Date: Feb 3, 2013 9:02 PM Author: Graham Cooper Subject: Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle<br> to Resolve Several Paradoxes 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

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AS A SUBSET

{ n | n e N } C { n | m>n }

every n here --- has a bigger m here

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AS A PROLOG PREDICATE

subset( N, nat(N), bigger(M,N) )

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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

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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

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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

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