To: |
sowa@xxxxxxxxxxx, "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx> |
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From: |
"Rob Freeman" <lists@xxxxxxxxxxxxxxxxxxx> |

Date: |
Sun, 12 Oct 2008 19:49:59 +0800 |

Message-id: |
<7616afbc0810120449w35e53ccfycbeb7723bff3f05@xxxxxxxxxxxxxx> |

John, (01) Thanks for your analysis. That puts it nicely. Anything you can say in any logic, you should be able to say in English. In counting the meanings of English we need to count them all. (02) I don't know if your observation that the number would be countably infinite is true also. What matters to me is that, even for finite sets, the number of possibly "meaningful" relations could be very large (unless we impose an artificial limitation in some way.) (03) As Pat H. says: (04) "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.)" (05) Now, Pat deals with this by limiting the relations he considers: (06) "...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." (07) In short Pat assumes some kind of other semantic model, and that other model limits the relations he needs to consider, so this explosion of possibly meaningful sets "doesn't matter", QED. (08) Pat doesn't like the explosion of relations. He needs some other system of meaning to tell him which ones are meaningful: (09) "Yes, these sets are all distinct. I don't know if they are distinct in "meaningful ways" because I don't know what you count as a 'meaningful way'." (010) But Pat, nobody knows what to count as a "meaningful way". That is what is at issue. What right do we have to be throwing out all the combinatorial explosion of possible meaningful sets suggested by set theory? (011) I don't blame you. It is normal to look at this combinatorial explosion as a problem. Perhaps that is because of the "intractability" of such a system from the point of view of logic. But that same "intractability" has a flip side. It can also be seen as a power. What it tells us is that even for a finite "base set", you might elaborate a very large (countably infinite?) number of meanings. (012) So a finite number of examples can specify an infinite number of categories. Not just infinite strings of bits. Finite strings specifying infinite numbers of categories too. (013) Am I the only one to see that as quite powerful? (014) What I don't understand is why we are not using finite sets of examples to specify (very large or countably) infinite sets of categories in this way. Say, use one set of examples to represent numerous (all?) possible logics as relations over it. (015) If you are looking for a point to what I have been saying, that is one expression of it. (016) Pat Cassidy: By "Guo" do you mean Guo Jin formerly of Singapore National University? I can't imagine what methodology he used to decide if a word no. 1401 were "required", or if word no. 1400 was really the last one...!! (017) -Rob (018) P.S. John, that "countable infinity" of yours is a nice result. How did you get it? I've guessed that if you can find more than n sets over n examples then in principle you could go on for ever generating n+p sets over n, and then (n+p)+q sets over those n+p etc. So I guessed an infinity, but I had no evidence the expansion would not top out at some point (much larger than the set used to specify it.) (019) On Sat, Oct 11, 2008 at 9:01 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote: > |

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