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Formal Systems, Logics, and Programs

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Raymond Smullyan on Self Reference

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 14))

Abstract

This article connects certain writings of Raymond Smullyan on logic and formal systems with current research on constructive type theory. It also considers aspects of teaching logic to computer science students. Those students require logic in order to precisely define programming tasks and demonstrate that programs meet logical specifications of these tasks. Smullyan’s book First-Order Logic has been used for many years to teach logic to computer science students at Cornell. This article includes a brief account of an elegant result from this book. It is based on an extension of a fourteen page technical report that analyzes two pages on Boolean valuations in chapter one of his classic book. The analysis and results are formulated in a formal system of type theory and proved using the Nuprl proof assistant. The article also briefly considers the role of partial types, showing how they provide a new way to prove unsolvability results, another topic on which Smullyan has written with remarkable clarity. Partial types also provide a simple and faithful semantics for Hoare’s partial correctness logic, widely used to reason about programs and taught in computer science as an important applied logic. They also make it possible to state a clean rule for defined functions in Hoare logic.

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Notes

  1. 1.

    I had also assisted my uncle who was a magician, so I was fascinated by this connection to Smullyan already as an undergraduate.

  2. 2.

    In the extreme, some articles claim that eventually “intelligent” machines will use these tools to create mathematics that is beyond the abilities of unaided humans Aron (2015).

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Authors and Affiliations

  1. Cornell University, Ithaca, USA

    Robert L. Constable

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  1. Robert L. Constable
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Correspondence to Robert L. Constable .

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Editors and Affiliations

  1. City University of New York (CUNY) System, City University of New York, New York City, New York, USA

    Melvin Fitting

  2. NYC Department of Education, Brooklyn, New York, USA

    Brian Rayman

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Constable, R.L. (2017). Formal Systems, Logics, and Programs. In: Fitting, M., Rayman, B. (eds) Raymond Smullyan on Self Reference. Outstanding Contributions to Logic, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-68732-2_2

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