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Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Mikko <mikko.levanto@iki.fi> Newsgroups: sci.logic Subject: Re: How a True(X) predicate can be defined for the set of analytic knowledge Date: Sat, 29 Mar 2025 12:26:00 +0200 Organization: - Lines: 106 Message-ID: <vs8hro$13ns0$1@dont-email.me> References: <vrfvbd$256og$2@dont-email.me> <vrh432$39r47$1@dont-email.me> <vrhami$3fbja$2@dont-email.me> <vrj9lu$1791p$1@dont-email.me> <vrjn82$1ilbe$2@dont-email.me> <vrmpc1$bnp3$1@dont-email.me> <vrmteo$cvat$6@dont-email.me> <vru000$33rof$1@dont-email.me> <vrug71$3gia2$6@dont-email.me> <vs0e9v$1cg8n$1@dont-email.me> <vs1fda$296sp$3@dont-email.me> <vs3b1d$3aoq$1@dont-email.me> <vs3iap$9lob$1@dont-email.me> <vs63ue$2ngoo$1@dont-email.me> <vs6vdt$39556$18@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 29 Mar 2025 11:26:01 +0100 (CET) Injection-Info: dont-email.me; posting-host="03f9301939ae027bb2a626106409e7da"; logging-data="1171328"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+kHymj9+02W1fBdh3k3pKO" User-Agent: Unison/2.2 Cancel-Lock: sha1:zsaYZjIyWhABxkkegoO2JMMaJ2M= Bytes: 5640 On 2025-03-28 20:05:17 +0000, olcott said: > On 3/28/2025 7:16 AM, Mikko wrote: >> On 2025-03-27 13:03:21 +0000, olcott said: >> >>> On 3/27/2025 5:58 AM, Mikko wrote: >>>> On 2025-03-26 18:01:14 +0000, olcott said: >>>> >>>>> On 3/26/2025 3:36 AM, Mikko wrote: >>>>>> On 2025-03-25 14:56:33 +0000, olcott said: >>>>>> >>>>>>> On 3/25/2025 5:19 AM, Mikko wrote: >>>>>>>> On 2025-03-22 17:53:28 +0000, olcott said: >>>>>>>> >>>>>>>>> On 3/22/2025 11:43 AM, Mikko wrote: >>>>>>>>>> On 2025-03-21 12:49:06 +0000, olcott said: >>>>>>>>>> >>>>>>>>>>> On 3/21/2025 3:57 AM, Mikko wrote: >>>>>>>>>>>> On 2025-03-20 15:02:42 +0000, olcott said: >>>>>>>>>>>> >>>>>>>>>>>>> On 3/20/2025 8:09 AM, Mikko wrote: >>>>>>>>>>>>>> On 2025-03-20 02:42:53 +0000, olcott said: >>>>>>>>>>>>>> >>>>>>>>>>>>>>> It is stipulated that analytic knowledge is limited to the >>>>>>>>>>>>>>> set of knowledge that can be expressed using language or >>>>>>>>>>>>>>> derived by applying truth preserving operations to elements >>>>>>>>>>>>>>> of this set. >>>>>>>>>>>>>> >>>>>>>>>>>>>> A simple example is the first order group theory. >>>>>>>>>>>>>> >>>>>>>>>>>>>>> When we begin with a set of basic facts and all inference >>>>>>>>>>>>>>> is limited to applying truth preserving operations to >>>>>>>>>>>>>>> elements of this set then a True(X) predicate cannot possibly >>>>>>>>>>>>>>> be thwarted. >>>>>>>>>>>>>> >>>>>>>>>>>>>> There is no computable predicate that tells whether a sentence >>>>>>>>>>>>>> of the first order group theory can be proven. >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> Likewise there currently does not exist any finite >>>>>>>>>>>>> proof that the Goldbach Conjecture is true or false >>>>>>>>>>>>> thus True(GC) is a type mismatch error. >>>>>>>>>>>> >>>>>>>>>>>> However, it is possible that someone finds a proof of the conjecture >>>>>>>>>>>> or its negation. Then the predicate True is no longer complete. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> The set of all human general knowledge that can >>>>>>>>>>> be expressed using language gets updated. >>>>>>>>>>> >>>>>>>>>>>>> When we redefine logic systems such that they begin >>>>>>>>>>>>> with set of basic facts and are only allowed to >>>>>>>>>>>>> apply truth preserving operations to these basic >>>>>>>>>>>>> facts then every element of the system is provable >>>>>>>>>>>>> on the basis of these truth preserving operations. >>>>>>>>>>>> >>>>>>>>>>>> However, it is possible (and, for sufficiently powerful sysems, certain) >>>>>>>>>>>> that the provability is not computable. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> When we begin with basic facts and only apply truth preserving >>>>>>>>>>> to the giant semantic tautology of the set of human knowledge >>>>>>>>>>> that can be expressed using language then every element in this >>>>>>>>>>> set is reachable by these same truth preserving operations. >>>>>>>>>> >>>>>>>>>> The set of human knowledge that can be expressed using language >>>>>>>>>> is not a tautology. >>>>>>>>>> >>>>>>>>> >>>>>>>>> tautology, in logic, a statement so framed that >>>>>>>>> it cannot be denied without inconsistency. >>>>>>>> >>>>>>>> And human knowledge is not. >>>>>>> >>>>>>> What is taken to be knowledge might possibly be false. >>>>>> >>>>>>> What actually <is> knowledge is impossibly false by >>>>>>> definition. >>>>>> >>>>>> What is presented as the body of human knowledge either is a very small >>>>>> part of actual knowledge or contains false claims. >>>>>> >>>>> >>>>> I am NOT referring to what is merely presented as the body >>>>> of general knowledge, I am referring to the actual body of >>>>> general knowledge. Within this hypothesis it is easy to see >>>>> that True(X) would be infallible. >>>> >>>> In that case your True(X) is uncomputable and any theory that contains >>>> it is incomplete. >>>> >>> >>> The body of general knowledge that can be expressed >>> using language is defined to be complete. >> >> That doesn't prevent us from presenting general knowledge that is not >> in that "complete" body. >> > > The problem of incompleteness is not inherent. The human instinct to find and report new knoledge is. -- Mikko