Consistency notions in illative combinatory logic

Abstract

In this paper we show that certain notions of consistency currently in use in illative combinatory logic are not always necessary nor always sufficient for the acceptability of the system.The three notions of consistency we will consider are:(A) Not all terms of the system are assertible.(B) Not all propositions of the system are assertible.(C) The system is Q-consistent, i.e. if Q (for equality) is in or is added to the system as a primitive with axioms such asPost defined a system of propositional calculus to be inconsistent if all propositions were assertible. In a more general system where there may be terms which are not propositions (although all propositions are terms), we need to distinguish notions (A) and (B) as two possible consistency notions “in the sense of Post”.In [3] Curry and Feys assume that Q is in the system defined as λxλy.Ξ(C*x)(C*y) or as a primitive, but Seldin in [5] also talks about Q-consistency of a system (F22) to which Q has to be added. We therefore consider the definition of Q-consistency that includes all three possibilities.In [3] Curry and Feys state that Q-consistency is essential for acceptability. This is because there the purpose was to have an assertional system of illative combinatory logic in which combinatory equality was represented by the assertion of QXY. However if Q is to represent some other form of equality, Q-consistency may become inessential for acceptability or even incompatible with it.