Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal-logic model model-category-theory monads monoid monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 31st 2020

    Starting something on Mike’s ideas here, as I wanted to refer to it.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 31st 2020

    Do we have anything on why “classical simple type (2-)theories” due to their multiple consequents deserve to be called “classical”, other than at sequent calculus

    Any of these logics may be presented using any sort of sequent, but Gentzen’s original sequent calculi presented each logic using only corresponding sequents.

    Is it down to the kind of issue covered here?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 1st 2020

    If ’classical’ can be added to ’simple type 2-theory’ (allowing multiple consequents) what to say about the next 3-theory, namely, the ’first-order logic’ 3-theory? Could there be a ’classical’ version of that?

    Isn’t this going to get messy? Ordinary classical first-order logic may be specified as a particular 2-theory in the ’first-order logic’ 3-theory.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 2nd 2020
    • (edited Sep 2nd 2020)

    Ah, here is Mike speaking to the issue in #3:

    I think of a 3-theory as specifying the arity and dependency structure. The 3-theory of “first order logic” has a base whose arity is many-to-one with no internal dependency and then another layer that depends on that. (The upper layer could be many-to-one or many-to-many, so actually there are two different 3-theories for first-order logic, just as there are two different 3-theories for propositional logic / simple type theory. The many-to-one and many-to-many versions are traditionally called “intuitionistic” and “classical”, though that’s confusing too.)

    Very confusing.

    So the dependency of the 3-theory of first-order logic can be mixed with the ’classical’ arity of multiple consequents, but we don’t know how to do this with the full dependency of the 3-theory of dependent type theory?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 6th 2020

    Added an explanation of ’classical’

    The term ’classical’ owes its origin to the contrast between systems LK and LJ of Gentzen’s sequent calculus, where LJ provides a proof system for intuitionistic logic via restriction to single sequents of the rules of LK for classical logic.

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 15th 2020

    Added comments on higher-order logic and modal logic.

    diff, v8, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)