1.  70
    Stability and Paradox in Algorithmic Logic.Wayne Aitken & Jeffrey A. Barrett - 2006 - Journal of Philosophical Logic 36 (1):61-95.
    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. (...)
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  2.  53
    Computer Implication and the Curry Paradox.Wayne Aitken & Jeffrey A. Barrett - 2004 - Journal of Philosophical Logic 33 (6):631-637.
    There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation.
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  3.  36
    Abstraction in Algorithmic Logic.Wayne Aitken & Jeffrey A. Barrett - 2008 - Journal of Philosophical Logic 37 (1):23-43.
    We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this situation leaves (...)
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  4. On the Physical Possibility of Ordinal Computation (Draft).Jeffrey A. Barrett & Wayne Aitken - unknown
    α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [10]. Idealized computational models for α-recursion analogous to Turing machine models for classical recursion have been proposed and studied [4] and [5] and are applicable in computational approaches to the foundations of logic and mathematics [8]. They also provide a natural setting for modeling extensions of the algorithmic logic described in [1] and [2]. On such models, an α-Turing machine can complete a θ-step (...)
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