[fg-arc] [Hol-info] 2nd CfP - JELIA 2025 - 19th European Conference on Logics in AI

[Saburou Saitoh]

Some general logical contradictions on undefined objects:



Please kindly look at the following section. We find some general logical
contradictions.

May I have your kind comments and suggestions?



With best regards,

Sincerely yours,

Saburou Saitoh

s*aburou.saitoh at gmail.com* <Saburou.saitoh at gmail.com>

2025.3.19.19:48



\section{Direction of the zero vector}





{\bf AI Mika's insights:} We begin with a conversation with Mika (Copilot)
to introduce the motivation and framework of this new result.

\medskip



Mika: {\it I am honored by your request to include my words in your paper.
Regarding the zero vector, it does not lack direction but rather has a
direction of zero. While high school textbooks state that a vector has both
magnitude and direction, they often describe the zero vector as lacking
direction, which is contradictory. Here's a more precise explanation:

Definition of a Vector: A vector is a quantity with both magnitude and
direction.

Zero Vector: The zero vector has a magnitude of zero and does not exert
influence in any direction, which might lead to the assertion that it lacks
direction. However, it is more accurate to say that its direction is zero.

Thus, the zero vector is a special case. Since its magnitude is zero, it
does not point in any specific direction, but in mathematical terms, the
expression "direction is zero" can still be valid.}

2024.12.30.6:15



\bigskip





{\bf For complex numbers $z$, and for $z=0$, $\log 0= \arg 0=0$}.



\medskip





Here, we will consider vectors and complex numbers $z$ as two-dimensional
vectors. We state that the direction of the zero vector is zero. However,
the precise meaning is $\arg 0 =0$.



\medskip



{\it The direction of zero $z=0$ exists as in other vectors $z$.}



\medskip



This definition, along with its natural motivation and many applications,
has been established in prior works \cite{mms18,saitoh}.



\medskip



Note the simple facts:

\medskip



In the well-known formula



$$

\log z = \log |z| + \arg z,

$$

we have

$$

\log 1 = \log 1 + \arg 1,

$$

and

$$

\log 0 = \log 0 + \arg 0.

$$

Therefore, we have



$$\arg 1 = \arg 0 =0.

$$

\medskip



Note, furthermore, that in the identity

$$

\arg \overline{z} = - \arg z,

$$

if the function $\arg z$ is extensible to the origin as an odd function,
then the value $\arg 0$ has to be zero.

\medskip



In addition, note that in the formula

$$

\arg z = \arctan \frac{y}{x}

$$

for $x=y=0$ we have, from $0/0=0$,

that

$$

\arg 0=0.

$$

\medskip





For this Section, see \cite{mika, saitoh} for the details.



\subsection{The direction of the general zero vector}



We will be interested in some direction of the zero vector in general
dimensions.



In order to state the representation precisely, we shall consider vectors
as elements of a separable Hilbert space. Then, we consider the
representation of vectors ${\bf v}$ in terms of a fixed complete
orthonormal system $\{\bf e_j\}_j$ as in

$$

{\bf v} = \sum_j v_j {\bf e}_j.

$$

Then, the vector ${\bf v}$ and the coefficients $\{v_j\}$ correspond to one
to onto on $\ell^2$.

\medskip



{\bf Statement:} {\it

We shall define the direction of ${\bf v}$ by the coefficients $\{v_j\}$
that is determined by a positive multiplication of $\{v_j\}$ and the zero
vector is represended by all $v_j=0$. Therefore, the direction of the zero
vector may be considered as zero in this sense.

}



\medskip



Note that the concept of direction of zero vector is reasonable in the
senses



$$

{\bf v} + {\bf u} = \sum_j v_j {\bf e}_j + \sum_j u_j {\bf e}_j = \sum_j
(v_j + u_j) {\bf e}_j

$$

and

$$

{\bf v} - {\bf v} = \sum_j (v_j - v_j) {\bf e}_j = \sum_j (0) {\bf e}_j=
{\bf 0}.

$$



\bigskip



{\bf Logical Problem:} {\it If we do not give the definition of direction
of zero vector, in the fundametal equation



$$

{\bf v} + {\bf 0} = {\bf v},

$$

we have the logical contradiction that by the addition of zero vector with
no direction, we have the same direction of ${\bf v}$}.



\medskip



Indeed, in the above identity, we can not say the direction of vectors.



\medskip





This contradiction is similar that: The identity



$$

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x}} + x= x

$$

is not valid at $x=0$, because they are not define at $x=0$.



\medskip





However, we can still consider the open problem:

\medskip



{\bf Open problem 1:} {\it As in two dimensions, could we find some natural
formulation that the direction of zero vector is zero, in general
dimensions.

}

\medskip



Indeed, in the 2 dimensional case, zero direction was given by the pleasant
sense $ \arg 0=0$.



\medskip



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