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nLab > Latest Changes: Lindenbaum-Tarski and Boolean algebras.

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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeNov 3rd 2010
    • (edited Nov 3rd 2010)

    Just now I needed a definition and discussion of term algebra for the new entry on Lindenbaum-Tarski algebra. I noted we have Lindenbaum algebra in several places with no explanation. I am no logician and have very few logic books available. Are these the same and what generality should be used for the term algebra.

    I also looked at the entry on Boolean algebra and was a bit surprised to find there was no elementary algebraic version given. This is the (for dummies) version perhaps, but seeing one of the usual algebraic description and examples (although these are in the Wikipedia page I’m sure) might enable the ideas about Heyting algebras, lattices, etc., there to be more useful. I’m not sure what level to pitch any additions to that entry, any ideas or thoughts anyone?

    • CommentRowNumber2.
    • CommentAuthorFinnLawler
    • CommentTimeNov 4th 2010

    I’m pretty sure they’re the same thing — just the usual term model.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeNov 4th 2010

    As I understand it for a logic Λ\Lambda you identify terms in the underlying language if ’one iff t’other’ is in the logic, so there is some confusion. (I think L and L-T are probably names for the same thing.)

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 5th 2010

    I also looked at the entry on Boolean algebra and was a bit surprised to find there was no elementary algebraic version given.

    Done.

    Wikipedia leaves out the first two of my identities, but I don’t see how to prove them.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeNov 5th 2010

    Thanks. I did not want to suggest a notation or a viewpoint as the BAO entry is not that central to the modal logics yet.

    Mathematical Query: a Boolean with operators which is a closure algebra and a ’monadic’ algebra models S5. This is one of the epistemic logics and its frames are sets with an equivalence relation on them. The multimodal logic S5(n) has Kripke models based on a set with n-equivalence relations defined on it. Similarly the modal algebra has n-closure operators all of which are ’monadic’. A set with n-equivalence relations naturally suggests an n-fold groupoid structure.

    What is the algebraic dual of an n-fold groupoid?

    NB. The modal operators in the BAO preserve one of the structures but not both in general, so may be ’additive’ and th dual operation multiplicative. The usual duality theory at the set/BA level allows one to switch between the two forms of semantics. But what about replacing the equivalence relations with groupoids. David C. led a Café discussion on this area some time ago but I’m not sure that we got anywhere. Thoughts anyone?

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeNov 5th 2010

    I have added the construction of the canonical model to the entry on maximal consistent sets.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2013

    Our entry on Lindenbaum-Tarski algebra only discusses modal logics, not even mentioning the simpler cases of propositional logi that correspond to Heyting and Boolean algebras. Is this just because no one has written about them, or is there a deeper reason?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 22nd 2013

    I reckon it’s just because we were in a modal logic phase at the time.

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeJun 22nd 2013

    Yes, I wrote some of that and was trying to get some modal stuff down to give a slightly richer view of logic. Then David took up the thread, and it went on from there. We should have more there.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2013

    Okay, I added a brief comment at the top giving a broader scope. I don’t have time to do more at the moment.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2013
    • (edited Jun 23rd 2013)

    I find the statement of the Examples confusing. It says:

    The Lindenbaum–Tarski algebra of intuitionistic propositional logic is a Heyting algebra.

    The Lindenbaum–Tarski algebra of a classical propositional logic is a Boolean algebra.

    I would understand it either if it were

    The Lindenbaum–Tarski algebra of a theory in intuitionistic propositional logic is a Heyting algebra.

    or else

    The Lindenbaum–Tarski algebra of intuitionistic propositional logic is Heyting algebra.

    (wihtout the “a”).

    Which one is it?

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 23rd 2013
    • (edited Jun 23rd 2013)

    I would say “of a theory”. (The latter could then be interpreted as saying that the Lindenbaum-Tarski algebra of pure intuitionistic propositional logic (i.e., of a theory given by an empty collection of axioms) is a free Heyting algebra.)

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2013

    Fixed.

    • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeJun 25th 2013

    I fixed some formatting issues.

1 to 14 of 14

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