The Virgin Birth of Points
in [Physics]
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From: Schlock on 5 Dec 2007 13:35 On Tue, 4 Dec 2007 16:47:35 -0800 (PST), Randy Poe <poespam-trap(a)yahoo.com> wrote: >> For if "square circles" is not a predicate combination then what is >> it? > >A noun phrase. It's also thirteen letters of the alphabet. Hardly definitive for one so schooled in the black arts of dialectical evasion as yourself.
From: Wolf Kirchmeir on 5 Dec 2007 22:33 Shrew_D wrote: > On Wed, 05 Dec 2007 09:49:18 -0500, Wolf Kirchmeir > <ElLoboViejo(a)RuddyMoss.com> wrote: > >> Shrew_D wrote: >>> On Tue, 04 Dec 2007 15:48:02 -0500, Wolf Kirchmeir >>> <ElLoboViejo(a)RuddyMoss.com> wrote: >>> >>>> Shrew_D wrote: [...] >>> But then what is a definition? >> A statement that a term stands for some description of a >> class/entity/etc. You've used them in math class: "Let a be...." > > Except I meant in reference to propositions and how they differ other > than pro forma conventions. Those pro forma conventions are the essence of the issue. >>>>> In fact this would seem to be born out by your own example of a >>>>> definition for unicorns with various predicates in combination. > >>>> There are no predicates in my definition, merely descriptors. The "is" >>>> in my definition is an equivalence sign: it means you can replace >>>> "unicorn" with the definition wherever "unicorn" occurs. The descriptors >>>> can be used in predicates, such as "X has a horn at its front end." That >>>> may or may not be true about X - you'd have to take a look. >>> But aren't horns predicated of unicorns in your definition? They seem >>> to be. >> No, the definition applies to the term "unicorn", _not_ to the entity >> /unicorn/. I didn't say "All unicorns are wormlike critters...", which >> would be a proposition. > > Are you saying "unicorns are wormlike critters" is not a proposition? > If so I would strenuously disagree. Sorry, you haven't understood my point. You seem to think I'm making some sort of claim about unicorns and their features. I'm not. I'm just equating a term with a string of other terms, is all. If I say that 'unicorn' is defined "a green wormlike critter...", then that's a definition of the word 'unicorn', not a proposition about an animal, real or imaginary. If I build a system using 'unicorn' and other terms, by means of axioms and definitions, then I could _interpret_ the term 'unicorn' as referring to an animal. Or to a bus. Or the snowpile at the end of my driveway. Or whatever. You see, a term is just that. If you like, replace 'unicorn' by T, and the terms in its definition by X{g,w,h}. Then my definition becomes T = X{g,w,h}. Which you may read as "T refers to a member of a class X characterised by w, c and h". (And that reading assumes some conventions about notation, BTW, which may or may not be th usual conventions.) [...] > I think there might be a definitive way to settle the issue. Consider > the following: There is no issue, really. I've just told you what I mean by 'definition' and indirectly what I mean by 'proposition' and 'predicate' You're looking for some real or correct meanings of these terms. There aren't any. There are just meanings that we may or may not agree on, by convention. > Are there any words or word combinations used as "descriptors" in > definitions which cannot be used as predicates in propositions and > conversely are there any words or word combinations which can be used > as predicates in propositions which cannot be used as descriptors in > definitions? If not then I see no viable distinction between the terms > whether we call them descriptors or predicates and I see no reason > both cannot be evaluated for truth, whatever that means. Let's assume you mean the usual grammatical sense for 'predicate'. Then the answer is, no, the terms on _either_ side of a definition can be used in propositions. So why is a definition important? Because it enables us to distinguish between true and false propositions. If a definition is true or false, in what sense is that so? For it to be true/false, we must interpret a definition as a proposition. There are in my view two interpretations of a definition that will answer the case: a) the definition is shorthand for "When people use T, then (in some context) they mean X{a,b,c...}." This is the form of a dictionary definition, and it's easy to determine whether a definition is true in this sense - which is why I deliberately made up an unconventional definition of 'unicorn.' b) the definition asserts the existence of the referent of T. IOW, my definition would not only define 'unicorn', it would also say something about unicorns. That can only be so if the definition implicitly asserts the existence of unicorns. But how can we decide of the definition is true in this sense? Let's hunt for unicorns. You find something that you thinmk is one. You yell, "Hey, Wolf, this 'ere thing 'ere wot I'm lookin' at has a horn at its front end." I say, "Thass nice, is it green??" You say, "Yes indeedy, and it's of a wormlike shape and configgerashun, too!" And I say, "Is it really? Shrew, my friend, you've found a unicorn!" "Yass, that I 'as", you say, and receive the congratulations of the assembled multitudes. Does this event make the definition true? Before you answer with an eager Yes, consider this alternative case. We've hunted high and low, and flushed every critter in our little corner of the universe, and we have found not a single example of a green, wormlike one a horn at front end. We've some with