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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJul 26th 2016

    I’m trying to think of a good name for the following categorical structure, which is a direct categorical representation of the type-theoretic structure of first-order logic.

    Start with a cartesian multicategory SS. Now consider what structure is possessed by the contravariant “representable functors” Hom S(,B)Hom_S(-,B): there is one such set Hom S(A 1,,A n;B)Hom_S(A_1,\dots,A_n;B) for each list of objects of SS, and they are acted on by the morphisms in SS. Together all this structure forms something that one might call a “contravariant functor” from SS to SetSet, and writing it all down diagrammatically yields a notion of “contravariant functor” from SS to any category CC. (Equivalently, it’s an ordinary contravariant functor to CC from the category with products freely generated by SS.) Now the structure I’m interested in is such a contravariant functor PP from SS to the category of cartesian multicategories (or perhaps cartesian multiposets, or perhaps noncartesian ones; all of those could be indicated with adjectives on a basic notion).

    This represents the “judgmental structure” of first-order logic: the objects of SS are the types or sorts, its morphisms are the terms, the objects of P(A 1,,A n)P(A_1,\dots,A_n) are the propositions in context A 1,,A nA_1,\dots,A_n, and the morphisms of P(A 1,,A n)P(A_1,\dots,A_n) are the entailments ϕ 1,,ϕ nψ\phi_1,\dots,\phi_n \vdash \psi in that context. All the logical connectives and quantifiers can be characterized (independently) with universal properties inside such a structure.

    The extant definition that this is closest to is probably a first-order hyperdoctrine; the difference is that I want the unadorned word not to assume objects with any of these universal properties exist, including products in the base multicategory. So one possibility would be something like “multi-hyperdoctrine” or “virtual hyperdoctrine”; but aside from their intrinsic unloveliness, I’ve never really liked the word “hyperdoctrine” anyway (I haven’t been able to see any sense in which it is a “hyper” version of anything called a “doctrine”), and moreover it doesn’t convey in any way the connection to logic.

    Any suggestions for a good name? Are there any good words we can borrow from philosophy? Or portmaneau words we can coin?

    (I’m actually surprised that no one has written down something like this before. But in general, cartesian multicategories seem to be have been shockingly under-appreciated, so maybe it’s not very surprising. Of course, if anyone has written such a thing down, I would love to see it.)

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 26th 2016

    Would you ask on the categories mailing list? Or is that a can of worms?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2016

    Perhaps Peirce is your man. He coined scores of new terms and, as you know, worked diagrammatically in this area.

    Is this interest of yours part of that ’product type as positive’ idea?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 26th 2016

    I’m not really inclined to ask on the categories list; not yet anyway. (-:

    Yes, this is part of the general idea that multicategorical structures should represent the judgmental structure of a type theory. Do you know offhand which papers of Peirce I should be looking at?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2016

    Peirce’s work is notoriously scattered over manuscripts. There is a commented manuscript on his graphs available here, where you find such gems as ’endoporeutic’. His theory of signs had him coin: rheme, dicent, delome, qualisign, sinsign, legisign,…

    But maybe it wouldn’t be so easy to find something to apply to the whole structure as you’d like. Perhaps Todd would know.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 26th 2016

    Hmm… yes, a lot of his neologisms seem to refer to aspects of individual signs rather than to an entire “system of signs”. But seeing the word semiosis for “the process of signification” makes me ponder coining a word like semios for “a universe of things that are signified”. It even ends with “-os” like topos, logos, and tripos! (-:

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2016
    • (edited Jul 26th 2016)

    But those three are proper Greek words. Time to end the tyranny of masculine words, what about the neuter

    σημεῖον

    or feminine

    σῆμα ?

    • CommentRowNumber8.
    • CommentAuthorwellsoberlin
    • CommentTimeJul 26th 2016
    What about "foltt" == First Order Logic Type Theory. That makes it analogous to "HoTT".
    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2016

    Just testing. Of course ’tripos’ is not a Greek word, but a corruption of the Latin tripus. Ah, I hadn’t realised the origin of tripos in the category theoretic sense.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2016

    I actually think it’s better if it isn’t a real Greek word (at least, not a modern Greek word), since then it’s less likely to collide with other mathematical terms when translated into Greek.

    Would your suggestions in #7 transliterate into “semeion” and “sema”?

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2016

    How about combining “sem” with another root indicating a world or universe, like “semacosm”?

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 27th 2016
    • (edited Jul 27th 2016)

    Re #10, yes. But we tend to turn ’ei’ into just ’i’ when forming new words, such as dinosaur from deinos.

    Semata is the plural of sema. ’Sematography’ is a word.

    Or from here

    semeiography, semeiograph, semeiographic, semiography, semiograph

    1. A description of the signs of disease or a written system of symbolic notations.

    2. The doctrine of signs; as in, pathology, a description of the marks or symptoms of diseases.

    Back to ’doctrine’.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2016

    “Sema” appears to be a real Greek word meaning sign, so I think we should avoid that. “Semion” isn’t too bad, I suppose.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJul 27th 2016

    Although “semion” does kind of sound like half of a subatomic particle… (-:

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 27th 2016
    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 28th 2016

    I couldn’t put my finger on it, but I realised what it made me think of: sémillon grapes, which are one of the varieties grown in the regions near Adelaide.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2016

    Turning these words over in my head, I really don’t like that “semios” and “semion” appear to involve the root “semi-” (which is much more common in English) rather than the intended “sema-“. Maybe we can coin a similar word that actually starts with “sema-“, like “semantios” or “semantos” or “semantion” or “semanton”, to hopefully evoke the intended association with “semantics” in an English reader.

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 29th 2016

    Semanton sounds like a reasonable word, if a little like a fundamental particle. What sort of derivative words do you anticipate?

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJul 29th 2016

    “The fundamental particle of meaning”… hmm…

    Derivative words is a good question. I suppose one might want to make it into an adjective. “Semantonic”? “Semantonal”?

    • CommentRowNumber20.
    • CommentAuthorTim_Porter
    • CommentTimeJul 29th 2016

    Semantons should provide a link with Infons (as Keith Devlin talked about some years ago) which were ‘fundamental’packets of information’ in situation theory!!!! (I am sort of joking but you never know!)

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeJul 30th 2016

    I actually don’t like the “fundamental particle”connotation of “-on”, since the structure in question is not really a “fundamental particle” in any way; it’s more like a whole universe!