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_=28NATURE_OF_TRUTH_ITSELF=29?= Date: Sun, 12 Jul 2020 18:07:10 -0600 Organization: Christians and Atheists United Against Creeping Agnosticism Lines: 92 Message-ID: References: <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: Mon, 13 Jul 2020 00:07:13 -0000 (UTC) Injection-Info: reader02.eternal-september.org; posting-host="6e7a9754d85c63fa264627d995bced54"; logging-data="13775"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19HwGANJlVFwfjitnCpC8Hu" User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.14; rv:68.0) Gecko/20100101 Thunderbird/68.10.0 Cancel-Lock: sha1:mQDqN/qyIhuM11yN8lokcQ49TPc= In-Reply-To: Content-Language: en-US Xref: csiph.com comp.theory:21598 comp.ai.philosophy:21930 comp.ai.nat-lang:2346 On 2020-07-12 17:53, olcott wrote: > On 7/12/2020 6:21 PM, André G. Isaak wrote: >> On 2020-07-12 17:04, olcott wrote: >>> On 7/12/2020 4:37 PM, André G. Isaak wrote: >>>> On 2020-07-12 15:24, olcott wrote: >>>>> On 7/12/2020 3:31 PM, olcott wrote: >>>>>> On 7/12/2020 2:36 PM, André G. Isaak wrote: >>>>>>> On 2020-07-12 12:51, olcott wrote: >>>> >>>>>>>> ∀x ∀y (x + y = y + x) is neither true nor false in Q because it >>>>>>>> lacks an interpretation in Q. >>>>>>> >>>>>>> Which makes no sense at all unless you are using the term >>>>>>> 'interpretation' in some manner completely removed from its >>>>>>> meaning in logic. >>>>>>> >>>>>>> André >>>>>>> >>>>>> >>>>>> ∀x(Ix → Dx) (where D = dairy product and I = ice cream) is a >>>>>> simple, straightforward example of a contingent statement. >>>>>> >>>>>> ∀x(Ix → Dx) (where D and I have not been defined) is neither true >>>>>> nor false in FOL. >>>>> >>>>> The above example proves that the following definition is correct: >>>> >>>> How on earth is an example supposed to prove a definition? >>> >>> It proves that the above simple definition is sufficient (as shown >>> below) and the six page long defintion provided by Mendelson is not >>> required: >>> >>> -- Interpretation (logic) >>> -- An interpretation is an assignment of meaning to the >>> -- [non-logical] symbols of a formal language. >>> -- https://en.wikipedia.org/wiki/Interpretation_(logic) >>> >>> An interpretation is the assignment of meaning to the {D, I} >>> [non-logical] symbols of a formal language that you gave them. >>> >>> When we plug your example ∀x(Ix → Dx) >>> (where D = dairy product and I = ice cream) into that definition we >>> get a simple example of a sentence that is a truth bearer. >>> >>> When we plug an adaptation of your example ∀x(Ix → Dx) >>> (where D and I are undefined) into that definition we get a simple >>> example of a sentence that is NOT a truth bearer. >> >> Whether D is interpreted as 'dairy product', 'demon', or is simply an >> interpreted symbol has nothing to do with the meaning of >> 'interpretation'. >> >> An interpretation is simply a possible assignment of truth values to >> the entities present in a set of formulae. >> >> (A ∨ B) → C >> >> has eight possible interpretations because it has three different >> propositional symbols, each of which can be true or false. Under six >> of those interpretations, the above statement is true. Under the >> remaining two the above statement is false. >> >> Whether I intend A to stand for 'apples are red' or 'aardvarks are >> mammals' or nothing at all isn't part of the interpretation. It isn't >> something logic cares about. >> >> André > > Pick exactly one of {a,b,c}: > So then do you say that ∀x(Ix → Dx) is > (a) true, > (b) false > (c) undefined > when x,I, and D are undefined? x in the above is a bound variable. What would it even mean for it to be 'undefined'? (d) none of the above. It is true under some interpretations and false under other interpretations. Again, you simply do not understand what the term 'interpretation' means. Whenever you introduce a symbol such as I or D, you expand the set of interpretations. Nothing requires 'defining' for this to occur. André -- To email remove 'invalid' & replace 'gm' with well known Google mail service.