On 2015-08-09 05:10, Fred E.J. Linton wrote:
Not wishing to broadcast my illiteracy in the matter ... So I ask you now, in public, where my shame can be greatest: what do you mean by "lativity"?
Thank you, Fred, for your questions. We were actually nervously waiting for somebody to ask that question, so we will remember you always for having done that. In logic we typically have signatures, terms, sentences, structured sets of sentences, entailment, models, satisfactions, axioms, theories and proof calculi. We cannot e.g. define entailment before we have a notion of sentences, and we should not define sentences before we have a notion of terms. The latter is a bit more controversial. In first-order logic I would see P(x), where P is a "predicate symbol", as a term, and not as a sentence, whereas putting a quantifier in front of it, Ex.P(x), makes it no longer a term. This is why I have difficulties e.g. to accept that the two 'not's in expressions like "not Ex.P(x)" and "Ex.not P(x)" would be the same. I am starting to think they are only informal as symbols, a bit similar as Church said lambda is and informal symbol, so actually not part of the formal syntax. Am I wrong or am I wrong? In logic we indeed need a signature (sorts and operators) in order to construct the categorical object of terms. Construction is important. We need terms to categorically construct sentences, which appear because of a sentence functor not being extendable to a monad. Otherwise sentences are terms, aren't they, because then we could substitute sentences within sentences, and that does not comply with our traditional view of sentences. Traditional first-order pretty much doesn't care, and neither did Aristotle about these things. Aristotle's and first-order logic are therefore very "illative" and also very unsorted, I would like to add. You are not broadcasting illiteracy at all, and your shame couldn't even be small because no shame whatsoever is justified to exist, at least not on your side. There may, however, be some of it now or eventually on my side, but let us see what happens if/when/how this dialogue develops. It may indeed turn out that at least some members of this catlist will see me just as a devoted soldier "seeking the bubble reputation, even in the canon's mouth". When we were searching for a name describing what we try to explain, we wanted to be careful not to use a "reserved word" that is more easily misunderstood than not well understood. In the latter case, we thought we can always try to explain, as I am about to do now. In the former case it would be a differentiation, which is more tricky. So here goes. 'Lative' is related to motion, and more specifically, motion 'to' and 'from', so when terms appear in sentences, terms 'move into' sentence, and 'appear within' sentences. At the same time, sentences 'move away from' terms, and separates terms from sentences. In comparison, 'ablative' is motion 'away', and nominative is static. The lative locative case (casus) indeed represents "motion", whereas e.g. a vocative case is identification with address. We want to underline the need not to have "mixed bags", so that we can ensure that a term does not appear in the bag of sentences, or a structured set of sentences would appear in the bad of entailment. I shouldn't compare with waste sorting, because then somebody might say "Patrik Eklund said Kurt G??del didn't care about waste sorting". Obviously, I do respect the work of G??del, even if at the same time I do find his approach "illative". From categorical point of view, G??del also complies only with the underlying category of sets, but as we have shown (e.g. in http://www.sciencedirect.com/science/article/pii/S0165011413000997), we may have other underlying categories for the the term monad. The term constructor and the use of category theory as the metalanguage for logic is here important. Logic developed "hand-in-hand" with set theory and as being a metalanguage for category theory should then not be confused with the lativity of logic we explain using categorical notions. Let me also again underline that nomenclatures and classifications in health care is one of our motivation areas of examples and applications. Nomenclatures exist also elsewhere, but he ones appearing in health we find very motivating. At this point of our "research program" we believe we have a fair understanding of the lativity as related to signatures, terms and sentences, and we hope we have a fair intuition about how we now proceed to investigate the lativity of that with respect e.g. to to entailment and models. Thanks again, and indeed, possible shame in whatever form or magnitude is all mine. Best, Patrik [For admin and other information see: http://www.mta.ca/~cat-dist/ ]