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## Re: [ontolog-forum] Axiomatic ontology

 To: "Rob Freeman" "[ontolog-forum]" Pat Hayes Wed, 8 Oct 2008 10:54:28 -0500 <4E764D34-9223-42D6-87E9-6E88CE69018B@xxxxxxx>
 On Oct 7, 2008, at 11:59 PM, Rob Freeman wrote:Pat,Thanks for addressing my complexity point.On Tue, Oct 7, 2008 at 11:26 PM, Pat Hayes wrote:On Oct 7, 2008, at 2:44 AM, Rob Freeman wrote:My argument has been that for many systems there are more ways ofgeneralizing examples than there are examples. This is a game changingassumption. Up to now it has always been assumed that there will befewer generalizations than examples (rules, classes, etc.) At worstthat you need to make one generalization per example (analogy.) If itis possible to make more generalizations than examples, the gamechanges completely. No one set of generalizations will suffice.OK, that sounds really interesting. But Im having trouble understanding it.Can you make the point more concrete, perhaps with an example/sketch?  Whatleads you to this conclusion, that there will be this overwhelming number ofgeneralizations in many systems? What kind of systems, and why will theyhave the top-heavy quality?It is easy to prove there could be more meaningful combinations of nelements than n, for any system. More combinations, of course. This is why for example classical higher-order logic is undecidable, because its semantics requires that its interpretations contain all relations in this mathematical sense (and when the base set is infinite, this makes the set of relations much more infinite.) But (1) you have to be more specific about what you mean by a 'system' in order to see what this means in practice; and (2) not all combinations are 'meaningful'. The maximum number is something likethe binomial coefficient {n choose k} = {n!}/{k!(n-k)!}. Withvariations according whether the order of elements matters etc.For sets its just 2|nThe only question is whether all these possible sets are distinct inmeaningful ways.No, that is not the only question. A more important question is, so what? Why does this fact of combinatory mathematics matter to a given system? Take Common Logic, for example (which isn't a system, exactly, but never mind): the number of possible relations over a universe is a much larger cardinality than the universe (see above), but this doesn't matter because the logic isn't obliged to consider all of these relations, only enough to provide denotations for all the relation-naming terms in the language. So this just isn't an issue in CL. Let me turn the question around. How do we know they are not? Hasanyone ever looked into it?Well, the question as posed isn't precise enough to look into. What kind of system are you talking about? For the case of cardinality in logical interpretations, the answer is certainly yes, in fact its been a major focus of work in model theory for the past decade. The only way to model that would be to base analysis on sets of examples.There is an old idea (due I believe to Whitehead in the early 20th century)that may be relevant here. If all you have are categories (generalizations,classes,..), then you can define individuals to be maximal sets ofintersecting categories. Mathematically, they are ideals or ultrafilters(same thing looked at upside-down.) Its a very elegant and powerfultechnique: I've used for example to show that any notion of 'context' thatsatisfies some basic assumptions can be reduced to sets of individualcontext-points. It can be used to define time-point in terms oftime-interval, and other apparently impossible "backward" creations ofindividual- or concrete- ish objects from clouds of vaguer ones.Right, yes, basing categories on sets has a history. We can feelcomfortable thinking of categories in this way. Others have done it.Using sets to represent categories is not a crazy proposal at all. Um...no, its not. In fact, virtually all of modern mathematics is 'based on sets'. Thequestion is whether you can ignore sets and work only with categories.? Why is that an interesting question or an interesting ambition? What do you gain by ignoring sets? It is really a question of complexity. For a given set of examples ifthere are fewer distinct sets than examples you will be able to labelall the sets and get along fine with the labels. On the other hand ifthere are more distinct sets than elements you will have noalternative but to work directly with the sets.I completely fail to follow you here. Of course there are not fewer sets than examples, since there is a singleton set for each example. Pat------------------------------------------------------------IHMC                                     (850)434 8903 or (650)494 3973   40 South Alcaniz St.           (850)202 4416   officePensacola                            (850)202 4440   faxFL 32502                              (850)291 0667   mobilephayesAT-SIGNihmc.us       http://www.ihmc.us/users/phayes
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 Current Thread Re: [ontolog-forum] Axiomatic ontology, (continued) Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, Pat Hayes Re: [ontolog-forum] Axiomatic ontology, John F. Sowa Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, Pat Hayes Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, Pat Hayes <= Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, Christopher Menzel Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, Patrick Cassidy Re: [ontolog-forum] Axiomatic ontology, Christopher Menzel Re: [ontolog-forum] Axiomatic ontology, Pat Hayes Re: [ontolog-forum] Axiomatic ontology, John F. Sowa [ontolog-forum] Scientific American article on "natural" word order, Rich Cooper Re: [ontolog-forum] Axiomatic ontology, Rob Freeman Re: [ontolog-forum] Axiomatic ontology, John F. Sowa