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Jan 21, 2024, 11:42:38 AMJan 21

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I have just finished something like logic (first try, should be very

raw). Hope

this can help in dealing with several logical puzzle (e.g. Russell's

paradox), at least, in the moment.

Mostly, I would like to gather response form you to improve what

is typed, thank you.

In my opinion, many C++ programs written are actually from

'intuition', not even close to C++'s original idea

(class, concept,..,so called RAII). The is an higher level thinking

about using C++, e.g. how to translate/solve problems in C++, or

'modeling problems' and solve in C++.

----------------------------------------------------

Description::= Description is composed of a series of terms which are

composed

of discrete symbols. The form of description is indefinite, it can

be like

computer programs, mathematical expression,..,or the sentense in

this file.

Description associates to the object being described, called its

semantics.

Axiom0: Description and its semantics object are physical entities,

both occupy

different position in spacetime domain.

Axiom1: Description exists via manufraturing.

Ex: "A tree falls in a forest": This description must be made in

someone's

head. If no one is there, this description (and its object) is

a made up

in latter time (Thus, most of our knowledge are

archaeological).

Ex: "This sentence is false": Ans: Undecidable (borrow from Halting

Prolem

Theorem). Self-referential descriptions like this, including

"This

sentence is true", suffer from circular reasoning. Although the

latter

seems not to cause paradox.

Normalized Description::= A description of concated terms like

"P(a,b,..)" or

(P,a,b,..). Normalized description is like a natural sentence, but

all the

ingredient must be defined. Thus, the form is mostly regular.

The object of normalized description can be another description but

cannot

be the description itself (would cause circular reasoning). Thus,

from this

definition, we will have the chain "description ->

object(description) ->

... -> object(description) -> object". The last one is called a

terminal

description. Thus, for simplicity reasons, the description that has

no

object is also called a terminal description.

Substitute::= The term in a description that functions as a placeholder

for

another description or a parameter of the description. Thus,

substitute may

also be called as argument, parameter, or variable. In any

condition,

description that contain substitute are not terminal description.

(In

reality, terminal description may also has its object except such

object

is inexpressible).

Partial Solution::= Part of a whole solution.

Theorem: If a (whole) solution is built procedurally by accululating

parts, a

partial solution will appear in the process of creating the

solution.

+-------------+

| Proposition |

+-------------+

Proposition::= Any description that can be classified as true or false

(note

that true/false are arbitrary. This article is only concerned with

the

manipulation of symbols, not the attribution of true or false).

Proposition

can be any form of description as long as it can be classified as

true (T)

or false (F). Since we are more concerned with normalized

propositon, in the

formal presentation, some token should be added to indicate the

notion of

proposition, e.g. Prop(x,y,..).

Sometimes, descriptive proposition may be called as 'abstract

proposition'.

In contrast, the object of that description can be called the

object

proposition. If a descriptive proposition does not contain

substitute, it

can be called a terminal or atomic proposition.

+-------------------------+

| Creation of Proposition |

+-------------------------+

From Axiom1, proposition also exists via manufraturing (inversely,

proposition does not exist before manufratured).

From the observation of the descriptions we used, our knowledge can

all be

be called partial description. Such observation indicates that we

should

add procedure into 'description'. If so, the most suitable choice

of the

language for procedure should be C++, because it has the concept of

'object' and 'construction'(manufacture). Therefore, the following

article,

pseudo-C++ will be used in the sense of computation theory.

+-----+

| Set |

+-----+

Substitute in normalized description is often the substitute of the

element

of a set, e.g. the n in Prop(n∈ℕ).

Set is often expressed by using proposition, and is often defined

in way of

procedural description like the example in Peano Axioms.

+------------------------+

| Procedural Proposition |

+------------------------+

Procedural Proposition::= Proposition whose semantics is a program,

e.g.

decision function.

Postulate: Precedure is the only way to express infinite instances.

