An extension of the Quantified Propositional Calculus1 obtained by the addition of two binary propositional functions is put forward as an inheritor of E. Schröder's “Algebra der Logik”. The formal system is itself not new, in fact it forms part of A. P. Morse's “A Theory of Sets”; although the latter is considered as a first-order system. Since the additional propositional functions are not invariant under the logical biconditional, this system–and many others naturally obtained from it–give us a collection of (...) examples of non-standard, but mathematically meaningful, propositional systems. (shrink)

A conservative extension of first order logic, suitable for circumscriptive inference, is introduced.

A reduction algebra is defined as a set with a collection of partial unary functions (called reduction operators). Motivated by the lambda calculus, the Church-Rosser property is defined for a reduction algebra and a characterization is given for those reduction algebras satisfying CRP and having a measure respecting the reductions. The characterization is used to give (with 20/20 hindsight) a more direct proof of the strong normalization theorem for the impredicative second order intuitionistic propositional calculus.