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Each new user of a new system uncovers a new class of bugs. -- Kernighan

devel / comp.theory / About building a "general logic based on computation"

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.
----------------------------------------------------

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))
-----------------------------------------------------------------------
------

On Mon, 2024-01-22 at 00:10 +0800, wij wrote:
> 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.
> ----------------------------------------------------
>
> 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))
> ---------------------------------------------------------------------
> --
> ------
>

Click here to read the complete article

On Mon, 2024-01-22 at 03:15 +0800, wij wrote:
> On Mon, 2024-01-22 at 00:10 +0800, wij wrote:
> > 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.
> > ----------------------------------------------------
> >
> > 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))
> > -------------------------------------------------------------------
> > --
> > --
> > ------
> >
>
> +-------------------------------+
> > Aux. explanation in Paradoxes |
> +-------------------------------+
>     These are some explanation for paradoxes commonly seen (in *THIS
> logic).
>
> pseudo-Zeno Paradox: "While shooting the target, the arrow will pass
> infinite
>     number of points,..., the arrow won't hit the target".
>     Ans: Correct. Because the *problem statement* says so. From the
> problem
>     statement, the arrow is always at the position before the target.
> The
>     premise does not contain the information of "hitting the target",
> no valid
>     logic can lead to conclude "hit the target" is true. Basically,
> this
>     paradox has nothing to do with physics. (If related to the
> physical
> reality,
>     "pass infinite number of points" might not be true)
>
>     Note: Zeno Paradox has many versions. Like all paradoxes, the
> real
> answer
>           depends on (formal) modeling.
>     Note: 1+1/2+..+1/2^^∞≈2
> (
> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-zh.t
> xt/download
> )
>
> Rabbit Can't Outrun Turtle: Ans: Similar to the above. (some version
> actually
>     contains two premises)
>
> Supertask: Ans: NoSolution. The reason is similar to the above: The
> information
>      in the problem statement does not contain the condition at T1.
>
> Liar's Paradox: (Alread memtioned)
>
> Halting Paradox: Depend on modeling.
> -------------------------------------------------------------------
>

Click here to read the complete article

Some errors corrected, esp. in the last post. The updated version can
be downloaded from:
https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download

+------------------------------------+
| General logic based on computation |
+------------------------------------+

Description::= Description is composed of a series of finite 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, i.e. description usually comes in pairs.

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", controdict Axiom0 (known as circular
reasoning).
Although the latter seems not to cause paradox.

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, adding some token to indicate the notion of
proposition
might be necessary, e.g. (x,y,(..),..) or Prop(x,y,Prop(..),..).

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, nor
operators, it can be called a terminal or atomic proposition.

Note: Propositions can be said in 'contingency' status.
https://en.wikipedia.org/wiki/Contingency_(philosophy)

+-------------------------+
| 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
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
will use pseudo-C/C++ in the sense of computation theory.

+------------------------+
| Procedural Proposition |
+------------------------+
Procedural Proposition::= Proposition whose semantics is a program,
e.g.
decision function.

Postulate: Precedure, composed of finite symbols, is the only way to
express
infinite instances.

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;
}

+---------------------+
| Paradox Explanation |
+---------------------+
These are some explanations for paradoxes commonly seen (in *THIS
logic).

Pseudo-Zeno Paradox: "While shooting the target, the arrow will pass
infinite
number of points,..., the arrow won't hit the target".
Ans: Correct. Because the *problem statement* says so. From the
problem
statement, the arrow is always at the position before the target.
The
premise does not contain the information of "hitting the target",
no valid
logic can lead to conclude Prop("hit the target") is true.
Basically, this
paradox has nothing to do with physics. (If related to the physical
reality,
"pass infinite number of points" might not be true)

Note: Zeno Paradox has many versions. Like all paradoxes, the real
answer
depends on modeling.
Note: 1+1/2+..+1/2^^∞≈2
(https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-zh.txt/download
)

Click here to read the complete article

On 2024-01-21 16:10:34 +0000, wij said:

> 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.
> ----------------------------------------------------
>
> 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.

You should start with more fundamental concepts before you introduce
"description". Description is a character string so you should start
with "character" or "string".

> 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.

Before any axioms you should introduce the concepts that used in
axioms. Usually "exists" comes from logic but if you are creating
a new system of logic you cannot use that.

> 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.

Mikko

On Sat, 2024-01-27 at 11:49 +0200, Mikko wrote:
> On 2024-01-21 16:10:34 +0000, wij said:
>
> > 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.
> > ----------------------------------------------------
> >
> > 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.
>
> You should start with more fundamental concepts before you introduce
> "description". Description is a character string so you should start
> with "character" or "string".

'description' (whatever it is), is actually not definable as a basic
term. It can be described, though. I used 'term', but did not describe
what it is. In general, many things used in 'general logic' are not
clearly specified, e.g. This 'general logic' is based on many that
can be find in "Boolean Algebra"
https://en.wikipedia.org/wiki/Boolean_algebra

If "character" is used as the basic, I am afraid, "description" will be
about a collection of characters, nothing else. And this part is
already well described by "formal language". That is what I thought.

> > 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.
>
> Before any axioms you should introduce the concepts that used in
> axioms. Usually "exists" comes from logic but if you are creating
> a new system of logic you cannot use that.

But where is the "exist" in the logic you said from?
(I am already defining some kind of logic)

> > 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.
>
> Mikko
>

Subject	Author
About building a "general logic based on computation"	wij
Re: About building a "general logic based on computation"	wij
Re: About building a "general logic based on computation"	wij
Re: About building a "general logic based on computation"	wij
Re: About building a "general logic based on computation"	Mikko
Re: About building a "general logic based on computation"	wij