horns, and some without. We've found critters of all colours imaginable, and some colours we couldn't imagine. We've found critters of ever possible shape. But not one combines the three features in my definition of 'unicorn.' So we may conclude that there are no unicorns in our corner of the universe. Does that make the definition false? In short, the definition allows us to say something about the existence of unicorns, but it does not itself assert the existence of unicorns. Thus it is neither true nor false. > Frankly I can't think of any examples of words and word combinations > which aren't applicable to both definitions and propositions (other > than pro forma conventions). Nor does your definition for unicorns > show evidence of any. Well, those pro forma conventions are crucial. They make the difference between conversation and logical argument. Of course, usually we assume the pro from conventions, and don't make them explicit. And that implicitness may mislead us. > So it looks to me like definitions are really mini propositions in the > respect both definitions and propositions use the same vocabulary. And > if there were categorical differences in vocabulary we'd have to have > two different vocabularies with one vocabulary for definitions and one > for propositions. Vocabulary is not sufficient to make a proposition true or false. Hence we may use the same vocabulary for definitions and propositions. Consider: 1) "Socrates is a man, and he is mortal." 2) "Socrates is a man, or he is mortal." These propositions use the same vocabulary, but they have different forms. Hence one can be true while the other is false. Write the truth tables for them, and you'll see. (There are 3 such tables, since 'or' has two distinct meanings.) Footnote: we create a definition when it gets tedious to use a long string of terms that keeps turning up in our arguments. This is such a handy trick, that mathematicians use it even prior to constructing arguments with such a string of terms. That string of terms may of course be a proposition, and usually is, but the definition isn't. But the term used to replace the string of terms is then a proposition... ;-) Maybe that's what Lester had in mind - who knows? > However I think my original point regarding Randy's use of "short- > hand" still stands because clearly the "shorthand" in a definition has > to be "shorthand" for something and I think Lester is correct on this > issue. Since I cannot figure out what Lester means by "predicates" etc, I can neither agree nor disagree. HTH
From: Lester Zick on 6 Dec 2007 00:02 On Wed, 5 Dec 2007 11:28:34 -0800 (PST), Randy Poe <poespam-trap(a)yahoo.com> wrote: >On Dec 5, 1:46 pm, lwal...(a)lausd.net wrote: >> So here, Zick is actually correct. There is a >> consistent geometry with square circles. > >Except that Zick never made such a statement. I did. > >http://groups.google.com/group/sci.math/msg/7894e2daf6e59cab > >Zick's position is that definitions are either true >or false, and "a squircle is a square circle" is >a "false definition". Well it's certainly a self contradictory definition. Whether that means it's a false definition in your neomathspeak argot is anyones guess. We might also consider four sided triangles from a similar perspective. In my lexicon self contradictory means false. In yours who knows. ~v~~
From: Schlock on 6 Dec 2007 00:03 On Wed, 5 Dec 2007 10:44:18 -0800 (PST), Randy Poe <poespam-trap(a)yahoo.com> wrote: >On Dec 5, 1:35 pm, Schlock <schl...(a)truthless.net> wrote: >> On Tue, 4 Dec 2007 16:47:35 -0800 (PST), Randy Poe >> >> <poespam-t...(a)yahoo.com> wrote: >> >> For if "square circles" is not a predicate combination then what is >> >> it? >> >> >A noun phrase. >> >> It's also thirteen letters of the alphabet. Hardly definitive for one >> so schooled in the black arts of dialectical evasion as yourself. > >Perhaps you would have gotten more information >relevant to your query had you included the >rest of the paragraph, the part that said >"one third of a predicate". > >And therefore not a predicate. I didn't realize predicates came in fractions. Is that anything like "a little bit pregnant"?
From: Lester Zick on 6 Dec 2007 00:03
On Wed, 5 Dec 2007 10:46:12 -0800 (PST), lwalke3(a)lausd.net wrote: >On Dec 4, 4:47 pm, Randy Poe <poespam-t...(a)yahoo.com> wrote: >> Among other things, predicates can stand alone as >> complete sentences. I will presume that you have >> a native language, and that you understand that >> "square circles" is not a complete sentence. > >It's been a while since I posted in this thread, but >let me jump into this conversation at this point. > >Square circles? > >Sure, under the standard norm, circles are not squares, >but under other norms, such as the Manhattan norm or >the maximum norm, circles are squares: > >http://en.wikipedia.org/wiki/Taxicab_geometry > >"Taxicab circles are squares with sides oriented at a >45 degree angle to the coordinate axes. A circle of >radius r for the Chebyshev distance (L_infinity >metric) on a plane is also a square with side length >2r parallel to the coordinate axes, so planar >Chebyshev distance can be viewed as equivalent by >rotation and scaling to planar taxicab distance." > >See also: > >http://en.wikipedia.org/wiki/Uniform_norm > >So here, Zick is actually correct. There is a >consistent geometry with square circles. Well as much as I appreciate the effort I was speaking in reference to Euclidean plane geometry in order to show that self contradictory predicates can be false in purely definitional contexts. ~v~~ |