The Definition of Points
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From: Lester Zick on 24 Apr 2007 18:28 On Tue, 24 Apr 2007 16:14:56 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Mon, 23 Apr 2007 21:47:51 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> 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. >>> May I address that? When I say that true() and false() are "zero-place" >>> logical operators, that really means that they take no "logical >>> parameters". That is, what is passed to the function for evaluation >>> includes neither "true" nor "false" but is rather a statement which is >>> to be evaluated, in the context of whatever "theory" we are considering, >>> as either "true(x)" or "false(x)". >> >> Wouldn't "true(1)" represent such a statement? > >"1" doesn't constitute a statement, because there is no predicate. Why not? Because you say so? > "1 is >a natural" or "1eN" is a statement and is true. So, true(1eN) and >false(not(1eN)) and not(false(1eN)). Yeah it looks like you're back to doing conjunctions again, Tony. So I don't know what to say. Just go ahead and say whatever you want. >>> The truth of x or y depends on what >>> rules we consider valid in out language. >> >> But just "considering rules valid" doesn't make them or their results >> true, Tony, and that's what I'm trying to ascertain. You claim to have >> proven certain things about the nature of truth true but those things >> rest on assumptions of truth regarding the nature of truth in specific >> contexts which aren't necessarily true in general. Really all you seem >> to have done is describe rules you consider true versus proving them >> true. >> >> ~v~~ > >Well, I exhausted all the self-contradictory, or redundant, alternatives. :) Well I can't tell if you exhausted all the self contradictory or redundant alternatives, Tony, but you certainly exhausted me. ~v~~
From: Lester Zick on 24 Apr 2007 18:31 On Tue, 24 Apr 2007 16:29:13 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>>> 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. >>> 1=not(0) >>> 0=not(1) >>> x<>not(x) >>> >>> That says it all. "not" defines the relationship between 0 and 1. In >>> arithmetic terms, not(x) can be equated with 1-x, for obvious intuitive >>> reasons. If 1 is the universe, and x is some portion of that universe, >>> everything that is not(x) is everything in the universe, minus what's in x. >>> >>> "not not" then equates to 1-(1-x), which is just x, which doesn't have >>> any truth value until you assign it one. >> >> Okay, Tony. Basically you're agreeing with what I've been saying. But >> I'm still bothered by this binary arithmetic approach to the mechanics >> of "not" and "not" compounded in terms of itself. > >not(means_anything(without("not",predicate))) > >not(x) is a one-place logical operator. Without a predicate x, and a >truth value for that predicate, "not" doesn't have a truth value. "Not" >simply negates itself into oblivion. > >> >> I think you really need a symbolic rather than arithmetic approach to >> the problem. In other words you've got the idea of differences but the >> differences are not between binary logic values. They're taken between >> real things in logical terms which can result in any and all kinds of >> differences any and all of which can be regurgitated tautologically in >> terms of subsequent differences. The 0 stays but the binary 1 goes. We >> can't just take the truth of one set of circumstances and compare it >> to the truth of another set of circumstances. Both sets may be true >> but may be true in different ways which preclude analyzing the truth >> of both together in exactly common terms. > >That depends how well you analyze what you're talking about. For all >intents and purposes, there is some finite number of differences >required to distinguish any object from any other, if nothing else, >limited by our ability to distinguish things. For any of those >differences, some finite specification suffices to determine them, or >they are not different in that respect. But, that's all on a pre-logical >level. > >> >> This is why I analyze the problem in terms of predicate combinations >> and not in strictly binary terms. We might have a "large apple" and a >> "green apple" both of which predicate combinations could be true of >> one specific real object without having any common "truth value" in >> terms of each other. This is why I analyze the problem in symbolic >> terms rather than strictly binary terms of common truth values. >> >> ~v~~ > >Okay, sure. We can have an apple A, and we may have green(A) or >not(green(A)), and we may have large(A) or not(large(A)). So, as far as >greenness and largeness, we have four possibilities. > >We may relate the two attributes probabilistically, and say most green >apples are little, and many little apples are green, and many red apples >are large, and many large apples are red, in whatever proportions >between 0 and 1. If those proportions are all 50%, there's no >correlation. In any case, all the mutually exclusive colors of apples >add to all the apples, and all the mutually exclusive sizes of apples >add to all the apples. Sorry, Tony, but this is enough for me. My point was that "apples are apples" but "green" is not "large" so we're left with ambiguous truth values. If the point escapes you then I don't know what else to say. ~v~~
