Stability and Paradox in Algorithmic Logic

Type-free systems of logic are designed to consistently handle significant instances of self-reference. Some consistent type-free systems also have the feature of allowing the sort of general abstraction or comprehension principle that infamously leads to paradox in naive set theory. Because type-free systems possess these features, and avoid the hierarchy of types that is felt to be unnatural in some contexts, they have the potential to play an important role in the foundations of mathematics, the theory of classes (producing a richer notion of class than that currently used in set theory and category theory), property theory, natural language semantics, the theory of truth, and theoretical computer science. Clearly, type-free systems must depart from classical logic in some way, but there is little agreement on what kind of type-free system to use, and which departures from classical logic should be allowed. Our approach to type-free logic is to study a naturally occurring type-free system that we believe is in some sense prototypical of systems that will ultimately prove useful. The logic studied in this paper, called algorithmic logic, concerns certain basic statements involving algorithms and the algorithmic rules of inference between such statements. This paper studies the propositional properties of algorithmic logic. A future paper will show that algorithmic logic possesses a general abstraction principle.