Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: =?UTF-8?B?QW5kcsOpIEcuIElzYWFr?= Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics Subject: =?UTF-8?Q?Re=3a_Simply_defining_G=c3=b6del_Incompleteness_and_Tarsk?= =?UTF-8?Q?i_Undefinability_away_V24_=28axiomatic_basis_of_truth=29?= Date: Sun, 5 Jul 2020 23:59:51 -0600 Organization: Christians and Atheists United Against Creeping Agnosticism Lines: 184 Message-ID: References: Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 6 Jul 2020 05:59:53 -0000 (UTC) Injection-Info: reader02.eternal-september.org; posting-host="510942d036ebb86d8e5d93351f8aba58"; logging-data="3225"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+vicosauqC6yyEMaKK/A1v" User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.14; rv:68.0) Gecko/20100101 Thunderbird/68.10.0 Cancel-Lock: sha1:xP+AghTOMsnff6u6FI/KcJ2DCQQ= In-Reply-To: Content-Language: en-US Xref: csiph.com comp.theory:21471 comp.ai.philosophy:21673 comp.ai.nat-lang:2226 On 2020-07-05 23:41, olcott wrote: > On 7/5/2020 11:58 PM, André G. Isaak wrote: >> On 2020-07-05 22:33, olcott wrote: >>> On 7/5/2020 11:06 PM, André G. Isaak wrote: >>>> On 2020-07-05 21:52, olcott wrote: >>>>> On 7/5/2020 5:28 PM, David Kleinecke wrote: >>>>>  > On Sunday, July 5, 2020 at 2:08:57 PM UTC-7, olcott wrote: >>>>>  > >>>>>  > Then Goedel showed that that there were propositions that were true >>>>>  > but not provable. >>>>>  > >>>>> >>>>> He could not have possibly shown this because the lack of >>>>> provability makes the expression unsound thus untrue. >>>> >>>> Soundness and Truth are not the same thing. Unsound does not mean >>>> untrue. >>>> >>> >>> Valid: an argument is valid if and only if it is necessary that if >>> all of the premises are true, then the conclusion is true; >>> >>> Sound: an argument is sound if and only if it is valid and contains >>> only true premises. >> >> Neither of those definitions equate soundness and truth. >> > > Soundness guarantees that the conclusion is true. > > To precisely paraphrase exactly what it says: > True premises + valid argument necessitates a true conclusion. > >>> https://web.stanford.edu/~bobonich/terms.concepts/valid.sound.html >>> >>> Even though the conclusion may be found to be true on some other >>> basis besides the unsound argument, the conclusion of any unsound >>> argument does not count ever count as true within the same chain of >>> inference. >> >> That claim is not made in the webpage you refer to, nor is it >> supported by that webpage, nor is it correct. > > Here is a more precise statement: > > Within the chain of inference from premises to conclusion ONLY a sound > argument guanatees a true conclusion. Yes. That is true. It doesn't remotely mean that true and sound are synonyms. 'true' applies to WFFs, not arguments. Sound applies to arguments, not WFFs. A sound argument is guaranteed to have a true conclusion. It does not follow from that that the conclusion of an unsound argument is untrue. > >>>>> (1) Unless you start with premises known to be true (or axioms >>>>> essentially stipulated to be true) >>>>> >>>>> (2) and have a complete inference chain from these premises (or a >>>>> formal proof) >>>>> >>>>> (3) to the conclusion (or consequence) >>>>> >>>>> (4) then the whole argument (or WFF) is >>>> >>>> Soundness is a property of arguments, not of WFFs. A WFF can be true >>>> or false, valid or invalid, but it cannot be sound or unsound. >>> >>> When mathematical logic is required to conform to the sound deductive >>> inference model then (within the same formal system) unprovable means >>> untrue. That it does not means this now only indicates that it >>> currently diverges from the sound deductive inference model. >> >> You have never formally defined this thing you call the "sound >> deductive inference model". > > Never less than 500 times. Here it is simplified: > > A sentence φ of theory T is true in T if and only if sentence T is a > theorm of theory T: True(T, φ) ↔ T ⊢ φ That isn't a definition. It is an *assertion* and one which is demonstrably false. >> And if you ever do, it will be of no relevance to any claims made by >> Gödel, Tarski, Church, or any of the other people you like to talk >> about because their claims pertain to formal systems as they are >> defined within mathematical logic. Not to "sound deductive inference >> models". > > Tarski "proved" that True(T, φ) cannot possibly ever be fully defined in > any formal system what-so-ever. He did this (crazy as it sounds) on the > basis that his theory could not prove the liar paradox. > > IT NEVER OCCURRED TO ANYONE THAT SELF-CONTRADICTORY SENTENCES ARE NOT > TRUTH BEARERS ??? You need to define 'self-contradictory' sentence and redefine 'truth-bearer' to get the above claim to work. You also need to learn the difference between completeness and consistency. A sentence such as the liar paradox is *not* something that leads to incompleteness. It leads to inconsistency. > It would then be possible to reconstruct the antinomy of the liar in the > metalanguage, by forming in the language itself a sentence x such that > the sentence of the metalanguage which is correlated with x asserts that > x is not a true sentence. > > In doing this it would be possible, by applying the diagonal procedure > from the theory of sets, to avoid all terms which do not belong to the > metalanguage, as well as all premisses of an empirical nature which have > played a part in the previous formulations of the antinomy of the liar. The above paragraph is gibberish. > http://www.liarparadox.org/247_248.pdf > >> >>>>> (5) unsound (or untrue). >>>>> >>>>> (6) The sound deductive inference model forms the correct axiomatic >>>>> basis of truth making true and unprovable totally impossible. >>>>> >>>>> Because this is as obvious as a pie in the face and I have been saying >>>> >>>> "obvious as pie in the face" and "flat-out wrong" are not the same >>>> thing. What you are claiming is flat-out wrong. Sound and True are >>>> not the same things. Provable and True are not the same things. >>>> >>>> André >>>> >>> >>> A sound argument derives a true conclusion and an unsound or invalid >>> argument does not derive a true conclusion. >>> >>> When mathematical logic is required to conform to the sound deductive >> >> Required by whom? Certainly not by you given that you have no >> authority to require anything of anyone. You can propose non-standard >> definitions to your hearts content. You can make proclamations about >> how maths *should* work to your hearts content. Mathematics will >> proceed happily along using the standard definitions and completely >> ignoring your various proclamations. >> >> André > > When the sound deductive inference model is the architectural basis of > the notion of formal system but it isn't, so this is irrelevant. > Tarski's claim that no formal system can > possibly fully define the notion of True(T, φ) is proven to be false. > > Furthermore when the sound deductive inference model is the > architectural basis of the notion of formal system True(T, φ) and but again, it isn't, so this is irrelevant. > unprovable(T, φ) cannot possibly co-occur. > > So like I have been saying for 24 threads: > Simply defining Gödel Incompleteness and Tarski Undefinability away > Your personal beliefs on what words ought to mean has absolutely no bearing on what they actually mean. It is very clear that in English, as well as in logic and maths, true and provable are not synonyms. In Robinson arithmetic, x + y = y + x is true. But it is not provable, at least not from within Q. The overwhelming majority of people believe that P = NP is false. But those people acknowledge that neither P = NP nor P ≠ NP have been proven to date. If true and provable/proven meant the same thing then the sentence 'most people believe P ≠ NP to be true though it has not been proven' would be a contradiction. Clearly it isn't. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.