+---------------+

| Some examples |

+---------------+

Prop(∀x,P(x))::= P(x1)∧P(x2)∧..∧P(xn) (x∈{x1,x2,..})

Equ. to "∀xP(x)" in many books. If defined in Pseudo-C, then:

bool f() {

for(int x=1; x<=S.size(); ++x) {

if(P(x)==false) {

return false;

}

}

return true;

};

Universal quantifier itself is also a proposition, therefore, from

definition, its negation exists:

~Prop(∀x,P(x))= ~(P(x1)∧P(x2)∧..∧P(xn)= ~P(x1)∨~P(x2)∨..∨~P(xn)∨

= Prop(∃x,~P(x))

Prop(∃x,P(x))::= P(x1)∨P(x2)∨..∨P(xn) (x∈{x1,x2,..})

Equ. to "∃xP(x)" in many books. If defined in Pseudo-C, then:

bool f() {

for(int x=1; x<=SetX.size(); ++x) {

if(P(x)==true) {

return true;

}

}

return false;

};

Existential quantifier itself is also a proposition, therefore,

from

definition, its negation exists:

~Prop(∃x,P(x))= ~(P(x1)∨P(x2)∨..∨P(xn))= ~P(x1)∧~P(x2)∧..∧~P(xn)

= Prop(∀x,~P(x))

Prop(∃x,∀y,P(x,y))::= Prop(∀x,Prop(∃y,P(x,y))) // Concatenation:

∃x∀y:P(x,y)

= (P(x1,y1)∧P(x1,y2)∧..∧P(x1,yn))∨

(P(x2,y1)∧P(x2,y2)∧..∧P(x2,yn))∨

...

(P(xn,y1)∧P(xn,y2)∧..∧P(xn,yn))

From procedual definition:

bool f() {

for(int x=1; x<=SetX.size(); ++x) {

for(int y=1; y<=SetY.size(); ++y) {

if(P(x,y)==false) {

break;

}

}

if(y>=SetY.size()) {

return true;

}

}

return false;

}

"∃x∀y:P(x,y)" itself is also a proposition, therefore, from

definition, its

negation exists: ~(∃x∀y:P(x,y))

= (~P(x1,y1)∨~P(x1,y2)∨..∨~P∨(x1,yn))∧

(~P(x2,y1)∨~P(x2,y2)∨..∨~P∨(x2,yn))∧

...

(~P(xn,y1)∨~P(xn,y2)∨..∨~P∨(xn,yn))

= Prop(∃y,~P(x1,y))∧

Prop(∃y,~P(x2,y))∧

...

Prop(∃y,~P(xn,y))∧

= Prop(∀x,Prop(∃y,~P(x,y))) = (∀x,∃y,~P(x,y))

Prop(∀x,∃y,P(x,y))::= Prop(∀x,Prop(∃y,P(x,y))) // Concatenation:

∀x∃y:P(x,y)

= (P(x1,y1)∨P(x1,y2)∨..∨P(x1,yn))∧

(P(x2,y1)∨P(x2,y2)∨..∨P(x2,yn))∧

...

(P(xn,y1)∨P(xn,y2)∨..∨P(xn,yn))

From procedual definition:

bool f() {

for(int x=1; x<=SetX.size(); ++x) {

for(int y=1; y<=SetY.size(); ++y) {

if(P(x,y)==true) {

break;

}

}

if(y>=SetY.size()) {

return false;

}

}

return true;

}

"∀x∃y:P(x,y)" itself is also a proposition, therefore, from

definition, its

negation exists: ~(∀x∃y:P(x,y))

= (~P(x1,y1)∧~P(x1,y2)∧..∧~P(x1,yn))∨

(~P(x2,y1)∧~P(x2,y2)∧..∧~P(x2,yn))∨

...

