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nLab > Latest Changes: excluded middle

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMay 11th 2016

    I added to excluded middle a discussion of the constructive proof of double-negated LEM and how it is a sort of “continuation-passing” transform.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2016
    • (edited May 11th 2016)

    We have a stub entry continuation-passing style. I have now made “continuation-passing” redirect to it, so that one more of your links points somewhere.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 11th 2016

    Thanks!

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 12th 2016

    My coding skills are so bad. It would be good to have one example at least at continuation-passing style, but I can’t read even that first simple pyth program at the wikipedia page.

    • CommentRowNumber5.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMay 12th 2016
    • (edited May 12th 2016)

    The connection between the double-negation translation and the continuation-passing transform is indeed quite intriguing! Also note that, while there is only one transformation on propositions/types, there are actually a few variants of the transformation on proofs/terms, corresponding to the different kinds of the continuation-passing transform.

    David, the key idea is the following. Ordinarily, you call a subroutine/procedure/function and it returns to you some result:

    y = f(x).

    In contrast, if you employ continuation-passing style, then subroutines never return. Instead, when calling a subroutine, you pass an additional argument (a so-called continuation). This argument is itself a subroutine; it expects the result of the computation as its argument:

    f(x,(y↦⋯)). f(x, (y \mapsto \cdots)). f(x,λy.⋯). f(x, \lambda y. \cdots).

    The continuation-passing transform is a mechanical procedure which transforms a program written in direct style into the equivalent program written in continuation-passing style.

    I have some slides on this material; but they are directed at a general computer science audience. Therefore they explain intuitionistic logic, but not continuations.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2016

    These are nice slides of yours. I have now referenced them here.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2016

    Yes, great slides.

    I’m slowly getting there with continuation-passing. The naive factorial example here helped.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2019
    • (edited Jun 6th 2019)

    added publication data for

    • Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, 1996

    added also pointer to

    • {#Diaconescu75} Radu Diaconescu, Axiom of choice and complementation, Proceedings of the American Mathematical Society 51:176-178 (1975) (doi:10.1090/S0002-9939-1975-0373893-X)

    • {#GoodmanMyhill78} N. D. Goodman J. Myhill, Choice Implies Excluded Middle, Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik 24:461 (1978)

    And then I added pointer to a stand-alone entry Diaconescu-Goodman-Myhill theorem, which I am splitting off so that we may disambiguate from Diaconscu’s theorem

    diff, v27, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 17th 2020

  10. added section on sharp excluded middle

    Anonymous

    diff, v28, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2022

    This addition would deserve to be equipped with a comment on why or when one would consider such a definition.

    Related discussion of ♯\sharp-modal types forming a Boolean subtopos is at Aufhebung (in the subsection here).

  12. added a section on other equivalent statements to the principle of excluded middle

    Quentin Adamson

    diff, v30, current

  13. adding the paper to the references section:

    Quentin Adamson

    diff, v30, current

  14. also added a section on excluded middle in material set theory explaining Shulman’s distinction between excluded middle for all logical formulae and excluded middle only for Δ 0\Delta_0-formulae.

    Quentin Adamson

    diff, v30, current

    • CommentRowNumber15.
    • CommentAuthorGuest
    • CommentTimeMar 26th 2023

    added a section about excluded middle in type theory

    diff, v31, current

1 to 15 of 15

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