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Two versions of the Drinker's Paradox
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Dan Christensen
Jun 4, 2014, 12:47:54 PM6/4/14
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Here are presented two versions of the Drinker's Paradox/Theorem for comparison.
VERSION 1
From Wikipedia:
The drinker paradox (also known as drinker's principle, drinkers' principle or (the) drinking principle) is a theorem of classical predicate logic, usually stated in natural language as: There is someone in the pub such that, if he is drinking, everyone in the pub is drinking.
The actual theorem is
Ex:[D(x) -> Ay:D(y)]
where D is an arbitrary predicate.
The paradox was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book "What Is the Name of this Book?"
Source: "Drinker's Paradox," http://en.wikipedia.org/wiki/Drinker%27s_paradox
Here, D(x) means that x is drinking.
Implicit is that the domain of discussion is the people in some pub and that there is at least one person there. Version 2 makes these hidden assumptions explicit.
VERSION 2
Ex in P => Ey in P: [y in D => Az in P: z in D]
where P and D are arbitrary sets.
Here, P is the set of people in a pub (giving a name to the implicit domain of discussion in Version 1). D is the set of drinkers.
That the domain of discussion is non-empty is given explicitly by Ex in P.
We can also generalize to obtain:
A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
Disadvantage of Version 2:
1. More symbols. P, D, and set membership vs. D
Advantages of Version 2:
1. No hidden assumptions -- assumptions that most mathematics students would be unaware. Everything is made explicit to minimize confusion.
2. Result can be formally generalized in first-order logic. No implicit quantifying over predicates.
See "The Drinker's Paradox" at my blog http://www.dcproof.wordpress.com (dated June 3, 2014)
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my new Math Blog at http://www.dcproof.wordpress.com
VERSION 1
From Wikipedia:
The drinker paradox (also known as drinker's principle, drinkers' principle or (the) drinking principle) is a theorem of classical predicate logic, usually stated in natural language as: There is someone in the pub such that, if he is drinking, everyone in the pub is drinking.
The actual theorem is
Ex:[D(x) -> Ay:D(y)]
where D is an arbitrary predicate.
The paradox was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book "What Is the Name of this Book?"
Source: "Drinker's Paradox," http://en.wikipedia.org/wiki/Drinker%27s_paradox
Here, D(x) means that x is drinking.
Implicit is that the domain of discussion is the people in some pub and that there is at least one person there. Version 2 makes these hidden assumptions explicit.
VERSION 2
Ex in P => Ey in P: [y in D => Az in P: z in D]
where P and D are arbitrary sets.
Here, P is the set of people in a pub (giving a name to the implicit domain of discussion in Version 1). D is the set of drinkers.
That the domain of discussion is non-empty is given explicitly by Ex in P.
We can also generalize to obtain:
A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
Disadvantage of Version 2:
1. More symbols. P, D, and set membership vs. D
Advantages of Version 2:
1. No hidden assumptions -- assumptions that most mathematics students would be unaware. Everything is made explicit to minimize confusion.
2. Result can be formally generalized in first-order logic. No implicit quantifying over predicates.
See "The Drinker's Paradox" at my blog http://www.dcproof.wordpress.com (dated June 3, 2014)
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my new Math Blog at http://www.dcproof.wordpress.com
Peter Percival
Jun 4, 2014, 1:06:08 PM6/4/14
to
Dan Christensen wrote:
> Here are presented two versions of the Drinker's Paradox/Theorem for comparison.
>
> VERSION 1
>
> From Wikipedia:
>
> The drinker paradox (also known as drinker's principle, drinkers' principle or (the) drinking principle) is a theorem of classical predicate logic, usually stated in natural language as: There is someone in the pub such that, if he is drinking, everyone in the pub is drinking.
>
> The actual theorem is
>
> Ex:[D(x) -> Ay:D(y)]
>
> where D is an arbitrary predicate.
>
> The paradox was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book "What Is the Name of this Book?"
>
> Source: "Drinker's Paradox," http://en.wikipedia.org/wiki/Drinker%27s_paradox
>
> Here, D(x) means that x is drinking.
