The Definition of Points
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From: Tony Orlow on 18 Apr 2007 14:31 Lester Zick wrote: > On Tue, 17 Apr 2007 12:20:01 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> What question? You seem to think there is a question apart from >>> whether a statement is true or false. All your classifications rely on >>> that presumption. But you can't tell me what it means to be true or >>> false so I don't know how to answer the question in terms that will >>> satisfy you. >>> >>> ~v~~ >> A logical statement can be classified as true or false? True or false? > > A logical statement as opposed to what, Tony? As opposed to, say, an arithmetic formula. > >> In other words, is there a third option, for this or any other statement? > > Hard to tell without seeing the statement. > > ~v~~ No, it's up to you. A logical statement is one that has some measure of truth, from false to true. One can consider just false and true, or one can consider a multilevel logic like a scale from 1 to 10, or even a probabilistic logic with all real values from 0 through 1. Since you only speak of truth versus falsity, I imagine you are considering the first type, or Boolean binary logic.
From: Tony Orlow on 18 Apr 2007 14:39 Lester Zick wrote: > On Tue, 17 Apr 2007 12:20:59 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Fri, 13 Apr 2007 14:33:20 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Thu, 12 Apr 2007 14:35:36 -0400, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>> >>>>>> Lester Zick wrote: >>>>>>> On Sat, 31 Mar 2007 20:58:31 -0500, Tony Orlow <tony(a)lightlink.com> >>>>>>> wrote: >>>>>>> >>>>>>>> How many arguments do true() and false() take? Zero? (sigh) >>>>>>>> Well, there they are. Zero-place operators for your dining pleasure. >>>>>>> Or negative place operators, or imaginary place operators, or maybe >>>>>>> even infinite and infinitesimal operators. I'd say the field's pretty >>>>>>> wide open when all you're doing is guessing and making assumptions of >>>>>>> truth. Pretty much whatever you'd want I expect.Don't let me stop you. >>>>>>> >>>>>>> ~v~~ >>>>>> Okay, so if there are no parameters to the function, you would like to >>>>>> say there's an imaginary, or real, or natural, or whatever kind of >>>>>> parameter, that doesn't matter? Oy! It doesn't matter. true() and >>>>>> false() take no parameters at all, and return a logical truth value. >>>>>> They are logical functions, like not(x), or or(x,y) and and(x,y). Not >>>>>> like not(). That requires a logical parameter to the function. >>>>> Tony, you might just as well be making all this up as you go along >>>>> according to what seems reasonable to you. My point was that you have >>>>> no demonstration any of these characteristics in terms of one another >>>>> which proves or disproves any of these properties in mechanical terms >>>>> starting right at the beginning with the ideas of true and false. >>>>> >>>>> ~v~~ >>>> Sorry, Lester, but that's an outright lie. I clearly laid it out for >>>> you, starting with only true and false, demonstrating how not(x) is the >>>> only 1-place operator besides x, true and false, and how the 2-place >>>> operators follow. For someone who claims to want mechanical ground-up >>>> derivations of truth, you certainly seem unappreciative. >>> Only because you're not doing a ground up mechanical derivation of >>> true or false. You're just telling me how you employ the terms true >>> and false in particular contexts whereas what I'm interested in is how >>> true and false are defined in mechanically reduced exhaustive terms. >>> What you clearly laid out are the uses of true and false with respect >>> to one another once established. But you haven't done anything to >>> establish true and false themselves in mechanically exhaustive terms. >>> >>> ~v~~ >> Again, define "mechanics". > > Tony, time for you to do a little work for yourself. I've already gone > through this. You describe for me the mechanics of using binary truth > values and I explain to you I'm interested in truth not binary truth > values and how to ascertain truth in mechanical terms initially and > not how to work with truth values mechanically once ascertained. So, you are of the opinion that science can be performed without collecting any data, doing experiments or studies, and ascertaining truth from fact? Then you are in as much a religious limbo as the Cantorian ball replicators. How far has this Ivory Tower approach gotten you so far? If you get down the mechanics of deduction, then you can consider more readily what underlying principles may be causing whatever phenomena you're investigating. > > By the way what is the truth value of "square triangles" and how does > that differ from the truth value of "blue squares" and how do you know > the difference? > > ~v~~ "The triangle is square" is false because the definitions of "triangle" and "square" are mutually self-contradictory, because 3<4. "The square is blue" is not inherently self-contradictory, because there is no contradiction between "triangle" and "blue". So, the truth value depends on the particular triangle, and perhaps what one considers "blue" as opposed to "green" or "purple". 01oo
From: Tony Orlow on 18 Apr 2007 14:49 Lester Zick wrote: > On Tue, 17 Apr 2007 13:39:45 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Sure. Happens all the time. However if you're asking whether a >>> statement must be one or the other the answer is no. There are >>> problematic exceptions to the so called excluded middle. > >> Please eloborate. > > "Black is crows" is ambiguous in general terms and neither true nor > false since "crows are black". Hence we find that "crows are black" is > true but "black is not crows" is true too in general scientific terms. > > ~v~~ Okay, consider a universe where the ONLY black things are crows. In that universe, would not "black is crows" be true? What else could black be, besides some number of crows? I think you are assuming that all crows are black, and noting the proper subset relation. C is a proper subset of B, because all members of C are members of B, but not all members of B are members of C. So, indeed, black(crow) might be equal to 1, meaning 100% of crows are black, while crow(black) might only be equal to say 0.05, because only one out of twenty black things are crows. It's a 5% probability that if something is black, it has the property of crowness, while if it is a crow, it has 100% probability of being black. Make sense? 01oo
