Truth (Was:Re: PROOF INFINITY DOES NOT EXIST!...)
in [Math]
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From: Wolf K on 21 Jul 2010 10:13 On 21/07/2010 05:38, Vesa Monisto wrote: > "herbzet"<herbzet(a)gmail.com> wrote in message > news:4C463DA5.66429293(a)gmail.com... >> Vesa Monisto wrote: [a very nice distillation of how different types of relation between entities cannot be mutually mapped.] I think, however, that the proof-procedures used e.g. with transitive expressions still conform to the meta-axiom of logical truth: if some set of transitive transformations X produces state Q, and a different set of transitive transformations Y produces the same state Q, then X = Y. Eg, 3 right turns are equivalent to 1 left turn, hence 3N right turns are equivalent to N left turns. A trivial example from ordinary arithmetic asserts that x/y = Nx/Ny. That is, if you cut a pie into four equal pieces, then (ignoring errors) if you eat any two pieces, you'll eat the same as if you'd eaten the other two pieces. To put it another way: proof in any axiomatic system amounts to constructing tautologies. The proof procedures (inference rules, transformation rules, process rules, etc) are designed to make this possible. If they don't, or if they lead to inconsistencies, then the axioms and/or the proof procedures must be revised. Or so it seems to me. wolf k.
From: Vesa Monisto on 21 Jul 2010 11:38 "Wolf K" <wekirch(a)sympatico.ca> wrote in message news:7gD1o.22771$MG2.17(a)unlimited.newshosting.com... > ... > ... the proof-procedures used e.g. with transitive expressions still > conform to the meta-axiom of logical truth: > if some set of transitive transformations X produces state Q, and a > different set of transitive transformations Y produces the same state Q, > then X = > Y. Eg, 3 right turns are equivalent > to 1 left turn, hence 3N right turns are equivalent to N left turns. > ... Yes, thank you. -- Now I see that my mistake has been to take 'the transitivity' too literally, nearly at face value. -- Transverse waves do not move, they propagate. Cursors do not move, they propagate. Truth doesn't move, it propagates (some times left, some .. right ;). > Or so it seems to me. You propagated the image me, too. -- Many thanks! V.M.
From: herbzet on 22 Jul 2010 00:09 Vesa Monisto wrote: > "herbzet" wrote: > >Vesa Monisto wrote: > >> > >> Well, side-tracks, but surely you accept that there in X=Y is > >> needed even 'transitivity', If x > y And y > z Then x > z, even [if?] one > >> cannot find x containing a particular z (contained by y)). > > > > Can you give an example of what you're talking about here? > > > >> These kinds of problems > >> occur frequently in propramming -- so na�ve they seem to be. -- Nowadays > >> those old bugs are resolved e.g. by P(A(D))-structuring in C, C++, C#. > > Very kind of you to let me try to formulate my 'worries' better! Sure. > I think Witt (and Hintikka) exaggerated a bit. The picture theory of truth > in Witt's Tractatus is an aspect of experiencing 'truth', not totally > "absurd" > (Witt) or "crash" (Hintikka). I know that Hintikka had access (via von > Wright) > even to Witt's unpublished papers, so I'm confident on Hintikka's opinions. > > I tried to neglect my worry by "Nah, just a scope-resolution problem" > but didn't quite succeed. Relations can be reflexive, (anti)symmetrical, > functional, etc., and *transitive* (aRb & bRc) --> aRc, "carrying across > the middle term" (and here I had an association to MST = MetaSystem > Transition as a thread in c.ai.ph.). -- How concepts are carried to > pictures? > Do concepts really 'have, contain, carry' their 'meanings' or just trigger > receivers to decode / interpret / run them in actual (ostensive) situations? > > A simplistic (concrete) example to my toy-problem: > If a concrete closed box A contains another box B containing a 3rd box C > then I can't find the box C from the box A. (Contents are not at same > level.) > If set's A powerset P contains more elements (= combinations) than set A, > then I'm comparing cardinals of very different (exclusive sets) using very > shallow (flat) concepts (a la 'they are all just elements' or 'they all can > be > coded just by integers' or some other such an artificial 'overing' > principle). > > If I have three color-pens RGB in a cup then I have 'picking-possibilities' > called combinations 2^3, a derived set C = [RGB,RG,GB,RB,R,G,B,0] > or permutations 3! = 6, P = [RGB,RBG,GRB,GBR,BRG,BGR], but > those elements of derived sets C and P are not pens but 'picks' and 'perms'. > I could say 'who cares, cause all have countable cardinals for comparing', > but beyond the shallow overing/covering-codings there are deep-relations > programmers are trying to solve e.g. by graphs (trees). [Fof AI ;] > > If statements are seen as images of concepts, then I could take the word > image very abstractly a la "program is an image from input set to output > set". > Even 'true', I see that to be a very shallow 'picture' (a la Tractatus). > On the other hand we make pictures from input and output sets to objects. > Then we have more than just one dimension for picturing (imagination) and > if these freedoms are confused, we have serious problems (confusions). > > How to solve problems by P(A(D)) = Pointers(Addresses(Data))-structs. > Aristotle used with his syllogistics three lines picturing the scopes of > structs: > > ------------------ "All men are mortal" > ---------- "Socrates is a man" > ---- "Socrates is mortal" > > Venn used areas (disks) and I've used conics for to show even pointers: > > /\ Top = mortality (one could imagine the conic as a wigwam) > / /\ Subtop = man > / / .\ Point = Sokrates (lying on the ground of the wigwam ;) > > Now I can picture all the cases A, E, I, O by shifting the subtop (subwam); > > /\ > / /\/\/\ > / / /\/\ \ > ----------- Predication principle (dictum de omni et nullo) [I Anal. > 24b27-30] > > Now I can see pointers P as tops of those wigwams, addresses A as their > left groundpoints and data D as lying on ground, as a struct P(A(D)) or > P<)A<)D when using "<)" as an icon for 1-n -relations (run to > 1-1 -relations). > > Well, "I've Got Game" and liked to hear your wordings (concepts) to clear > my picture. (You know, I'm a yangster learned English as my 4th language.) > > -- I'll promiss not to be bothering you too much! ;) Ok -- I think your concerns as a computer guy are somewhat different than my somewhat narrower concerns as a logic hobbyist. But very well, carry on. -- hz
From: herbzet on 22 Jul 2010 02:09 Wolf K wrote: > herbzet wrote: > > Can you give an example of what you're talking about here? > > Tautologies. Note that a proof in math/logic amounts to demonstrating a > tautology: You show that X can be transformed into Y using whatever > proof procedures apply. I want to emphasize that: using whatever proof > procedures apply. Which ones apply depends on which axiomatic system you > are using. Ok -- here is a simple system, the MIU system: The MIU system consists of finite strings of the characters 'M', 'I', and 'U'; for example the following are elements of the MIU system: UIM, UUUU, IMIMU, M, IIIIII, MIU, UUUMMM, etc. There are four transformation rules: 1) If a string ends in 'I', you can append a 'U'. Examples: MII --> MIIU UI --> UIU I --> IU 2) If you have a string 'Mx', you can append 'x'. Ex: MIU --> MIUIU MUM --> MUMUM MU --> MUU MII --> MIIII 3) If 'III' occurs as a substring, you can replace it with 'U'. Ex: UMIIIMU --> UMUMU MIIII --> MUI MIIII --> MIU IMMIII --> IMU 4) If 'UU' occurs in a string, you can drop it. Ex: UUU --> U MUUI --> MI MUUUIII --> MUIII UU --> (empty string) (These four transformation rules cannot be run backwards.) There is one axiom of the MIU system: The string 'MI'. Here is a sample derivation in the system of the string "MUIIU': 1) MI axiom 2) MII rule 2 3) MIIII rule 2 4) MIIIIU rule 1 5) MUIU rule 3 6) MUIUUIU rule 2 7) MUIIU rule 4 Ok. So we have an axiomatic system with a proof procedure. Rhetorical question: is this system a system of tautologies, a system of logic, in any sense of the the terms? My answer to the rhetorical question (which, being rhetorical, does not require an answer): If there is an interpretation of this system which makes it in any sense a "logical" system, I sure don't see it. > You also, of course, assert that if X can be transformed into Y, then X > and Y are "logically equivalent". Hence "X = Y is logically true". The derivation above goes from X = MI to Y = MUIIU. The four transformation rules contain both shortening as well as lengthening rules, so it may be possible to make the reverse transformation from MUIIU to MI -- let us assume this has been done. Do we say, then, that "MI = MUIIU" is a logical truth? > NB that "true" is an undefined term: a statement is either "true" or > "false". The rules of inference in e.g. PQ (Boolean) logic are designed > to