Re: Notation for FOL and Set Thoery.

On Feb 27, 7:34 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
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- Hide quoted text -

- Show quoted text -

I admit that the notational system above didn't follow fixed rules.

The rules that I would like them to be present in a notational system
are:

1) symboles should not be oriented in opposite manner to the familier
symboles present in english.

According to this rule we see that conventional universal quantifier
which looks like upside down upper case 'A' , and also existential
quantifier which looks like upper case 'E' that is turned to the
opposite direction , are NOT acceptable.

2) symboles that reflect very primitive concepts should be composed of
one character

3) the more complex the concept the more characters the symbole that
represent it shall have

4) If a concept is symmetrical then the symbol that represent it
should be non directional.

5) if the concept is not symmetrical then it should be symbolized by a
directional symbole.

6) the symbole should have a certain explanation related to the
concept it symblizes.

7) the symbole '&' to represent conjunction is the ONLY exception of
the above rules.


Example: @ if we use it to represent membership, this symboles gives
the impression of containement which is the sole concept of
membership.

I mean if we have a concept like for example equality

a equal b is the same always to b equal a

then equality should be symbolized by a symbol that is not directional
like =

( a symbole is not directional if you turn it to the opposite
direction you will yield the same symbole, so symbols like: = , ! , /
\ , \/ , A, <> ,<-> , <=> , . , : , | are non directional symboles
while symbols like -> , \ , > , |' , ~ , & are directional symboles
since they don't look like <- , / , < , '| . )

Now let me answer of of Charlie-Boo questions:

|| to represent 'or' is composed of two characters because
disjunction is a more complex concept than negation and conjunction.

Now exclusive or should be composed of three characters since it is a
more complex concept than or, but since disjunction is non directional
then it should be symbolized by a symbole that is not direction so the
symbol ||| would be better than
the one I proposed as ||' since this is direction.

So Correction: xor : |||

The problem is that and is symbolized by a directional symbole that &,
while conjunction is not a directional concept, but this will be an
exception of the rules since & is one symbol and since & gives the
impression of tieying things together and since it is used a lot to
symbolize 'and' so we can use it.

Negation is not directional in my opinion therefore it is correctly
symbolized here.

Implication is a more complex concept than conjunction and negation
and it has the same degree of complexity of disjunction therefore it
should be composed of two characters.

biconditional is build from the uniconditional ( implication ) and it
is non directional so it is correctly symbolized in the system above.

About Membership it is incorrectly symbolized since it is asymmetrical
relation and it was symbolized by :: which is non direction and this
is false, it should be symbolized by a direction one character
symbole. the only one that I can see it suitable seems to be actually
@. since it is directional and it gives the impression that
containement.

So I will change membership symbole to @

Correction:

Membership : @

Also subsethood is not directional and it should be symbolized by a
non directional symbole But it should be composed of more than two
characters, since its definition depends on membership ,
quantification , implication
So it should be symbolized by a non directional symbole composed of
Four characters.

I should revise the system according to the new rules above.

Let me see what I can come with.

Zuhair










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