Abstract
A formal presentation of the ramified theory of types of the Principia Mathematica of Russell and Whitehead is given (along with the simplified theory of types of Ramsey). The treatment is inspired by but differs sharply from that in a recent paper of Kamareddine, Nederpelt and Laan. Algorithms for determining whether propositional functions are well-typed are described, including a complete algorithm for the ramified theory of types, which is unusual in requiring reasoning about numerical inequalities in the course of deduction of type judgments. Software implementing these algorithms has been developed by the author, and examples of the use of the software are presented. The approach is compared with that of Kamareddine, Nederpelt and Laan, and some brief observations are made about use of the type checker in a proof checker for the ramified theory of types under development.
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Bibliography
M. Crabbé. On the consistency of an impredicative subsystem of Quine’s NF. Journal of Symbolic Logic 47 (1982), pp. 131–136.
M. Randall Holmes. Subsystems of Quine’s New Foundations with Predicativity Restrictions, Notre Dame Journal of Formal Logic, vol. 40, no. 2 (spring 1999 ), pp. 183–196.
M. Randall Holmes. software files (in standard ML) rtt. sml (source for the type checker) and rttdemo. sml (demonstration file), accessible at http://math.boisestate.edu/~holmes/holmes/rttcover.html.
F. Kamareddine, T. Laan, and R. Nederpelt. Types in mathematics and logic before 1940, Bulletin of Symbolic Logic, vol. 8, no. 2, June 2002.
R. Milner. A theory of type polymorphism in programming, J. Comp. Sys. Sci., 17 (1978), pp. 348–375.
A. F. Peressini. Cumulative versus noncumulative ramified types, Notre Dame Journal of Formal Logic, vol. 38, no. 3, summer 1997.
A. N. Whitehead and B. Russell. Principia Mathematica (to *56), Cambridge University Press, 1967.
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Holmes, M.R. (2003). Polymorphic Type-Checking for the Ramified Theory of Types of Principia Mathematica . In: Kamareddine, F.D. (eds) Thirty Five Years of Automating Mathematics. Applied Logic Series, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0253-9_8
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DOI: https://doi.org/10.1007/978-94-017-0253-9_8
Publisher Name: Springer, Dordrecht
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