Re: Notation for FOL and Set Thoery.

On Feb 27, 7:29 pm, Zaljo...@xxxxxxxxx wrote:
Hi all,

The notation that is in print for FOL looks ugly and clumzy.

Yes. They admit it for Godel's notation, but they just change the
syntax and not the semantics. The real problem is that it is at the
wrong level of abstraction. A higher level is much easier to use.

here I will present better notations.

1) The logical connectives:

Negation: ~

Conjunction: &

Disjunction: ||

No, keep it 1 character as you started.

Exclusive disjunction: ||'

3 characters?? UGH

Implication: ->

Biconditional: <->

3 characters. You don't believe in brevity?

2) The universal quantifier:

The symbole will be the slash \

The slash gives the BEST representation of the universal quantifier
humans ever thought of, it just fit the mession, that no other symbole
can be imagined to do that job as well.

Any other unambiguous symbol would do. Why is it (the class of single
symbols) best? If brevity, why have you ignored brevity above?

\x  means for all x , or every x , or each x.

There is no comparison whatsoever between the clumzy symbole used in
print to represent this quantifier and the simplicity and the beauty
of the slash.

In some contexts in CBL, the syntax (choice) of the variable (name)
indicates how it is quantified, requiring no additional characters to
indicate the quantifiers. (Normal logic just lets the variations be
redundant equivalent syntaxes.)

3) The existential quantifier

The symbole will be the underscore _

\x _y

the above mean for all x, there exist y.

Now the uniques existential quantifier ( there exist exactly one )
is symbolized by putting ! after the underscore.

_!z

Why 2 symbols?

the above means  ' there exist exactly one z '

The multiple existential quantifer ' there exist more than one z '
is represented as below:

is redundant, isn't there already enuf of that?


_>1z

4) Set Theory

Membership : this is represented by the symbole ::

symbols

y :: x

should symmetrical symbols represent commutative operators?

means y is a member of x.

I like to start with the biggest in general: SET . Element

Subsethood: this is represented by the symbole :::

Why the hood? Now three??

y ::: x

means y is a subset of x.

Union: this is represented by the symbole |_|

How about putting something between the bars (instead of going to them
as you seem)?

y |_| x

menas y union x

also

|_| x

means union x.

Intersection: this is represented by the dot .

A.B

means A intersection B.

Complementation: this is represented by the symbole `

x` is the complementary of x

So lets see how Replacement can be writtin using this notation:

[ \x _!y P(x,y) ] -> \c _b  \y ( y :: b <-> _x :: c P(x,y) )

Set brackets { }

Ordered pair of x and y represented as: (x,y)

5) Mereology:

5.a: Parthood: this is represented by the symbole :.

y :. x

the above means y is a part of x

5:b:Proper parthood: this is represented by the symbole :..

y :.. x

this means y is a proper part of x.

5.c:Atomic parthood: this is represented by the symbole :-.

y :-. x

y is atomic part of x

Now we have the following definition:

\x \y ( y :-. x <-> ( y :. x & y is an atom ) )

5.d: Overlap, this is represented by symbole ,,

 ,, xy

means Overlap of x and y.

5.e: Underlap: this is represented by the symbole ;;

;; xy

means the Underlap of x and y

Example: the axioms of ZF

Extensionality

\x \y ( \z ( z :: x <-> z :: y ) -> x=y )

Empty

_x \y ~ ( y :: x )

Pairing

\r \s _x  \y ( y :: x <-> ( y=r || y=s ) )

Union

\r _x  \y ( y :: x <-> _z ( z :: r & y :: z ) )

Infinity

\x ( 0 :: x & \y ( y :: x -> y |_| {y} :: x ) )

Power

\r _x  \y ( y :: x <-> y ::: r )

Replacment

[ \x _!y P(x,y) ] -> \c _b  \y ( y :: b <-> _x :: c P(x,y) )

Regularity:

\x ( _y :: x -> _y ( y :: x & ~_c ( c :: x & c :: y ) ) ).

Or can be writtin as:

\x ( ~x=0 -> _y ( y :: x & y.x = 0 ) )

As can be seen clearly, there is no comparison between the clarity and
simplicity of this system, and the clumzyness of the conventional
notation.

Oh *** no. YOU DIDN'T CHANGE THE LEVEL OF ABSTRACTION!!!!!!

YOU JUST FIDDLED WITH THE SYNTAX!!!!

BLAH!!!!!!!!!!!

But OLD habbits die hard!

CAREFULLY, NOT HARD

I shall call the notational system depicted above Zuhair's notational
system and I shall represent it by the symbole
[Z_

So my conclusion is that

[Z_ is simpler and clearer than [convensional_

Proof?

Zuhair

How would you develop a higher level than that stupid set theory
notation, 'Z (if I may)?

You are right if you say it is stupid and clumsy. That's what
inbreeding brings us. But what are the steps in creating its
replacement?

(As a computer programmer, I have developed hundreds of formal systems
for 50 different organizations. Mathematical systems simply need to
follow the same guidelines as computerized formal systems.)

C-B
***!!!!! SAVE ME
.