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Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 3, 2013 9:02 PM
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On Feb 4, 9:12 am, Charlie-Boo <shymath...@gmail.com> wrote:
> On Feb 3, 4:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > On Feb 4, 7:18 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > > On Feb 3, 4:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > > > On Feb 4, 3:01 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > > > > On Feb 1, 3:35 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > > > > > On Feb 2, 4:09 am, Charlie-Boo <shymath...@gmail.com> wrote:
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> > > > > > > There is a peculiar parallel between Semantic Paradoxes, Set Theory
> > > > > > > Paradoxes and ordinary formal Arithmetic.
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> > > > > > > Consider the following 3 pairs of expressions in English, Set Theory
> > > > > > > and Mathematics:
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> > > > > > > A
> > > > > > > This is false.
> > > > > > > This is true.
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> > > > > > > B
> > > > > > > 1/0
> > > > > > > 0/0
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> > > > > > > C
> > > > > > > {x | x ~e x} e {x | x ~e x}
> > > > > > > {x | x e x} e {x | x ~e x}
> > > > > > > {x | x ~e x} e {x | x e x}
> > > > > > > {x | x e x} e {x | x e x}
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> > > > > > > A is the Liar Paradox, B is simple Arithmetic, and C is Russell?s
> > > > > > > Paradox.
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> > > > > > This is Russells Paradox
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> > > > > > {x | x ~e x} e {x | x ~e x}
> > > > > > <->
> > > > > > {x | x ~e x} ~e {x | x ~e x}
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> > > > > > To make a consistent set theory the formula { x | x ~e x }
> > > > > > must be flagged somehow.
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> > > > > How do you define a wff - precisely? That is the problem. Frege was
> > > > > right, Russell was wrong, and all you need is an exact (formal)
> > > > > definition of wff.
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> > > > > C-B
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> > > > in the usual manner by Syntactic construction.
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> > > > IF X is a WFF
> > > > THEN ALL(Y) X is a WFF
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> > > > and so on.
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> > > The problem isn't with the connectives. What can X be for starters -
> > > the most primitive wffs from which we build others?
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> > > C-B
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> >http://en.wikipedia.org/wiki/First_order_logic#Formation_rules
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> > In PROLOG we use lowercase words for TERMS
> > and uppercase words for VARIABLES
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> > ATOMIC PREDICATE
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> ATOMIC PREDICATE meaning relation?
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> C-B
RELATION
p(a, b, e)
ATOMIC PREDICATE
p(a, b(c,d), e(f,g))
NON-ATOMIC PREDICATE
a(b) -> d(c)
NON-ATOMIC PREDICATE
All(a) p(a, b(c,d), e(f,g))
Relational Algebra is generally used to refer to ordinary tuples of
terms. e.g SQL Tables.
QUANTIFIED LOGIC
ALL(n):N EXIST(m):N m>n
------------------
AS A SUBSET
{ n | n e N } C { n | m>n }
every n here --- has a bigger m here
-----------------
AS A PROLOG PREDICATE
subset( N, nat(N), bigger(M,N) )
------------------
subset() is not easy to program though...
You can use the LISP addon to PROLOG
or my record management addon.
PROLOG+ A CONCEPT DATABASE MANAGEMENT LANGUAGE (DBML)
We add simple breadth first extensions to PROLOG CLAUSES.
Y> next record
Y< prev record
Y>> last record
Y<< 1st record
AXIOM OF FINITE SUBSETS
-----------------------
subs(A,X,Y) <- e(A>>,X) ^ e(A,Y).
subs(A,X,Y) <- e(A,Y) ^ e(A>,X) ^ subs(A,X,Y).
ss(X,Y) <- e(A<<,X) ^ subs(A,X,Y).
This is just using PROLOG RECURSION to do a FOR LOOP
A>> last record
A> next record
A<< first record
AXIOM OF FINITE SET EQUALITY
----------------------------
equals(X,Y) <- ss(X,Y) ^ ss(Y,X).
Now you can do Set Theory and Logic all in Atomic Predicates!
In BLOCK PROLOG the above would be a rule:
equals X Y
ss X Y
ss Y X.
Herc
--
www.BLoCKPROLOG.com
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