(~P(xn,y1)∧~P(xn,y2)∧..∧P~(xn,yn))

= (∃x,∀y,~P(x,y))

-----------------------------------------------------------------------

------

raw). Hope

this can help in dealing with several logical puzzle (e.g. Russell's

paradox), at least, in the moment.

Mostly, I would like to gather response form you to improve what

is typed, thank you.

In my opinion, many C++ programs written are actually from

'intuition', not even close to C++'s original idea

(class, concept,..,so called RAII). The is an higher level thinking

about using C++, e.g. how to translate/solve problems in C++, or

'modeling problems' and solve in C++.

----------------------------------------------------

Description::= Description is composed of a series of terms which are

composed

of discrete symbols. The form of description is indefinite, it can

be like

computer programs, mathematical expression,..,or the sentense in

this file.

Description associates to the object being described, called its

semantics.

Axiom0: Description and its semantics object are physical entities,

both occupy

different position in spacetime domain.

Axiom1: Description exists via manufraturing.

Ex: "A tree falls in a forest": This description must be made in

someone's

head. If no one is there, this description (and its object) is

a made up

in latter time (Thus, most of our knowledge are

archaeological).

Ex: "This sentence is false": Ans: Undecidable (borrow from Halting

Prolem

Theorem). Self-referential descriptions like this, including

"This

sentence is true", suffer from circular reasoning. Although the

latter

seems not to cause paradox.

Normalized Description::= A description of concated terms like

"P(a,b,..)" or

(P,a,b,..). Normalized description is like a natural sentence, but

all the

ingredient must be defined. Thus, the form is mostly regular.

The object of normalized description can be another description but

cannot

be the description itself (would cause circular reasoning). Thus,

from this

definition, we will have the chain "description ->

object(description) ->

... -> object(description) -> object". The last one is called a

terminal

description. Thus, for simplicity reasons, the description that has

no

object is also called a terminal description.

Substitute::= The term in a description that functions as a placeholder

for

another description or a parameter of the description. Thus,

substitute may

also be called as argument, parameter, or variable. In any

condition,

description that contain substitute are not terminal description.

(In

reality, terminal description may also has its object except such

object

is inexpressible).

Partial Solution::= Part of a whole solution.

Theorem: If a (whole) solution is built procedurally by accululating

parts, a

partial solution will appear in the process of creating the

solution.

+-------------+

| Proposition |

+-------------+

Proposition::= Any description that can be classified as true or false

(note

that true/false are arbitrary. This article is only concerned with

the

manipulation of symbols, not the attribution of true or false).

Proposition

can be any form of description as long as it can be classified as

true (T)

or false (F). Since we are more concerned with normalized

propositon, in the

formal presentation, some token should be added to indicate the

notion of

proposition, e.g. Prop(x,y,..).

Sometimes, descriptive proposition may be called as 'abstract

proposition'.

In contrast, the object of that description can be called the

object

proposition. If a descriptive proposition does not contain

substitute, it

can be called a terminal or atomic proposition.

+-------------------------+

| Creation of Proposition |

+-------------------------+

From Axiom1, proposition also exists via manufraturing (inversely,

proposition does not exist before manufratured).

From the observation of the descriptions we used, our knowledge can

all be

be called partial description. Such observation indicates that we

should

add procedure into 'description'. If so, the most suitable choice

of the

language for procedure should be C++, because it has the concept of

'object' and 'construction'(manufacture). Therefore, the following

article,

pseudo-C++ will be used in the sense of computation theory.

+-----+

| Set |

+-----+

Substitute in normalized description is often the substitute of the

element

of a set, e.g. the n in Prop(n∈ℕ).

Set is often expressed by using proposition, and is often defined

in way of

procedural description like the example in Peano Axioms.

+------------------------+

| Procedural Proposition |

+------------------------+

Procedural Proposition::= Proposition whose semantics is a program,

e.g.

decision function.

Postulate: Precedure is the only way to express infinite instances.