From: Lester Zick on 24 Apr 2007 18:50 On Tue, 24 Apr 2007 16:44:21 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> 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. >>> See the mechanism for their mutual definition, above. >>> >>> As far as the two truth values being "exhaustive", it's simply declared >>> in Boolean logic that these are the only two values allowed, true() and >>> false(). Now, these are not really all the possibilities, if one looks >>> at truth in the context of probability. Where a probability of truth of >>> x is 100%, the truth value is 1, meaning "definitely true". Where a >>> statement has a 0% probability of truth, the value is 0, meaning >>> "definitely false". Between these two lie a continuous set of possible >>> probabilities. So, in this context, {0,1} is NOT an exhaustive set of >>> all alternatives. But, this requires that you accept partial truth, >>> which doesn't seem to be acceptable for you. What other alternatives do >>> you see besides true(x) and false(x) for a statement x? >> >> Well as noted in the preceeding reply segment there are certain subtle >> complexities in relations between true predicate combinations which >> obscure the idea of binary truth values but have nothing to do with >> probabilities whatsoever. I don't reject the idea of probability but I >> think the idea of truth in general has to be straightened out before >> we can get to any mechanically exact understanding of probability. >> >> I agree the common boolean notion of truth only allows for two truth >> values but I don't agree that actually reflects what truth means in >> strict mechanical terms. > >You mean determining the truth of any given statement? You have to see >under what conditions it holds. Don't know what you're trying to say here, Tony. My comment was simply an observation. >> Taking two symbolic predicate combinations such as "green apple" and >> "large apple" both of which are true of one specific real thing we are >> then left to consider the truth of each in relation to the other. And >> what we find are that there are complex subordinated differences >> between them not represented in simple binary 1's. It's a sequential >> combinatorial logic problem where the truth of each predicate has to >> be considered individually and in combination. > >Okay, so you're talking about what I mentioned in my last post, the >relationship between "green" and "large" in relation to "apple". This >involves the probabilistic approach, I think. One might say that 10% of >large apples are green, so green(large(apple))=0.1 and >not(green(large(apple)))=0.9. You might say 99% of green apples are not >large, so large(green(apple)) would be 0.01 and not(large(green(apple))) >is 0.99. You might then be able to say that, given that an apple is >large or not, that the "truth" of it being green is 0.1. Oh Jesus, Tony. Let's just forget it. >> Let's suppose for example we have two predicates "animal" and "fox" >> both of which are true with respect to one specific real thing. And we >> represent the truth of each with a binary 1. Then we further consider >> the truth of the predicates in combination and we can't tell whether >> the thing is an "animal fox" or a "fox animal" from the representation >> of truth in each instance. Thus the binary arithmetic representation >> is completely misleading even though it correctly identifies the truth >> in each instance. It just can't tell us how the true but different >> predicates "fox" and "animal" relate to each other. And if they're >> related incorrectly the resulting predicate combination will not be >> true even though each predicate in the combination is. >> >> ~v~~ > >Okay, but the probability that a fox is an animal is 100%, whereas the >probability that an animal is a fox is, say, 0.01%. Since fox(x) implies >animal(x), given the 100% inclusion, we can eliminate the bit for >"animal". This really becomes a statistical correlation issue, at this >point. How do we determine which value between 0 and 1 denotes the >correlation between two predicates? We gather data, I guess, and maybe >discover some logical or causal relationships in the process. Tony, this is the last time around. If we have a " large green apple" and a "large red apple" "apple" and "large" are true and according to your boolean scheme would have a truth values of 1 but there is no probability whatsoever that "green" is "red" so it would have a truth value of 0 in your boolean logic and the net truth value for "large green apple" in comparison to "large red apple" is ambiguous. Boolean logic just doesn't account accurately and correctly for truth in mechanical terms. You can imagine whatever you want because that's what you've been taught. When you get around to considering the truth values of complex predicate combinations maybe things'll change. Maybe not. I just don't know what else to tell you. ~v~~
From: Tony Orlow on 24 Apr 2007 19:01 Lester Zick wrote: > On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> 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. >> I'm trying to keep it simple, and just discuss the mechanics of the most >> basic kind of logic, where absolute "truth" exists. It doesn't, in real >> science. > > I don't understand where you think absolute truth exists if not in > real science. > > ~v~~ Absolute truth underlies the universe. Science only confirms a theoretical truth to within some degree of accuracy, or disproves it. 01oo
From: Tony Orlow on 24 Apr 2007 19:08
Lester Zick wrote: > On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> 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. >>> >> Again, please comment on my use of "not" to define the relationship >> between true() and false(). > > Okay that's an improvement. But one of the things I don't see is how > you produce and "truth values" between 0 and 1 using not. > > ~v~~ If you already have truth values between 0 and 1, as in a probabilistic model, then not(x) is defined as 1-x, which is between 0 and 1 for x between 0 and 1. In that kind of system, not(x) is arithmetic. For uncorrelated x and y, and(x,y)=x*y, and or(x,y)=not(and(not(x),not(y))), or 1-(1-x)(1-y), or x+y-x*y. Where there is a correlation between x and y, it becomes hairier. I think that's what you're trying to address with your large and/or green apples? 01oo |