>
> Implicit is that the domain of discussion is the people in some pub and that there is at least one person there. Version 2 makes these hidden assumptions explicit.
>
> VERSION 2
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> where P and D are arbitrary sets.
>
> Here, P is the set of people in a pub (giving a name to the implicit domain of discussion in Version 1). D is the set of drinkers.
>
> That the domain of discussion is non-empty is given explicitly by Ex in P.
>
> We can also generalize to obtain:
>
> A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
>
> Disadvantage of Version 2:
>
> 1. More symbols. P, D, and set membership vs. D
>
> Advantages of Version 2:
>
> 1. No hidden assumptions -- assumptions that most mathematics students would be unaware. Everything is made explicit to minimize confusion.
Mathematics has nothing to say about drinkers, pubs, etc.
> Here are presented two versions of the Drinker's Paradox/Theorem for comparison.
>
> VERSION 1
>
> From Wikipedia:
>
> The drinker paradox (also known as drinker's principle, drinkers' principle or (the) drinking principle) is a theorem of classical predicate logic, usually stated in natural language as: There is someone in the pub such that, if he is drinking, everyone in the pub is drinking.
>
> The actual theorem is
>
> Ex:[D(x) -> Ay:D(y)]
>
> where D is an arbitrary predicate.
>
> The paradox was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book "What Is the Name of this Book?"
>
> Source: "Drinker's Paradox," http://en.wikipedia.org/wiki/Drinker%27s_paradox
>
> Here, D(x) means that x is drinking.
>
> Implicit is that the domain of discussion is the people in some pub and that there is at least one person there. Version 2 makes these hidden assumptions explicit.
>
> VERSION 2
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> where P and D are arbitrary sets.
>
> Here, P is the set of people in a pub (giving a name to the implicit domain of discussion in Version 1). D is the set of drinkers.
>
> That the domain of discussion is non-empty is given explicitly by Ex in P.
>
> We can also generalize to obtain:
>
> A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
>
> Disadvantage of Version 2:
>
> 1. More symbols. P, D, and set membership vs. D
>
> Advantages of Version 2:
>
> 1. No hidden assumptions -- assumptions that most mathematics students would be unaware. Everything is made explicit to minimize confusion.
> 2. Result can be formally generalized in first-order logic. No implicit quantifying over predicates.
1. Irrelevances introduced.
2. More complicated than it needs to be.
3. Goes belong pure predicate calculus and introduces set theory.
Your version needs the language of set theory. Pure first order logic
may have binary predicates but none of them is properly read as "is an
element of". Rather, they are read "R", "Q", etc. Just that without
interpretation. Meanwhile the drinker's paradox only needs one monadic
predicate.
It seems to me that the "paradox" is (Ex)(D(x) -> (Ay)D(y)) and talk of
drinkers just allows it to be expressed in familiar language. Once
drinkers are mentioned people in a pub follow on. But if you forget
about the familiar expression it is just (Ex)(D(x) -> (Ay)D(y)) and your
version is far to elaborate. What is paradoxical about (Ex)(D(x) ->
(Ay)D(y)) is inherited, so to speak, from the "paradoxes" of material
implication
--
Nam Nguyen in sci.logic in the thread 'Q on incompleteness proof'
on 16/07/2013 at 02:16: "there can be such a group where informally
it's impossible to know the truth value of the abelian expression
Axy[x + y = y + x]".
Dirk Van de moortel
Jun 4, 2014, 1:14:23 PM6/4/14
to
The formal statements
Ex in P => Ey in P: [y in D => Az in P: z in D]
and
A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
do not really explicitly give that the domain of discussion is non-empty.
They just say what follows *if* the domain of discussion is non-empty.
Right?
Dirk Vdm
Message has been deleted
Dan Christensen
Jun 4, 2014, 1:44:57 PM6/4/14
to
On Wednesday, June 4, 2014 1:06:08 PM UTC-4, Peter Percival wrote:
> > Advantages of Version 2:
>
> >
>
> > 1. No hidden assumptions -- assumptions that most mathematics students would be unaware. Everything is made explicit to minimize confusion.
>
>
>
> Mathematics has nothing to say about drinkers, pubs, etc.
