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| Date | 2023-05-29 08:53 -0700 |
| References | (16 earlier) <b5a0b313-6e00-456b-a54a-319224932608n@googlegroups.com> <u3l2ve$gb4s$1@solani.org> <7a8f9c8f-3ee8-4ef2-bf30-e4d5e2f383e3n@googlegroups.com> <a05e76db-1b0b-4e9d-90a9-3f09522e169an@googlegroups.com> <e5ccd941-cd2d-4baa-a6d2-0601411e93cen@googlegroups.com> |
| Message-ID | <ecec6216-0c4c-412f-860e-bd538403a67cn@googlegroups.com> (permalink) |
| Subject | Re: Smullyan's Proof of the Drinkers Principle V3 |
| From | Mild Shock <bursejan@gmail.com> |
Why is this condition, in a set theoretic version, reasonable: s ⊆ U You can try yourself. When you do the non-set theoretic version, like below with predicate U(_) for the pun and predicate Drinker(_) for the drinkers. Then you can prove: ALL(x):U(x) => /* U is the universe */ ALL(x):[Drinker(x) => U(x)] Or symbolically using ⊆ between predicates: ALL(x):U(x) => /* U is the universe */ Drinker ⊆ U So we have again s ⊆ U respectively Drinker ⊆ U. Why was this swept under the rug, when you create your generalized drinker. Is this part of the generalization, sweeping stuff under the rug? BTW: Here is a DC Proof proof: 14 ALL(x):U(x) => ALL(x):[Drinker(x) => U(x)] Conclusion, 1 ---------------------------- cut here -------------------------------------- 1 ALL(x):U(x) Premise 2 ~ALL(x):[Drinker(x) => U(x)] Premise 3 ~~EXIST(x):~[Drinker(x) => U(x)] Quant, 2 4 EXIST(x):~[Drinker(x) => U(x)] Rem DNeg, 3 5 ~[Drinker(u) => U(u)] E Spec, 4 6 ~~[Drinker(u) & ~U(u)] Imply-And, 5 7 Drinker(u) & ~U(u) Rem DNeg, 6 8 Drinker(u) Split, 7 9 ~U(u) Split, 7 10 U(u) U Spec, 1 11 ~U(u) & U(u) Join, 9, 10 12 ~~ALL(x):[Drinker(x) => U(x)] Conclusion, 2 13 ALL(x):[Drinker(x) => U(x)] Rem DNeg, 12 14 ALL(x):U(x) => ALL(x):[Drinker(x) => U(x)] Conclusion, 1 Dan Christensen schrieb am Sonntag, 21. Mai 2023 um 17:22:30 UTC+2: > On Sunday, May 21, 2023 at 9:59:36 AM UTC-4, Mostowski Collapse wrote: > > What makes you think your nonsense of a generalized > > drinker paradox has even the slightest relevance to > > Smullyans Drinker paradox. > [snip] > > Wrong again, Mr. Collapse. Again, Smullyan informally states his Drinkers' Principle as follows: > > "There exists someone such that whenever he (or she) drinks, everybody drinks." > --Smullyan, "What is the name of this book?" p.209 > > You can formalize this statement in at least two different ways: > > (1) Using "standard" FOL implicitly that assumes a non-empty domain of discourse. If we make this assumption explicit in DC Proof, we would have something like: > > EXIST(a):U(a) => EXIST(a):[U(a) & [Drinker(a) => ALL(b):[U(b) => Drinker(b)]]] > > For any unary predicates U and Drinker. Note that every quantifier here is restricted by U. > > (2) Using ordinary set theory, can obtain: EXIST(a):[a in D => ALL(b):b in D] > > For any set D. > > Slightly more interesting might be: EXIST(a):[a in D => ALL(b):[Pub(b) => b in D] > > For any set D and unary predicate Pub. > > Ultimately you cannot prove: > > > > ALL(U):ALL(s):[Set(s) & Set(U) & Subset(s,U) => EXIST(a):[a e U & ~a e s]] > > > Again, what happens if U=s? > > There is no Russell Paradox for the Smullyan Riddle. > The resolution of Russell's Paradox is fundamental to set theory. Deal with it, Mr. Collapse. > Dan > > Download my DC Proof 2.0 freeware at http://www.dcproof.com > Visit my Math Blog at http://www.dcproof.wordpress.com
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Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 06:59 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 07:10 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-21 08:22 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 12:14 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 12:16 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 12:17 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-21 12:50 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 12:56 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-21 15:45 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-21 16:01 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 16:29 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 16:56 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-21 16:59 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-21 17:51 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-22 02:51 +0200
Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-26 14:13 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 08:53 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 08:56 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 09:03 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 09:34 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 09:38 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 09:39 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 10:24 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <janburse@fastmail.fm> - 2023-05-29 19:49 +0200
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 11:10 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 13:39 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 13:58 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 15:17 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 15:24 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-29 17:39 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 18:52 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 19:04 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-29 19:34 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-05-30 15:18 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-05 11:39 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-05 15:56 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-05 18:53 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-05 22:40 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-05 22:48 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-05 22:57 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-06 07:37 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-06 09:13 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-06 10:13 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-07 13:31 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-07 13:37 -0700
Re: Smullyan's Proof of the Drinkers Principle V3 Mild Shock <bursejan@gmail.com> - 2023-06-07 13:42 -0700
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