From: Tony Orlow on 18 Apr 2007 14:52 Lester Zick wrote: > On Tue, 17 Apr 2007 13:39:45 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Well your phrase "exploring the meaning of truth" is ambiguous, Tony, >>> because what you're really doing is exploring consequences of truth or >>> falsity given assumptions of truth or falsity to begin with, which is >>> an almost completely trivial exercise in comparison with the actual >>> determination of truth in mechanically exhaustive terms initially. >>> >> I am exploring the mechanics of truth, and its pursuit, which you are >> not, really, as far as I can tell. > > You're exploring the mechanics of using truth once established but I > see no indication you're exploring the mechanics of truth otherwise. > > ~v~~ I am trying to get there, but you're struggling against even defining in any mechanical terms what your statements mean, which is obviously deliberate. The mechanics of deduction are pretty straightforward. The mechanics of induction are a little harder to ascertain, because we doing it unconsciously by nature, but can be developed by looking at the deductive mechanics. 01oo
From: MoeBlee on 18 Apr 2007 14:55
On Apr 17, 9:56 pm, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > > On Apr 17, 11:33 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> But, I have a question. What, exactly, is the difference between > >> "equality" and "equivalence"? > > > As I told you, in first order, we cannot generalize (I'll say about > > exceptions in a moment) that identity can be "captured" by axioms. > > Thus in the general case, we require stipulating a fixed semantics for > > the identity symbol. Meanwhile, equivalence often pertains to an > > equivalence relation (reflexive, symmetric, transitive). Every > > equiavlance relation on set induces a partition of the set; that > > partition being a set of equivalence classes. Then two objects are > > equivalent iff they are both members of the same equivalence class, > > which is to say that they bear the equivalence relation (the > > reflexive, symmetrc, transitive relation) to each other. Notice that > > identity is a an equivalence relation but not all equivalence > > relations are the identity relation. > > Okay, distinction noted. It would seem to apply well to cardinality, > where we have classes of sets that match, even if they are not equal as > sets, eh? I wouldn't put it that way, but I think you're onto the basic idea. Yes, bijectability (equinumeroisity) is an equivalance relation on any set. > >> Is it not, in the tradition of Leibniz, a > >> matter of detecting a distinction between two objects, or not? That is, > >> if we can detect no difference between two objects, if we can find no > >> attribute which distinguishes them, then are they not "equal" or > >> "equivalent", or "identical"? Ultimately, if we can not say, in one > >> sense or another, that x<y or y<x, then do we not consider that x=y? > > > One direction of Leibniz's principle (the indiscerniblity of > > identicals) is acheived by an axiom that if x = y then whatever holds > > for x holds for y. But, in first order, we cannot generalize that the > > other direction (the identity of indiscernbiles) can be stated even as > > an axiom schema. Thus a stipulated semantics is given for the identity > > symbol. > > And this is because first order logic does not allow the universal > quantifier applied to sets or properties, but only elements, is that > right? Again, I wouldn't put it that way, but you're on to the right idea. In first order, we don't quantify over predicates (we don't follow the universal or existential quantifier by a variable that ranges over sets of n-tuples (n>0) of members of the domain of discourse, since we have no such variables). > Is it not possible to concoct a logic where the first order of > analysis is on the property, versus object, level? I'll have to dream > about that a bit...what do you think? I don't know what the point would be. If you are going to quantify over predicates, then why not just quantify over both predicates and individuals, which is what second order logic is. > >> Defining "=" depends, as far as I can tell, on defining "<". Is this > >> wrong, in your "opinion"? > > > In what context? In set theory, given an appropriate axiomatization, > > we can define '=' from just 'e' (the membership symbol). By the way, > > that does not contradict my remarks about first order not being able > > to "capture" identity, since what that says is that it is not the case > > that for ANY langauge or theory we can "capture" identity. For CERTAIN > > languages and theories (viz. those with only finitely many predicate > > symbols) we can "capture" so as to make a definition for the identity > > symbol that conforms to the basic semantics that 'x=y' is true iff > > denotation of x and the denotation of y are identical. > > > MoeBlee > > In a very real sense, for all xeX, x<X, so defining '=' from 'e' is > pretty similar to defining it from '<'. That's the definition of '<' for ordinals. It's not for sets in general. And it wouldn't work for equality of sets in general as long as we have regularity, since it is never the case that xey & yex. The actual definition can be made by the subset relation: x=y <-> (x subset_of y & y subset_of x). > I understand that this is not > pure first order logic, but first order with an added operator with its > own properties. First order by itself can't distinguish among anything > but truth values, I'm not sure what you mean by that. > so if those pertain to some objects or properties, the > additional relationships between those need to be handled with operators > that take as parameters whatever type of objects or properties one has, > rather than just truth values. In the context of whatever additional > operators are added to first order logic, indeed, if every statement > true of each is true of the other, then those two objects or properties > are considered identical. In set theory, since there are only finitely many predicate symbols, we CAN and we DO implement both directions of Leibniz's principle. > That's not to say that an additional operator > might not produce a difference between two previous equated elements, is it? I don't follow what you meant about operators. But why don't you get 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar? I'm not saying that challengingly (this time). I am sincerely suggesting that such books would give you start toward a solid basis upon which to develop and communicate your own ideas and possible innovations with things such as operators. MoeBlee |