>
Just beer mugs, tables and chairs? (Hilbert)
> > Advantages of Version 2:
>
> >
>
> > 1. No hidden assumptions -- assumptions that most mathematics students would be unaware. Everything is made explicit to minimize confusion.
>
>
>
> Mathematics has nothing to say about drinkers, pubs, etc.
>
>
>
> > 2. Result can be formally generalized in first-order logic. No implicit quantifying over predicates.
>
>
>
> Disadvantages:
>
> 1. Irrelevances introduced.
>
> 2. More complicated than it needs to be.
>
> 3. Goes belong pure predicate calculus and introduces set theory.
>
Dan Christensen
Jun 4, 2014, 2:32:21 PM6/4/14
to
On Wednesday, June 4, 2014 1:14:23 PM UTC-4, Dirk Van de moortel wrote:
> I have a quibble about something you wrote in version 2.
>
> The formal statements
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> and
>
> A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
>
> do not really explicitly give that the domain of discussion is non-empty.
>
True. The set P corresponds only roughly to the so-called domain of discussion in Version 1.
> I have a quibble about something you wrote in version 2.
>
> The formal statements
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> and
>
> A P,D:[Ex in P => Ey in P: [y in D => Az in P: z in D]]
>
> do not really explicitly give that the domain of discussion is non-empty.
>
> They just say what follows *if* the domain of discussion is non-empty.
>
> Right?
>
William Elliot
Jun 4, 2014, 10:42:33 PM6/4/14
to
On Wed, 4 Jun 2014, Dan Christensen wrote:
> Ex:[D(x) -> Ay:D(y)]
> where D is an arbitrary predicate.
Ex.[D(x) -> Ay.D(y)]
> Ex:[D(x) -> Ay:D(y)]
> where D is an arbitrary predicate.
iff
Ex.~D(x) v Ay.D(y)
iff
~Ax.D(x) v Ay.D(y)
iff
T
Lay off the drinking, you're already dead drunk.
Dan Christensen
Jun 4, 2014, 11:01:04 PM6/4/14
to
On Wednesday, June 4, 2014 10:42:33 PM UTC-4, William Elliot wrote:
> On Wed, 4 Jun 2014, Dan Christensen wrote:
>
>
>
> > Ex:[D(x) -> Ay:D(y)]
>
> > where D is an arbitrary predicate.
>
>
>
> Ex.[D(x) -> Ay.D(y)]
>
> iff
>
> Ex.~D(x) v Ay.D(y)
>
Should be Ex.[~D(x) v Ay.D(y)]
> On Wed, 4 Jun 2014, Dan Christensen wrote:
>
>
>
> > Ex:[D(x) -> Ay:D(y)]
>
> > where D is an arbitrary predicate.
>
>
>
> Ex.[D(x) -> Ay.D(y)]
>
> iff
>
> Ex.~D(x) v Ay.D(y)
>
> iff
>
> ~Ax.D(x) v Ay.D(y)
>
Should be ~Ax.~[~D(x) v Ay.D(y)]
I hope this helps.
William Elliot
Jun 4, 2014, 11:41:20 PM6/4/14
to
On Wed, 4 Jun 2014, Dan Christensen wrote:
> >
> > > Ex:[D(x) -> Ay:D(y)]
> >
> > > where D is an arbitrary predicate.
> >
> > Ex.[D(x) -> Ay.D(y)]
> > iff
> > Ex.~D(x) v Ay.D(y)
> Should be Ex.[~D(x) v Ay.D(y)]
Onloy because you don't know the disjunctive theorem.
> >
> > > Ex:[D(x) -> Ay:D(y)]
> >
> > > where D is an arbitrary predicate.
> >
> > Ex.[D(x) -> Ay.D(y)]
> > iff
> > Ex.~D(x) v Ay.D(y)
> Should be Ex.[~D(x) v Ay.D(y)]
> > iff
> > ~Ax.D(x) v Ay.D(y)
> Should be ~Ax.~[~D(x) v Ay.D(y)]
Oh, yea, yea. To spell it out for the remedial,
> > ~Ax.D(x) v Ay.D(y)
> Should be ~Ax.~[~D(x) v Ay.D(y)]
the proof of the disjuctionve theorem.
iff ~Ax.(D(x) & ~Ay.D(y))
iff ~[Ax.D(x) & Ax.~Ay.D(y))
iff ~[Ax.D(x) & ~Ay.D(y))
iff ~Ax.D(x) v ~~Ay.D(y)
iff ~Ax.D(x) v Ax.D(x)
iff T
> I hope this helps.
