[Agda] Re: Agda Digest, Vol 87, Issue 24

Dan Licata drl at cs.cmu.edu
Sun Dec 16 16:46:52 CET 2012

Hi Thorsten and all,

(sorry for the delay, missed this message)

To my mind, the most compelling reasons not to identify "proposition"
with "hprop" (proof-irrelevant proposition) are:

1) We want "ordinary" mathematicians to start using homotopy type
   theory, and to start thinking in a proof-relevant way.  To accomplish
   this goal, the (somewhat devious) strategy is to rebind the words
   this mathematician is familiar with to the type-theoretic concepts, 
   so that they default to the type-theoretic logic.  
   To someone who already understands props-as-types, and thinks of
   types as logic, it makes some sense to use "proposition", in contrast
   with "type", to mean a more traditional proof-irrelevant logic.  
   But if our hypothetical ordinary mathematician doesn't already
   understand the props-as-types view of logic, and sees something
   called a "proposition", they will probably use that to state and
   prove a theorem.  So, it's the wrong default.   

2) I don't remember whether Mike talks about this in his post, but 
   there are lots of "modalities" that come up, of which hprop is just
   one example, and you get a notion of logic for each.  So it seems
   wrong to have "proposition" default to hprop-logic.  


On Nov26, Altenkirch Thorsten wrote:
> Hi,
> I agree with most of what Mike is saying but I don't see the problem with
> the word "proposition". Thinking propositionally as opposed to
> type-theoretically is less expressive because we can express any statement
> about the choice of proofs. In particular the strong statement 2. he makes
> (maybe with the added proviso that the conclusion is propositional too) I
> would call the "propositional axiom of choice" which opposed to the
> "type-theoretic axiom of choice" (1) is not provable (and indeed entails
> excluded middle). And we do have already the different terminologies: e.g.
> Pi-types vs forall and more importantly Sigma-types vs. exists and
> disjoint union vs. or.
> Indeed, the use of adverbs may be a good idea as an alternative. So I
> could say "there exists, type-theoretically" (meaning Sigma) as opposed to
> "there exists, propositionally" (meaning "exists", I.e. the bracket of
> Sigma). 
> Btw, there are other examples where the limitation on propositions is
> inapproriate. I.e. there are strictly positive propositional formulas
> where the existence of an inductive or coinductive closure is not
> acceptable intuitionistically.
> Thorsten
> On 23/11/2012 13:54, "Dan Licata" <drl at cs.cmu.edu> wrote:
> >On Nov21, Martin Escardo wrote:
> >> In HoTT, and even before HoTT, people define a "proposition" not to be
> >>a 
> >> type, but instead a type with at most one element (suitably
> >>formulated). 
> >> One also considers the "propositional reflection" of any type (similar
> >>to 
> >> so-called bracket types), which hides (some part of the) computational
> >> content. Denote the propositional reflection of a type X by X* or [X].
> >> Intuitively, X* is empty if X is empty, and X* has just one element if
> >>X* 
> >> is inhabited (the trick is how to say this without using excluded
> >>middle).
> >
> >Just a HoTT-related terminological note: There's been some pushback on
> >using the word "proposition" exclusively to mean "hprop" (where a type A
> >is an hprop iff there is a term of type (x y : A) -> Id x y), which is
> >what Martin is doing above.  Mike Shulman has a nice summary here:
> >http://golem.ph.utexas.edu/category/2012/11/freedom_from_logic.html
> >
> >The reason is that both the proof-relevant interpretation of logic
> >(forall=Pi/exists=Sigma) and the hprop interpretation of logic (using
> >bracket types for exists) are useful, and it's linguistically helpful to
> >use logical terminology for both.  And while it's very important to be
> >clear about which of the two you mean in a given context, I think it's a
> >bad idea to accomploish this by using logical terminology only for the
> >hprop interpretation.  I think Mike's suggestion of adverbs is much
> >better.
> >
> >-Dan
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> >Agda at lists.chalmers.se
> >https://lists.chalmers.se/mailman/listinfo/agda
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