Predicate calculus and naive set theory in pure combinatory logic

In [-1] Goodman introduces a weak version of weak combinatory logic with equality and within it defines a truth functional implication (3 k), conjunction (~k), and disjunction (wk) between functions of k variables. He also defines (in a metasystem) the notion of a decidable predicate of k variables and then proves interesting introduction and elimination rules for his connectives where all the variables are restricted to being "defined" (equal to themselves) and some to being decidable predicates of k variables. He then briefly applies this to basing a propositional calculus by considering ~1, c~1 and w I as "three valued truth functions". In this paper we take a "decidable predicate of 0 variables" to be a proposition and show that 3o, no, and u o are more appropriate logical connectives which range over propositions as well as, in a restricted way, over defined terms. If all variables are taken to range over propositions the system obtained is classical. Then we show that within this framework universal and existential quantification can be defined but not quite all the usual properties can be proved. An extra axiom ((~)) which extends the system to a weak form of strong combinatory logic with equality (and still satisfies Goodman 's interpretation of his system) allows us to develop the full predicate calculus. Just as the notion of proposition (or decidable predicate of 0 variables) is defined metatheoretically and hence theorems such as: If a is a proposition then [a v Fa and : If a is a proposition then Fa is a proposition are metatheorems, we can define the notion of set metatheoretically as a decidable predicate of one variable and also prove theorems of set theory in the metatheory. ~1 and ~1 become set theoretic union and intersection and other set theoretic operators can be defined so that we have metatheorems giving us as sets not only the union and intersection of sets, but also the complement, power set and union set of a set. Similarly the Axioms of Infinity and Replacement can be shown to hold.