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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

  16. Added “the ring of integers is a principal ideal domain” to “Other equivalent statements” section.

    Anonymouse

    diff, v49, current

  17. Added “Given a prime number pp, the finite field ℤ/pℤ\mathbb{Z}/p\mathbb{Z} is a principal ideal domain” to “Other equivalent statements” section.

    Anonymouse

    diff, v50, current

  18. There are multiple definitions of a principal ideal domain, and the fact about integers and discrete fields being principal ideal domains is really about the equivalence of two definition of principal ideal domain:

    • as an integral domain whose ideals are principal

    • as a Bézout unique factorization domain

    It is the equivalence of these two definitions of principal ideal domain that is equivalent to excluded middle.

    Anonymouse

    diff, v51, current

  19. adding section on other versions of excluded middle using weaker versions of disjunction in intuitionistic logic.

    Anonymouse

    diff, v52, current

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