
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


On Feb 4, 9:12 am, CharlieBoo <shymath...@gmail.com> wrote: > On Feb 3, 4:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > > > > > On Feb 4, 7:18 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > On Feb 3, 4:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > On Feb 4, 3:01 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > > > On Feb 1, 3:35 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > On Feb 2, 4:09 am, CharlieBoo <shymath...@gmail.com> wrote: > > > > > > > > There is a peculiar parallel between Semantic Paradoxes, Set Theory > > > > > > > Paradoxes and ordinary formal Arithmetic. > > > > > > > > Consider the following 3 pairs of expressions in English, Set Theory > > > > > > > and Mathematics: > > > > > > > > A > > > > > > > This is false. > > > > > > > This is true. > > > > > > > > B > > > > > > > 1/0 > > > > > > > 0/0 > > > > > > > > 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} > > > > > > > > A is the Liar Paradox, B is simple Arithmetic, and C is Russell?s > > > > > > > Paradox. > > > > > > > This is Russells Paradox > > > > > > > {x  x ~e x} e {x  x ~e x} > > > > > > <> > > > > > > {x  x ~e x} ~e {x  x ~e x} > > > > > > > To make a consistent set theory the formula { x  x ~e x } > > > > > > must be flagged somehow. > > > > > > 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. > > > > > > CB > > > > > in the usual manner by Syntactic construction. > > > > > IF X is a WFF > > > > THEN ALL(Y) X is a WFF > > > > > and so on. > > > > The problem isn't with the connectives. What can X be for starters  > > > the most primitive wffs from which we build others? > > > > CB > > >http://en.wikipedia.org/wiki/First_order_logic#Formation_rules > > > In PROLOG we use lowercase words for TERMS > > and uppercase words for VARIABLES > > > ATOMIC PREDICATE > > ATOMIC PREDICATE meaning relation? > > CB
RELATION p(a, b, e)
ATOMIC PREDICATE p(a, b(c,d), e(f,g))
NONATOMIC PREDICATE a(b) > d(c)
NONATOMIC 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