+---------------+

| Some examples |

+---------------+

Prop(∀x,P(x))::= P(x1)∧P(x2)∧..∧P(xn) (x∈{x1,x2,..})

Equ. to "∀xP(x)" in many books. If defined in Pseudo-C, then:

bool f() {

for(int x=1; x<=S.size(); ++x) {

if(P(x)==false) {

return false;

}

}

return true;

};

Universal quantifier itself is also a proposition, therefore, from

definition, its negation exists:

~Prop(∀x,P(x))= ~(P(x1)∧P(x2)∧..∧P(xn)= ~P(x1)∨~P(x2)∨..∨~P(xn)∨

= Prop(∃x,~P(x))

Prop(∃x,P(x))::= P(x1)∨P(x2)∨..∨P(xn) (x∈{x1,x2,..})

Equ. to "∃xP(x)" in many books. If defined in Pseudo-C, then:

bool f() {

for(int x=1; x<=SetX.size(); ++x) {

if(P(x)==true) {

return true;

}

}

return false;

};

Existential quantifier itself is also a proposition, therefore,

from

definition, its negation exists:

~Prop(∃x,P(x))= ~(P(x1)∨P(x2)∨..∨P(xn))= ~P(x1)∧~P(x2)∧..∧~P(xn)

= Prop(∀x,~P(x))

Prop(∃x,∀y,P(x,y))::= Prop(∀x,Prop(∃y,P(x,y))) // Concatenation:

∃x∀y:P(x,y)

= (P(x1,y1)∧P(x1,y2)∧..∧P(x1,yn))∨

(P(x2,y1)∧P(x2,y2)∧..∧P(x2,yn))∨

...

(P(xn,y1)∧P(xn,y2)∧..∧P(xn,yn))

From procedual definition:

bool f() {

for(int x=1; x<=SetX.size(); ++x) {

for(int y=1; y<=SetY.size(); ++y) {

if(P(x,y)==false) {

break;

}

}

if(y>=SetY.size()) {

return true;

}

}

return false;

}

"∃x∀y:P(x,y)" itself is also a proposition, therefore, from

definition, its

negation exists: ~(∃x∀y:P(x,y))

= (~P(x1,y1)∨~P(x1,y2)∨..∨~P∨(x1,yn))∧

(~P(x2,y1)∨~P(x2,y2)∨..∨~P∨(x2,yn))∧

...

(~P(xn,y1)∨~P(xn,y2)∨..∨~P∨(xn,yn))

= Prop(∃y,~P(x1,y))∧

Prop(∃y,~P(x2,y))∧

...

Prop(∃y,~P(xn,y))∧

= Prop(∀x,Prop(∃y,~P(x,y))) = (∀x,∃y,~P(x,y))

Prop(∀x,∃y,P(x,y))::= Prop(∀x,Prop(∃y,P(x,y))) // Concatenation:

∀x∃y:P(x,y)

= (P(x1,y1)∨P(x1,y2)∨..∨P(x1,yn))∧

(P(x2,y1)∨P(x2,y2)∨..∨P(x2,yn))∧

...

(P(xn,y1)∨P(xn,y2)∨..∨P(xn,yn))

From procedual definition:

bool f() {

for(int x=1; x<=SetX.size(); ++x) {

for(int y=1; y<=SetY.size(); ++y) {

if(P(x,y)==true) {

break;

}

}

if(y>=SetY.size()) {

return false;

}

}

return true;

}

"∀x∃y:P(x,y)" itself is also a proposition, therefore, from

definition, its

negation exists: ~(∀x∃y:P(x,y))

= (~P(x1,y1)∧~P(x1,y2)∧..∧~P(x1,yn))∨

(~P(x2,y1)∧~P(x2,y2)∧..∧~P(x2,yn))∨

...

(~P(xn,y1)∧~P(xn,y2)∧..∧P~(xn,yn))

= (∃x,∀y,~P(x,y))

-----------------------------------------------------------------------

------

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