Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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 set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 8th 2017
   • (edited Nov 8th 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexReturning to the [Barcan formulas](https://nforum.ncatlab.org/discussion/4787/homotopy-barcan/), where what is at stake is a comparison between $\bigcirc \exists x P(x)$ and $\exists \bigcirc P(x)$. Given our [set up](https://ncatlab.org/nlab/show/necessity+and+possibility#InFirstOrderLogicAndTypeTheory), strictly these are ill-formed. The monad $\bigcirc_W$ sends $W$-dependent types to the same. A quantifier such as $\exists_{x:A}$ change context, sending $A$-dependent types to plain types. Two thoughts then. (1) Say I have $w: W, x: A(w) \vdash \phi(w, x): Prop$. Then I can form $\bigcirc_W \sum_{x: A(w)} \phi(w, x)$, the world-dependent (constant) type containing all possible $A$s which are $\phi$. Evaluated at a specific world, this amounts to $\sum_{w:W}\sum_{x: A(w)} \phi(w, x)$. Then I can also form $\sum_{(w, x):\sum_{w:W}A(w)}\phi(w, x)$. A version of the Barcan formulas then amounts to the equivalence of taking dependent sum in one or two stages. $$ \sum_{w:W}\sum_{x: A(w)} \phi(w, x) \simeq \sum_{(w, x):\sum_{w:W}A(w)}\phi(w, x) $$ It's just the rebracketing of the three-part terms in pairs: $(w, (a, p)) \leftrightarrow ((w, a), p)$. We might say: > Possibly there's an $A$ which is $\phi$ and there is a possible $A$ which is $\phi$. (2) Alternatively, we are imagining some trans-world identity of $A$s, perhaps by the pulling back to the constant world-dependent type $W^{\ast} A$. Then what should I say strictly? If I have $w: W, x: A$ as my context, I guess strictly I shouldn't rely on this being the 'same' as $x: A, w: W$, and so interchanging $\bigcirc_W$ and $\exists_{x:A}$ is not allowed. Maybe it just amounts to the previous solution but with $W^{\ast} A(w)$ instead of $A(w)$. If you find the metaphysics of possible worlds unintuitive, turning to temporal logic and the Future operator may help. Some people would want to deny the assertion "in the future there will be a ruler of the world" entails "there is something (now) which in the future will be ruler of the world". My solution (1) is about reworking the latter claim into "there is a future person who will be ruler of the world", and is presumably unexceptionable as the rebracketing account would suggest.

   Returning to the Barcan formulas, where what is at stake is a comparison between $\bigcirc \exists x P(x)$ and $\exists \bigcirc P(x)$. Given our set up, strictly these are ill-formed. The monad $\bigcirc_W$ sends $W$-dependent types to the same. A quantifier such as $\exists_{x:A}$ change context, sending $A$-dependent types to plain types.

   Two thoughts then.

   (1) Say I have $w: W, x: A(w) \vdash \phi(w, x): Prop$. Then I can form $\bigcirc_W \sum_{x: A(w)} \phi(w, x)$, the world-dependent (constant) type containing all possible $A$s which are $\phi$. Evaluated at a specific world, this amounts to $\sum_{w:W}\sum_{x: A(w)} \phi(w, x)$. Then I can also form $\sum_{(w, x):\sum_{w:W}A(w)}\phi(w, x)$.

   A version of the Barcan formulas then amounts to the equivalence of taking dependent sum in one or two stages.

   $$\sum_{w:W}\sum_{x: A(w)} \phi(w, x) \simeq \sum_{(w, x):\sum_{w:W}A(w)}\phi(w, x)$$

   It’s just the rebracketing of the three-part terms in pairs: $(w, (a, p)) \leftrightarrow ((w, a), p)$. We might say:

   Possibly there’s an $A$ which is $\phi$ and there is a possible $A$ which is $\phi$.

   (2) Alternatively, we are imagining some trans-world identity of $A$s, perhaps by the pulling back to the constant world-dependent type $W^{\ast} A$. Then what should I say strictly?

   If I have $w: W, x: A$ as my context, I guess strictly I shouldn’t rely on this being the ’same’ as $x: A, w: W$, and so interchanging $\bigcirc_W$ and $\exists_{x:A}$ is not allowed.

   Maybe it just amounts to the previous solution but with $W^{\ast} A(w)$ instead of $A(w)$.

   If you find the metaphysics of possible worlds unintuitive, turning to temporal logic and the Future operator may help. Some people would want to deny the assertion “in the future there will be a ruler of the world” entails “there is something (now) which in the future will be ruler of the world”. My solution (1) is about reworking the latter claim into “there is a future person who will be ruler of the world”, and is presumably unexceptionable as the rebracketing account would suggest.