Not at all. I hope the disjuctive theorem helps you out.
Dan Christensen
Jun 5, 2014, 12:44:27 AM6/5/14
to
On Wednesday, June 4, 2014 11:41:20 PM UTC-4, William Elliot wrote:
> On Wed, 4 Jun 2014, Dan Christensen wrote:
>
> > >
>
> > > > Ex:[D(x) -> Ay:D(y)]
>
> > >
>
> > > > where D is an arbitrary predicate.
>
> > >
>
> > > Ex.[D(x) -> Ay.D(y)]
>
> > > iff
>
> > > Ex.~D(x) v Ay.D(y)
>
> > Should be Ex.[~D(x) v Ay.D(y)]
>
> Onloy because you don't know the disjunctive theorem.
>
True. Never heard of it, but thanks.
> On Wed, 4 Jun 2014, Dan Christensen wrote:
>
> > >
>
> > > > Ex:[D(x) -> Ay:D(y)]
>
> > >
>
> > > > where D is an arbitrary predicate.
>
> > >
>
> > > Ex.[D(x) -> Ay.D(y)]
>
> > > iff
>
> > > Ex.~D(x) v Ay.D(y)
>
> > Should be Ex.[~D(x) v Ay.D(y)]
>
> Onloy because you don't know the disjunctive theorem.
>
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my new Math Blog at http://www.dcproof.wordpress.com
William Elliot
Jun 5, 2014, 4:38:25 AM6/5/14
to
On Wed, 4 Jun 2014, Dan Christensen wrote:
> VERSION 2
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
> where P and D are arbitrary sets.
Px for x in P, Dx for x in D.
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
> where P and D are arbitrary sets.
Ex.Px -> Ey.(Py & (Dy -> Az.(Pz -> Dz)))
Case: Az.(Pz -> Dz)
Case: Ez.(Pz & ~Dz)
Both of those are easy with Rule C of Rosseur's "Logic for Mathematicians."
I'm not searching though your stream of thought.
Give the exact location such as
www.dcproof.wordpress.com/Drinkers.Paradox
Dan Christensen
Jun 8, 2014, 2:32:25 PM6/8/14
to
On Thursday, June 5, 2014 4:38:25 AM UTC-4, William Elliot wrote:
> On Wed, 4 Jun 2014, Dan Christensen wrote:
>
>
>
> > VERSION 2
>
> >
>
> > Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> > where P and D are arbitrary sets.
>
>
>
> Px for x in P, Dx for x in D.
>
>
>
> Ex.Px -> Ey.(Py & (Dy -> Az.(Pz -> Dz)))
>
>
>
> Case: Az.(Pz -> Dz)
>
> Case: Ez.(Pz & ~Dz)
>
>
>
> Both of those are easy with Rule C of Rosseur's "Logic for Mathematicians."
>
Can't get much easier than the proof I posted. Again, see the entry "The Drinker's Paradox" (dated June 3, 2014) at my math blog.
> On Wed, 4 Jun 2014, Dan Christensen wrote:
>
>
>
> > VERSION 2
>
> >
>
> > Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> > where P and D are arbitrary sets.
>
>
>
> Px for x in P, Dx for x in D.
>
>
>
> Ex.Px -> Ey.(Py & (Dy -> Az.(Pz -> Dz)))
>
>
>
> Case: Az.(Pz -> Dz)
>
> Case: Ez.(Pz & ~Dz)
>
>
>
> Both of those are easy with Rule C of Rosseur's "Logic for Mathematicians."
>
Dan Christensen
Jun 10, 2014, 12:55:26 AM6/10/14
to
On Wednesday, June 4, 2014 12:47:54 PM UTC-4, Dan Christensen wrote:
>
>
> VERSION 2
>
>
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
>
Revised: Ex in P & D subset of P => Ey:[y in D => P = D]
>
>
> VERSION 2
>
>
>
> Ex in P => Ey in P: [y in D => Az in P: z in D]
>
A bit more streamlined.
