California State University, San Marcos

**Abstract:**

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural

language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. The threat of paradoxes, especially the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. Our work has led to the creation of rich library of inference rules that are unproblematic in algorithmic logic and have a property we call stability. We have also identified a number of traditional rules of logic that are problematic in algorithmic logic, and so should be viewed with suspicion in type-free logic. We conjecture that by freely using stable libraries, limiting the scope of the problematic rules, and using multiple libraries, we can create a type-free logic that is viable for a wide range of applications.

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