The Definition of Points
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From: Lester Zick on 23 Apr 2007 14:47 On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 20 Apr 2007 12:00:37 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>>> What truth have you demonstrated without positing first? >>>> And what truth have you demonstrated at all? >>>> >>>> ~v~~ >>> Assuming two truth values as 0-place operators, I demonstrated that >>> not(x) is the only functional 1-place operator, and then developed the >>> 2-place operators mechanically from there. That was truth about truth. >> >> And how have you demonstrated the truth of the two "truth values" you >> assumed? >> >> ~v~~ > >The "truth" of the two truth values is that I've declared them a priori >as the two alternative evaluations for the truth of a statement. Well there are two difficulties here, Tony. You declare two a priori alternatives but how do you know they are in fact alternatives? In other words what mechanism causes them to alternate from one to the other? It looks to me like what you actually mean is that you declare two values 1 and 0 having nothing in particular to do with "truth" or anything except 1 and 0. The second problem is what makes you think two "truth" alternatives you declare are exhaustive? This is related to the first difficulty. You can certainly assume one thing or alternative a priori but not two. And without some mechanism to produce the second from the first and in turn the first from the second exclusively you just wind up with an non mechanical dualism where there is no demonstration the two are in fact alternatives at all or exhaustive alternatives either. We already know you think there are any number of points in the interval 0-1 so apriori declarations do not erase that inconsistency between different sets of assumptions. A few days ago you asked about what I mean by "mechanics" or an "exhaustive mechanics" and this is what I mean. You're welcome to assume one mechanism but then you have to show how that one mechanism produces every other aspect of the "mechanics" involved. And I believe it obvious that the one "mechanism" has to be the process of "alternation" itself or there is no way to produce anything other than our initial assumption. A chain is no stronger than its weakest link and if you make dualistic apriori assumptions neither of which is demonstrably true of the other in mechanically exhaustive terms you already have the weakest link right there at the foundation. > That's >not a matter of deduction until you specify what statements you are >assuming true or false, and on what basis you surmise them to be one or >the other. Well this comment is pure philosophy, Tony, because we only have your word for it. You can certainly demonstrate the "truth" of "truth" by regression to alternatives to "truth" by the mechanism of alternation itself and I have no difficulty demonstrating the "truth" of "truth" by regression to a self contradictory "alternatives to alternatives". Of course this is only an argument not a postulate or principle but then anytime you analyze "truth" you only have recourse to arguments. > Truth tables and logical statements involving variables are >just that. If I say, 3x+3=15, is that true? No, we say that IF that's >true, THEN we can deduce that x=4. But here you're just appealing to syllogistic inference and truisms because your statement is incomplete. You can't say what the "truth" of the statements is or isn't until x is specified. So you abate the issue until x is specified and denote the statement as problematic. The difficulty with syllogistic inference and the truism is Aristotle never got beyond it by being able to demonstrate what if anything conceptual was actually true. The best he could do was regress demonstrations of truth to perceptual foundations which most people considered true, even if they aren't absolutely true. But even based on that problematic assumption he could still never demonstrate the conceptual truth of anything beyond the perceptual level. And here the matter has rested for mathematics and science in general ever since. Empiricism benefitted from perceptual appearances of truth in their experimental results but the moment empirics went beyond them to explain results in terms of one another they were hoist with the Aristotelian petard of being unable to demonstrate what was actually true and what not. The most mathematicians and scientists were able to say at the post perceptual conceptual level was that "If A then B then C . . ." etc. or "If our axiomatic assumptions of truth actually prove to be true then our theorems, inferences, and so forth are true". But there could never be any guarantee that in itself was true. > If I say 3x+3=3(x+1) is that true? >Yes, it's true for all x. How about for x=3/0? > If I say a=not b, is that true? Not if a and b >are both true. How do you arrive at that assumption, Tony? If a and b are both true of the same thing they can still be different from each other. > If I say a or not a, that's true for all a. a and b are >variables, which may each assume the value true or false. Except you don't assign them the value true or false; you assign them the value 1 or 0 and don't bother to demonstrate the "truth" of either 1 or 0. >If you want to talk about the truth values of individual facts used in >deduction, by all means, go for it. I don't; I never did. All I ever asked was how people who assume the truth of their assumptions compute the truth value of the assumptions. ~v~~
From: MoeBlee on 23 Apr 2007 15:00 On Apr 23, 10:29 am, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > Successor, in the x -> x u {x} sense, depends only on singleton sets, > > subsets and unions, all of which are defined strictly in terms of 'e', > > so what is left? Nothing. > > Only the implication from the existence of one element to the next, > which defines the successor relation as one between two elements, and in > this case, equivalent to 'e'. If we define successor in the x -> x+1 > sense, then it's not really based on 'e', but it does define the > naturals, in a quantitative way. Successor CAN be related to 'e', but > really depends on recursive implication of existence. "recursive implication of existence." Oh, brother! MoeBlee
From: Virgil on 23 Apr 2007 16:14 In article <462ceab3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > But what your 1. and 2. don't take into account is that we can (and > > "frequently" must have), given any two elelemnt x,y in S: > > > > S > > / \ > > / \ > > > > S-{x} S-{y} > > > > \ / > > \ / > > > > S-{x,y} > > > > > > So the "tree" criss-crosses like a chain link fence. This type of > > partial order is usually called a lattice. > > Yes, it becomes a sort of a lattice-looking thing from one level to the > next. It's actually the set of vertices of a |S|-dimensional cube, if > the same subset may only occur once. If you allow the redundancies, so > that S-[x,y] appears both as a child of S-{x} and of S-{y}, then you get > a tree, but not every element is unique. I guess on each level n, where > S is on level 0, one gets each unique subset n times, and the number of > unique elements generated at each level is 2^n-n? Something like that. In a tree structure, as defined in mathematics, no 'child' can have more than one 'parent'. What you are describing should not be called a tree, but perhaps you mean a lattice, in which a 'child' can have any number of patents, like in the lattice of subsets of a given set with the relation of 'subset of' as the 'child' relation.
From: Virgil on 23 Apr 2007 16:19 In article <462cecec(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4628df20$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >>> I do say so. Does this mean you're going to stop claiming that > >>> relations in set theory aren't based solely on 'e'? > >>> > >>> -- > >>> mike. > >>> > >> No, not when sequences are defined using a recursively defined successor > >> function, which is a relation between two elements, as opposed 'e', a > >> relation between an element and a set. The combination of the two is > >> what produces an infinite set, no? > > > > So that x -> x u {x} does not depend on 'e' ? > > > > On can define successor entirely in terms of 'e'. > > > > Successor, in the x -> x u {x} sense, depends only on singleton sets, > > subsets and unions, all of which are defined strictly in terms of 'e', > > so what is left? Nothing. > > Only the implication from the existence of one element to the next, > which defines the successor relation as one between two elements, and in > this case, equivalent to 'e'. If we define successor in the x -> x+1 > sense, then it's not really based on 'e' T here is a reasonable way of defining x+1 in tems of 'e', but I have never seen it done without ever using 'e'. Perhaps TO can provide an axiom system allowing 'x+1' for each 'x' but not ever relying on any sets or any membership, so that we can see what it looks like? > but it does define the > naturals, in a quantitative way. Successor CAN be related to 'e', but > really depends on recursive implication of existence. Which itself depends on 'e'.
From: Ben newsam on 23 Apr 2007 19:43
On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >If you want to talk about the truth values of individual facts used in >deduction, by all means, go for it. I would counsel you seriously not to attempt it, Lester will only obfuscate the "discussion", and then will start to hurl personal insults at you. |