Symbols in Principia Mathematica

“(x)φx” — “There exists an x for which φx is true,” or “there exists an x satisfying φx” page 16. 1st ed.

“(x)φx” — “the x which satisfies $\varphi(\hat{x})$ .” Page 31. 1st ed.

“E!(x)φx”  — “the x satisfying φx exists”. Page 32. 1st ed.

“𝜾 ‘x” — “the class whose only member is x.” page 37, 1st ed.

It is meaningless to say a set is or is not a member of itself.

A defining function $\phi(\hat{z})$ can be used to determine a set whose members are arguments that satisfy $\phi(\hat{z})$. Let $\hat{z}(\phi z)$ denote the set determined by $\phi(\hat{x})$. To say $\hat{z}$ is or is not a member of itself is to use $\hat{z}(\phi z)$ as an argument for $\phi(\hat{z})$, namely $\phi(\hat{z}(\phi(z)))$, thus violates vicious-circle principle.

An example of vicious-circle fallacy in proving identity

Define “$\large x$ is identical with $y$” as meaning “whatever is true of $x$ is true of $y$,” i.e.

$\phi(x)$ always implies $\phi(y)$.”            (1)

Dem.

If $x$ is identical with $a$ and “$x$ is identical with $y$, then $y$ is identical with a.

Although the conclusion is sound, the above reasoning embodies a vicous-circle fallacy. The following is the above demonstration written out in full:

$x$ is identical with $a$” is, by definition, “$(\phi)\cdot\phi(x) \supset \phi(a)$.”   (2)

$\vdash$.  [(1).  $\frac{ (\phi) \cdot \phi(x) \supset \phi(a) }{ \phi(x) }$ ]                        (3)

$\supset \vdash$ $(\phi) \cdot \phi(x) implies \phi(a)$ .implies.  $(\phi) \cdot \phi(y) implies \phi(a)$ .             (4)

$\supset \vdash y$ is identical to $a$

(1) is a function in the form of $f(\phi(\hat{z}), x, y)$

(2) is a function of $x$ and can be written as $\psi(x)$ in which $\phi$ is an apparent variable instead of a real variable.

In step (3), (2) is taken as a possible value of $\phi$,  but (2) involves the totality (all possible values) of $\phi$, thus a value of $\phi$ is determined by its totality and violates vicious-circle principle.

Note: A value of $\phi\hat{x}$ is obtained by assigning a $\phi$, namely $\phi(x), \psi(x), \chi(x)$, etc. A value of $\phi(x)$ is obtained by assigning an $x$, namely $\phi(a), \phi(b), \phi(c)$, etc. The latter determination is NOT used in this post.

The right sources of national pride

• Sparkling teeth;
• Beautiful women, frank and affectionate;
• Handsome men, frank and affectionate;
• Magnificent animals in natural habitats;
• High quality of living;
• Contributions to science and technology;
• Contributions to philosophy and art;
• Fresh air;
• Clean water;
• Blue sky.