See "Drinker's Paradox" at my blog, dated June 3 (revised June 10).
Dan Christensen
Jun 10, 2014, 8:48:39 AM6/10/14
to
On Tuesday, June 10, 2014 12:55:26 AM UTC-4, Dan Christensen wrote:
> On Wednesday, June 4, 2014 12:47:54 PM UTC-4, Dan Christensen wrote:
>
>
>
> >
>
> >
>
> > VERSION 2
>
> >
>
> >
>
> >
>
> > Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> >
>
>
>
> Revised: Ex in P & D subset of P => Ey:[y in D => P = D]
>
No, the rules of logic are not all screwed up!
> On Wednesday, June 4, 2014 12:47:54 PM UTC-4, Dan Christensen wrote:
>
>
>
> >
>
> >
>
> > VERSION 2
>
> >
>
> >
>
> >
>
> > Ex in P => Ey in P: [y in D => Az in P: z in D]
>
> >
>
>
>
> Revised: Ex in P & D subset of P => Ey:[y in D => P = D]
>
In mathematics, outside of pure set theory, an existentially quantified statement is almost always implicitly of the form Ex:[x in S & P(x))], not as above with Ex:[x in S => Q] where x is not free in Q. This unusual form should tip you off that something weird, even "paradoxical" may be happening.
Dan Christensen
Jun 11, 2014, 3:04:29 PM6/11/14
to
From the Conclusion to "The Drinker's Paradox" from my blog:
Are there, as some have suggested, some cases in which formal logic fails us even in mathematics? This is not the final word, I am sure, but as far as I can tell, our system of logic works perfectly, even if it is sometimes seems at odds with our casual and imprecise usage in everyday speech.
As far as mathematics is concerned, it seems to me that the "problem," if you can call it that, is centered on a form of existential generalization that is rarely used in mathematics:
Ex:[x in S => Q] (1)
where S is a set and Q is a logical expression with no reference to x (as in The Drinker's Theorem).
In mathematics, we are almost always talking about the elements of some underlying set, e.g. the set of natural numbers in number theory, or the set of points in the Euclidean plane in geometry. Existentially quantified statements are then almost always implicitly of the form:
Ex:[x in S & Q(x)] (2)
where S is some set and Q(x) is a logical expression making reference to x.
Note that, in form (1), we have an existential generalization on a conditional (or implication) statement - very unusual in mathematics. In form (2), we have the more usual form of existential generalization on a conjunction (or AND-statement).
Although you will probably never see statements of the form (1) in any published mathematical proof, readers should be aware of the seemingly paradoxical results and interpretations that can arise from them - remember The Drinker's Paradox.
Visit my Math Blog at http://www.dcproof.wordpress.com
Are there, as some have suggested, some cases in which formal logic fails us even in mathematics? This is not the final word, I am sure, but as far as I can tell, our system of logic works perfectly, even if it is sometimes seems at odds with our casual and imprecise usage in everyday speech.
As far as mathematics is concerned, it seems to me that the "problem," if you can call it that, is centered on a form of existential generalization that is rarely used in mathematics:
Ex:[x in S => Q] (1)
where S is a set and Q is a logical expression with no reference to x (as in The Drinker's Theorem).
In mathematics, we are almost always talking about the elements of some underlying set, e.g. the set of natural numbers in number theory, or the set of points in the Euclidean plane in geometry. Existentially quantified statements are then almost always implicitly of the form:
Ex:[x in S & Q(x)] (2)
where S is some set and Q(x) is a logical expression making reference to x.
Note that, in form (1), we have an existential generalization on a conditional (or implication) statement - very unusual in mathematics. In form (2), we have the more usual form of existential generalization on a conjunction (or AND-statement).
Although you will probably never see statements of the form (1) in any published mathematical proof, readers should be aware of the seemingly paradoxical results and interpretations that can arise from them - remember The Drinker's Paradox.
Visit my Math Blog at http://www.dcproof.wordpress.com
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