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_=28TRUTH_BEARER_DEFINED=29?= Date: Sun, 12 Jul 2020 00:37:29 -0600 Organization: Christians and Atheists United Against Creeping Agnosticism Lines: 69 Message-ID: References: <87k0zc8ps5.fsf@nosuchdomain.example.com> <8d2dnRULA_sojpTCnZ2dnUU7-X_NnZ2d@giganews.com> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sun, 12 Jul 2020 06:37:31 -0000 (UTC) Injection-Info: reader02.eternal-september.org; posting-host="bc099710899c5f6aca02469e30257505"; logging-data="15326"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18NUEV+YQa6zbNaiPA0z9nb" User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.14; rv:68.0) Gecko/20100101 Thunderbird/68.10.0 Cancel-Lock: sha1:ytcNCLdKQUV5Shs/OhpT+FkmD44= In-Reply-To: Content-Language: en-US Xref: csiph.com comp.theory:21582 comp.ai.philosophy:21904 comp.ai.nat-lang:2331 On 2020-07-11 21:58, olcott wrote: > On 7/11/2020 7:57 PM, André G. Isaak wrote: >> On 2020-07-11 18:24, olcott wrote: >>> φ = (∀x ∀y (x + y = y + x)) >>> (Q ⊬ φ ∧ Q ⊬ ¬φ) Means that φ is neither true nor false in Q. >> >> No. It doesn't. You can assert this as many times as you want, but >> true and provable are not synonyms. The fact that commutativity cannot >> be proven in Q does not mean it is not true of Q. Your own personal >> definitions of 'true' are of no interest to people working in the >> field of logic. >> > > How can it shown to be true in Q ? > Show me the steps in Q that indicate it is true in Q. As has been pointed out numerous times, one must step outside of Q to demonstrate this. That's what true but not provable in Q means. Your failure to grasp this is only because you refuse to actually learn what 'true' and 'provable' mean and instead pretend they both mean the latter. >>> If φ is neither true nor false in Q then it cannot be logically >>> equivalent to anything in Q. >> >> But φ (if defined as x + y = y + x), *is* true in Q. >> >>> It is easy to see that "ice cream is a dairy product" in neither true >>> nor false in first order logic because it is not even expressible in >>> first order logic. >> Why would that not be expressible in first order logic? >> > > Even if some abstract relation such as > Is_a_type_of(ice_cream, Dairy_Product) can be stipulated first order > logic lacks any means to show whether or not this abstract relation is > satisfied. It simply does not know about ice cream or dairy products. > > FOL could be augmented with an axiom so that it can determine whether or > not Is_a_type_of(ice_cream, Dairy_Product) is satisfied, yet without > this augmentation Is_a_type_of(ice_cream, Dairy_Product) does not map to > any Boolean values in FOL, thus us not true in FOL. No. Without the addition of an axiom it is simply a contingent statement. The law of the excluded middle demands that it be either true or false. > THIS SEEMS TO BE A BRAND NEW INSIGHT INTO THE NATURE OF TRUTH BEARERS If it seems to you like some 'brand new insight', that's only because you have never bloody bothered to read even the most basic, introductory textbook on the subject of first-order logic, a subject which you nonetheless feel qualified to pontificate on. Any intro text will distinguish between statements which are valid (true in all interpretations), invalid (false in all interpretations), and CONTINGENT (true in some interpretations, false in others). ∀x(Ix → Dx) (where D = dairy product and I = ice cream) is a simple, straightforward example of a contingent statement. Its truth depends on the interpretation, but it *must* be either true or false, because that's how two-valued logics work which you would know if you actually bothered to learn one. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.