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Re: Smullyan's Proof of the Drinkers Principle V3

Newsgroups sci.logic
Date 2023-05-29 19:04 -0700
References (18 earlier) <2f3bdb06-2660-4304-adf7-6baf015c3129n@googlegroups.com> <39d944fc-40a3-4b53-b910-758de1944545n@googlegroups.com> <5a63fe01-7e52-46b8-abdf-768d264a2ca1n@googlegroups.com> <9fb38325-1197-4eb8-8ab0-81576941ac5dn@googlegroups.com> <f7567415-7e88-43aa-b51d-277b8d5e8e8cn@googlegroups.com>
Message-ID <2d5043f7-260a-4756-ba9d-b9a785b8b819n@googlegroups.com> (permalink)
Subject Re: Smullyan's Proof of the Drinkers Principle V3
From Dan Christensen <Dan_Christensen@sympatico.ca>

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On Monday, May 29, 2023 at 8:39:08 PM UTC-4, Mild Shock wrote:

> > > Conclusion it requires more work, to go from predicates
> > > to sets, than blindly replacing each predicate by a set
> > > membership.

> > I never claimed otherwise. In set theory, you have more tools at your disposal.

> > > It requires more when you want to
> > > preserve validity and invalidity, i.e. when you want to
> > > prevent false positives in the new formalization with sets.
> > [snip]
> >
> > "False positives?" With more tools, you can supposedly prove things that you could not prove without them.

> But the term "generalization" is wrongly applied.

Wrong.

> You didn't do
> a generalization. A generalization doesn't have false positives
> and only false positives. A generalization can be specialized,
>
> so that the old result can be obtained.

This is precisely what I have done here: http://www.dcproof.com/STGeneralizedDrinkersThm.htm

Here again, is Smullyan's informal definition of his Drinkers' Principle:

"[T]here exists someone such that whenever he (or she) drinks, everybody drinks." (1)

Using only your "standard" FOL modelling of Smullyan's DP, this translates EXIST(x):[D(x) => ALL(y):D(y)] for any unary predicate D where D(x) means x is a drinker.

Using ordinary set theory and rules of logic (more tools!), in addition to (1), we can generalize:

"[T]here exists someone such that whenever he (or she) drinks, anything whatsoever can be true." (2)

More formally: EXIST(x):[x in d => Q] for any set d and proposition Q, be it true or false

The "old result" is just the SPECIAL CASE of Q = ALL(y):y in d. Again, Q could be ANY proposition whatsoever, even Q = ~ALL(y):y in d, the negation of special case! AFAICT this insight is impossible using only FOL.

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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