Szabo introduced in Szabo (1978, pages 242-243) a measure of complexity counting the number of contractions above a cut. However, Szabo's measure fails to show that the lower cut in ( * * ) will have a smaller measure of complexity, as can be seen in a counterexample presented in detail in the third section of Borisavljević (1999) † .…”

“…The goal of this paper is to present a cut-elimination procedure for intuitionistic propositional logic, in which cut is directly eliminated, without passing via mix, and in which pushing cut above contraction, that is, passing from ( * ) to ( * * ), is a reduction step. The cut-elimination procedure of Borisavljević (1999) also eliminates cut directly, and it involves pushing cut above contraction, but it is different and more entangled than the procedure we are going to present here. In a procedure envisaged by Carbone (1997), reminiscent of Curry's mix-elimination procedure (see Curry (1963, Chapter 5, D2)), cut should be directly eliminated, but without pushing it above contraction.…”

“…To describe this procedure, we introduce the following auxiliary implicationless sequent system called G z . (The index z stands for 'Zucker', from whose indexing of sequents in Zucker (1974), the indices of G z are derived; a measure analogous to these indices may be found in Borisavljević (1999). )…”