Truth (Was:Re: PROOF INFINITY DOES NOT EXIST!...)
in [Logic]
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From: herbzet on 20 Jul 2010 00:42 Hi, Wolf, always nice to see you in sci.logic. Wolf K wrote: > A) Formal (logical) and contingent truth > > I taught formal logic in high school, (I sneaked it in under the aim of > "teach critical thinking".) As you might expect, some students twigged > to the fact that "truth" is a vague, ambiguous, polysemous, slippery term. > > "Logical truth" is clearly defined: A statement is "logically true" when > it has the form X = Y, where X and Y are well-formed statements in some > language, and the rules of inference allow the transformation of X into > Y, and vice versa. Note that this is a characterisation of a statement. Not bad for a brief note. What we would want further is some characterisation of these inference rules (I'd just as soon call them transformation rules) that makes them "logical". I mean, one could conjure up any old transformation rule that would change a statement X to a statement Y. What is wanted is a transformation rule that preserves ... something ... throughout the transformation. The first thought is that such a logical transform from statement X to statement Y should preserve truth, or lack of it, but this would clearly be a circular procedure if our object were to define truth in the first place. Next thought is that the transform should preserve meaning -- surely, if sentence X is transformed into a sentence Y to which it is identical in meaning, then it will naturally fall out that X and Y are identical in truth value, no? Of course, "meaning" is a rather slippery concept as well -- what do we mean by "mean"? -- but actually, as far as formal logic is concerned, this has been pursued with some rigor in the branch of logic called model theory. 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. 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. 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. Hope you are well. -- hz
From: Vesa Monisto on 20 Jul 2010 04:34 "Wolf K" <wekirch(a)sympatico.ca> wrote in message news:jjZ0o.8273$Dy6.4663(a)unlimited.newshosting.com... This thread is cross-posted! -- I'll comment from a programmers point of view (comp.ai.philosophy) your text -- I mostly agree. > I don't think "exist" is a good word to use about truth. I prefer > "subsist" as the technical term. But that's a side issue. Russell neglected the Meinongian "subsistence" in his pamphlet "On Denoting", but subsistence is a term for *connoting*, of course, e.g. for "square circles". It is easy to write a program for squares having rounded vertices. -- If the side of the square is "a" and the radius for rounding corners is a/2, we have a circle, and if the radius is 0, we have a square. That's like a reaction to Berkeley's notion that we couldn't *denote* the universal triangle, only some particular exemplares. The universal triangle is [(x1,y1,z1);(x2,y2,z2);(x3,y3,z3)]. Well, you know all that, but just for fun ... (Photoshop & CoreDraw have the rounding procedure; so: nothing new.) -- As "a side issue". > "Logical truth" is clearly defined: A statement is "logically true" when > it has the form X = Y, where X and Y are well-formed statements in some > language, and the rules of inference allow the transformation of X into Y, > and vice versa. Note that this is a characterisation of a statement. Frege started from the notions A=A and A=B and got serious troubles inherited e.g. by Quine and Wittgenstein. > So, what do we mean when conceive "truth" as a property of statements? A > statement is an image of a concept. It has the same relationship to a > concept as a photograph has to its subject. Of both we say that they are > "true" if we apperceive some similarity between the statement and the > concept, the photograph and its subject. Ditto for a theory (model) and > the slice of universe it refers to. Sounds like Wittgensteinian "tautologies" being 'pictures' having their negations as 'negatives' of affirmative pictures (= 'positives'), (Tractatus). But: Later Witt said: "The assumption that a phenomenological language is possible and that only it would say what we must [want to] express in philosophy is -- I believe -- *absurd*. We must get along with ordinary language and merely understand it better." (MS 107, p.176, Oct 22. -29) "Thus the crucial assumption of the philosophy of Tractatus crashed the same week as the Wall Street -- two days before the Black Thursday." (Hintikka, "Ludwig Wittgenstein, Half-Truths and One-and-a-half-Truths", p. 112) -- 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 one cannot find x containing a particular z (contained by y)). 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#. Programmers can and will test -- even not always prove. V.M.
From: herbzet on 20 Jul 2010 20:21 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#. -- hz
From: Vesa Monisto on 21 Jul 2010 05:38 "herbzet" <herbzet(a)gmail.com> wrote in message news:4C463DA5.66429293(a)gmail.com... >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! 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! ;) V.M. (Vovon man nicht sprechen kann, dar�ber kann man ein Wigwam bauen.)
From: Wolf K on 21 Jul 2010 09:47
On 20/07/2010 20:21, 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)). Actually, the "even [if?}" condition is explicitly contradicted by Vesa's example, which is equivalent to: [(x > y) ^ (y > z)] = [x > z]. This is true for all values of x, y, z for which x > y and y > z are true. > 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. 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". 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. 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". 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. 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. IOW, axioms define the only "meanings" of symbolic statements. Any other meanings ascribed to them are _interpretations_, and that's a whole 'nother issue. A very difficult one, as it turns out: consider the various interpretations of QM for example. [...] cheers, wolf k |