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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeMay 15th 2012

    Created ternary frame, a class of models for substructural logic which are basically obtained from Day convolution for promonoidal posets. Presumably people who know things know this, but in a few minutes of looking I haven’t found anyone mentioning it. Girard’s “phase space” semantics for linear logic (no relation to phase spaces of physics) are just the special case of regarding a monoidal category as a promonoidal one.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 19th 2017

    I cleaned up ternary frame a bit, including adding some references (none of which I’ve read myself yet!). I still don’t quite understand the “compatibility relation” from the promonoidal/Day perspective.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2018

    Added a note about how regarding PCAs as ternary frames almost recovers the notion of implication from realizability.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorSam Staton
    • CommentTimeMay 24th 2018

    That’s a nice point. I suppose you say “almost” because it amounts to xPQx \Vdash P\multimap Q if for all yy with yQy \Vdash Q, either (xy)Q(xy)\Vdash Q or (xy)(xy) undefined. So it’s a kind of partial realizability?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2018

    Yeah, that’s what I had in mind by “almost”. I was wondering whether it could be made to coincide exactly by some trick, like introducing a formal bottom element \bot and defining xy=x y = \bot if it used to be undefined. Then xPQx \Vdash P\multimap Q iff either xPQx \Vdash P \to Q in the realizability sense or Q\bot \Vdash Q

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