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Path: 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: Tue, 25 Mar 2025 12:19:44 +0200 Organization: - Lines: 65 Message-ID: <vru000$33rof$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> MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 25 Mar 2025 11:19:45 +0100 (CET) Injection-Info: dont-email.me; posting-host="2f0cf999c00fadfadc7508b49d8fc2da"; logging-data="3272463"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19zGkjTKXjJ7zayyJplMIwJ" User-Agent: Unison/2.2 Cancel-Lock: sha1:/7V9FbqVjavwK0t3e8gngjbf+H8= 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. -- Mikko