Completeness proof of functional logic, a formalism with variable-binding nonlogical symbols

Abstract

We know extensions of first order logic by quantifiers of the kind "there are uncountable many ...", "most ..." with new axioms and appropriate semantics. Related are operations such as "set of x, such that ...", Hilbert's $ε$-operator, Churche's $λ$-notation, minimization and similar ones, which also bind a variable within some expression, the meaning of which is however partly defined by a translation into the language of first order logic. In this paper a generalization is presented that comprises arbitrary variable-binding symbols as non-logical operations. The axiomatic extension is determined by new equality-axioms; models assign functionals to variable-binding symbols. The completeness of this system of the so called "Functional Logic of 1st Order" will be proved.