prevent transformation of a "true" sentence into a "false" one. Yes. (The question, for me, is how this magic is performed!) > But > "true" and "false" are already interpretations of PQ logic. You could > just as well name these values "red" and "yellow", or "worble" and > "gump", or "1" and "0". Yes. > IOW, whatever you like. For convenience, we use > the same names. But "a rose by any other name would smell as sweet", as > Juliet said. Sure, fine. > The values are defined in terms of the axiom of negation: You assert > axiomatically that if X has the value 1, the not-X has the value 0, and > conversely that if X has the value 0, the not-X has the value 1. These valuations occur outside the system -- they are an interpretation of the system. Ordinarily, they are not present axiomatically. (They occur, if you like, in some meta-system, if you want to get systematic about it.) (Also, in a case like the MIU system, you're going to have to define the negate of X.) > IOW, > axioms define the only "meanings" of symbolic statements. This is true, but rather tricky. > Any other > meanings ascribed to them are _interpretations_, and that's a whole > 'nother issue. IMO, logic presupposes meanings, truth, falsehood -- logic is not logic without interpretations of the system of symbols. The point of logic is that ... something ... is preserved throughout the allowed syntactic transformations and, further, that this is invariant over different interpretations. > A very difficult one, as it turns out: consider the > various interpretations of QM for example. I'd rather not, at the moment. -- hz
From: herbzet on 22 Jul 2010 09:19
herbzet wrote: > Wolf K wrote: > > The rules of inference in e.g. PQ (Boolean) logic are designed > > to prevent transformation of a "true" sentence into a "false" one. > > Yes. (The question, for me, is how this magic is performed!) The broader question of this thread, of how truth or falsity is assigned to sentences in the first place, is not my concern here. Given that a truth-value has somehow been assigned to a sentence, the question then becomes, by what magic do these particular transformation rules, the "logical" transforms, manage to pull off the trick of preserving "truth" throughout all the allowed transformations of that sentence, regardless of what truth-value was originally assigned? Let us recall that one and the same set of transforms allow: 1) the transformation of a false statement into another false statement 2) the transformation of a false statement into a true statement 3) the transformation of a true statement into another true statement but do *not* allow 4) the transformation of a true statement into a false statement. How is this done? What is the common feature of these "logical inference" rules by which they can magically distinguish and filter out "truth" from "falsehood"? Let there be no mistake, this is a very non-trivial question -- the nature of "logical implication" has been a matter of debate for thousands of years, and is very much a live question still. My answer, arrived at after a great deal of consideration (trust me), is as I posted the other day: "The only thing further I would want to say here is to broaden somewhat the proposed definition of logical truth -- not only to have the logical equivalence of statements X = Y, but also to have the logical inclusion of one statement in another. This is typically indicated with an arrow '->' and read as "implies"; thus we have X -> Y (X implies Y) which IMO is the assertion that what is meant by statement X includes what is meant by statement Y. [We might say that Y is analytic with regard to X.] "For example, "John is a bachelor" includes as part of its meaning "John is a man", so we can infer the latter from the former. [I point out: this inference is valid /regardless/ of whether 'John is a bachelor' is assigned the value "true" or "false".] "Again, it naturally falls out that if X is true, so too will Y be true (though not necessarily the reverse). We can then define the concept X = Y as meaning that X -> Y and Y -> X. "The "logical" transforms of a sentence X to a sentence Y then are those transforms which preserve some (or all) of the meaning of X in the meaning of Y, without adding anything extra." Or to put it differently, the relation of logical implication holds between two statements X and Y when what X asserts includes all of what Y asserts. That's all I really have to say. -- hz |