'When Are Relations Neither True Nor False?
in [Math]
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From: Tim Golden BandTech.com on 3 Apr 2010 02:23 On Mar 20, 11:34 pm, Newberry <newberr...(a)gmail.com> wrote: > On Mar 20, 2:44 pm, r...(a)zedat.fu-berlin.de (Stefan Ram) wrote: > > > Newberry <newberr...(a)gmail.com> writes: > > >When Are Relations Neither True Nor False? > > > (I am late to this thread, so please excuse me if I > > should repeat something that was already written.) > > > Always. A relation is a set of pairs, and a set is > > never true nor false. Therefore, every relation is > > neither true nor false. > > Your point is not well taken. From the context we see that I mean > "sentences with more than one variable." But since "when are sentences > with more than one variable neither true nor false" sounds awkward I > instead use the expression above. Here you are Newberry caught in the dimensional paradigm. When one rises to the two dimensional problem is it not apparent that the logic might rise to a continuous state? Let's go slowly, and backtrack to the one dimensional logic, which alotted a true/false value. This is consistent with the sphere in one dimension, which will give just two values: +1, -1 . Upon generalizing to two dimensions should we consider the spherical concept? Perhaps a realistic example should be played out. Let's just suppose that we have a criterion as a functional measure upon a given statement: Truth(A) where A is the statement and Truth(A) then is the boolean measure of the statement A. Is the two dimensional version of this statement Truth( A + B ) accurate where Truth( A + B ) = Truth( A ) + Truth( B ) ? Now we have attempted to formalize the logic of a dimensional construction, yet insofar as we have the following qualms + 1 + 1 = + 2 , + 1 - 1 = 0 , - 1 - 1 = - 2 then we have entered a tritype. We are left with a conditional as sensible, so that if either A or B is not true, then the result is not true. This is neither a product nor a sum, and instead requires a logic operation AND. Is this any better? At least it has only two values. Yet if we entered a multidimensional realm, even just a two dimensional realm, then we are forced to address the problem more cleanly. I suggest heading back toward the spherical implication, which causes from geometrical interpretation a continuum logic, directly from the spherical constraint. The issue of independence is apparent here. When two concepts (or logic) are independent then what right do we have to intertwine them? In effect the logic is challenging the cartesian product. I believe that this is appropriate. The problem then becomes one of multidimensional logic, and we should admit that a two dimensional logic is starkly different than a one dimensional logic, or two one dimensional logics for that matter. I believe that the correct answer is that a two dimensional logic is a continuous logic, consistent with the spherical interpretation, and so appropriate to the subject of this thread. Yet, the perception of this two-dimensional logic may actually be beyond us in modernity, even while we hem and haw on the truth of complex issues in a continuous way. I believe that the human inherently does practice a continuous truth process, though we are caught in discrete language. This is an issue for Chomsky and his ilk, as much as it is for the namby-pamby usenetters, myself included. The independence problem that I mention above applies to geometry in general, and in hindsight of the polysign system we have the option of disavowing the cartesian product, which leaves a two-dimensional realm unique from a one dimensional realm, or two one dimensional realms. The implication of such a construction is not light hearted at all. We may be lacking the final calculus that sets things straight, or rayed, as the unidirectional quality of polysign geometry would have me believe: http://bandtechnology.com/PolySigned/Lattice/Lattice.html In that the AND statement and any such combinatorial logic must marry independent values do we have a cause? Can this cause be taken to the two dimension spherical continuum? I honestly do not know, but I think that this could be approched as an open problem. Whether there is any productive work here... You tell me. - Tim
From: Daryl McCullough on 3 Apr 2010 09:54 Newberry says... >If it absolutely certain that PA is consistent why don't we formalize >the reasoning? It has been. It's easily formalized in ZFC. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 3 Apr 2010 11:39 Daryl McCullough wrote: > Newberry says... > >> If it absolutely certain that PA is consistent why don't we formalize >> the reasoning? > > It has been. It's easily formalized in ZFC. But PA's consistency itself and the proof of PA's consistency in ZFC are 2 different and independent issues. (There's a chance ZFC could be syntactically inconsistent and in which case it'd prove anything).
From: Nam Nguyen on 3 Apr 2010 12:09 Newberry wrote: > On Apr 2, 7:00 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Daryl McCullough wrote: >>> Newberry says... >>>> On Apr 1, 5:59=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>>>> How do I know that Peano arithmetic is consistent? I know it the way I >>>>> know any mathematical theorem I have personally proved. >>>> You proved PA consistent? >>> It's easy to prove in ZF. >> Is ZF _syntactically_ consistent? > > Such a proof in ZF that PA is consistent is obviously wothless. We had > a long discussion about this a while ago. Ultimately inconsistency proof is just 1st order _syntactical & finite_ proof, while consistency is neither syntactical nor finite: it's a meta proof requiring knowledge of FOL, subjective intuitions about "truth", about the natural numbers, or all of those. But in pursuing consistency proof in general, we tend not to realize a *subtle* relationship between intuition, truth, and syntactical proof. We often say a formula is a theorem iff there's a "finite" proof for it. But that isn't quite precise; it should be: "iff there's a _non-zero_ finite proof for it"! The distinction is necessary because intuition would still accept the following scenarios as valid: 1. - There's a proof which is of non-zero finite length. (This would be a typical proof). 2. - All proofs are of the zero finite length. This would correspond to to there's no proof! 3. - There's no proof of non-zero finite length length, but intuition might perceive a proof of infinite length! This is a enigma in reasoning: on the one hand intuition could have reason to believe the underlying concept _might be_ sound, but on the other hand neither the concept or its negation can be a syntactical theorem of FOL! So, if a "proof" falls into the 3rd category, the underlying formula would be neither true nor false, model-theoretically speaking or otherwise! And non-relative consistency "proofs" would fall into this 3rd category.
From: Newberry on 3 Apr 2010 12:42
On Apr 2, 3:19 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > >On Mar 31, 5:12=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> In general, any *procedure* used to evaluate the truth of sentences > >> in a self-referential language is incomplete, in the sense that there > >> are true sentences that are not evaluated as true by the procedure. > > >> This is easy to see: Let P be some procedure to evaluate the truth > >> of sentences. Then consider the sentence > > >> "When procedure P is applied to this sentence, the result is not true" > > >And what procedure would it be? > > Well, for example, the search for a proof for the statement. Or > Gaifman's procedure. > > >It cannot be Gaifman's procedure because the sentence above does > >not have the form "The sentence written in/on ... is true" > > Then, as I said, it's a true sentence that Gaifman's procedure does > not return true for. Gaifman's procedure is not applicable. It cannot take your sentence as an argument. > > -- > Daryl McCullough > Ithaca, NY |