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. As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally.
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References
Aitken, W. and Barrett J. A. (2004): Computer implication and the curry paradox, J. Philos. Logic 33, 631–637.
Aitken, W. and Barrett, J. A.: Abstraction in Algorithmic Logic (preprint).
Cantini, A. (1996): Logical frameworks for truth and abstraction: An axiomatic study, North-Holland. ISBN: 0-444-82306-9.
Cantini, A. (2003): The undecidability of Gri\(\breve{{\text{s}}}\)in's set theory, Stud. Log. 74(3), 345–368.
Feferman, S. (1984): Toward useful type-free theories, J. Symb. Log. 49, 75–111.
Field, H. (2004): The consistency of the naïve theory of properties, Philos. Q. 54(214), 78–104.
Fitch, F. B. (1969): A method for avoiding the Curry paradox, in N. Rescher (ed.), Essays in Honor of Carl G. Hempel, pp. 255–265.
Girard, J.-Y. (1998): Light linear logic, Inform. Comput. 143(2), 175–204.
Link, G. (ed.) (2004): One hundred years of Russell's paradox: Mathematics, logic, philosophy, de Gruyter. ISBN 3-11-017438-3.
Myhill, J. (1984) Paradoxes, Synthese 60, 129–143.
Orilia, F. (2000): Property theory and the revision theory of definitions, J. Symb. Log. 65(1), 212–246.
Petersen, U. (2000): Logic without contraction as based on inclusion and unrestricted abstraction, Stud. Log. 64(3), 365–403.
Restall, G. (1994): On logics without contraction, doctoral dissertation, University of Queensland.
Terui, K. (2004) Light affine set theory: A naive set theory of polynomial time, Stud. Log. 77(1), 9–40.
Weir, A. (1998): Naïve set theory, paraconsistency and indeterminacy. I., Log. Anal. (N.S.) 41(161–163), 219–266.
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Aitken, W., Barrett, J.A. Stability and Paradox in Algorithmic Logic. J Philos Logic 36, 61–95 (2007). https://doi.org/10.1007/s10992-005-9024-5
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DOI: https://doi.org/10.1007/s10992-005-9024-5