Neighbourhood semantics and labelled calculus for intuitionistic infinitary logic

Neighbourhood semantics for intuitionistic logic extended with countable conjunctions and disjunctions is introduced and shown equivalent to topological semantics, with an indirect completeness proof as payoff. The new semantics is used to obtain a labelled sequent calculus with good structural properties. In particular, admissibility of weakening and contraction, invertibility with preservation of height for each rule and cut elimination are shown. Finally, a direct Tait-Schutte-Takeuti style form of completeness via the extraction of a countermodel from a failed proof search is proved.

Automated summary

Neighbourhood semantics for intuitionistic logic extended with countable conjunctions and disjunctions is introduced and shown equivalent to topological semantics, with an indirect completeness proof as payoff. The new semantics is used to obtain a labelled sequent calculus with good structural properties, demonstrating admissibility of weakening and contraction, invertibility with preservation of height for each rule, and cut elimination. Finally, a direct Tait-Schutte-Takeuti style form of completeness via the extraction of a countermodel from a failed proof search is proved.