Wax On – Wax Off¶

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Author:	Wolfgang Scherer

The way of the hero: “wax on – wax off … and you will shine one day!” – Isn’t it such a sweet thought.

—Not The Karate Kid

Abstract

Understanding the structure of logic defies the narcissistic delusion that the human brain can do fabulous and wondrous things\(\tm\), which is particulary and painfully false.

Or as a dear relative phrased it:

Well, they say “You can lead a horse to the water …”, but these people – they do not just not drink – they turn around and shit in the water. At least by the time they reach their master’s degreee, they have slowly come to realize that they don’t know shit – that all their ideas are shit – and that they need to behave accordingly.

A Tautology To Prove a Paradox¶

I know, that I cannot know.

—Neither Socrates nor Kant

Hypothesis: The vast desert of NAND is utterly incomprehensible.

Let there be proof, let \(\neg\) be NOT, \(\wedge\) be AND, \(\vee\) be OR, \(\rightarrow\) be IF, \(\uparrow\) be NAND, as defined by the following excerpt of the truth tables for binary truth functions:

		\(f^{2}_{12}\)	\(f^{2}_{10}\)	\(f^{2}_{1}\)	\(f^{2}_{7}\)	\(f^{2}_{13}\)	\(f^{2}_{14}\)	\(f^{2}_{1} (a)\)	\(f^{2}_{7} (b)\)	\(f^{2}_{12} (c)\)	\(f^{2}_{13} (d)\)
\(p\)	\(q\)	\(\neg p\)	\(\neg q\)	\(\wedge\)	\(\vee\)	\(\rightarrow\)	\(\uparrow\)	\(\neg (p \uparrow q)\)	\(\neg p \uparrow \neg q\)	\(p \uparrow p\)	\(\neg p \vee q\)
0	0	1	1	0	0	1	1	0	0	1	1
0	1	1	0	0	1	1	1	0	1	1	1
1	0	0	1	0	1	0	1	0	1	0	0
1	1	0	0	1	1	1	0	1	1	0	1

Henceforth the compound phrase

(1)¶\[\begin{split}\begin{array}{l} \mbox{When the cock crows on the heap,}\\ \mbox{the weather will change}\\ \mbox{or it will just keep.}\\ \end{array}\end{split}\]

defined by its atomic member statements

\[\begin{split}\begin{array}{rl} p & = \mbox{the cock is crowing on the heap} ,\\ q & = \mbox{the weather is changing} ,\\ \neg q & = \mbox{the weather is keeping} , \end{array}\end{split}\]

gives rise to the symbolic representation

(2)¶\[p \rightarrow q \vee \neg q .\]

With a couple of well-known simple laws, namely

(3)¶\[\begin{split}\begin{array}{lclr} \hphantom{(}\hphantom{\neg} p \wedge q & = & \neg (p \uparrow \hphantom{\neg} q) , & (a) \\ \hphantom{(}\hphantom{\neg} p \vee q & = & \hphantom{(}\neg p \uparrow \neg q , & (b) \\ \hphantom{(} \neg p & = & \hphantom{(}\hphantom{\neg} p \uparrow \hphantom{\neg} p , & (c) \\ \hphantom{(}\hphantom{\neg} p \rightarrow q & = & \hphantom{(}\neg p \vee \hphantom{\neg} q , & (d) \\ \neg (p \vee q) & = & \hphantom{(}\neg p \wedge \neg q , & (e) \\ \end{array}\end{split}\]

proved by full induction over the truth tables, formula (2) is transformed to

(4)¶\[\begin{split}\begin{array}{lcll} & & p \rightarrow (q \vee \neg q) & | (d)\\ & = & \neg p \vee (q \vee \neg q) & | (b)\\ & = & p \uparrow \neg (q \vee \neg q) & | (e)\\ & = & p \uparrow ( \neg q \wedge q ) & | (a)\\ & = & p \uparrow \neg (\neg q \uparrow q) & | (c)\\ & = & p \uparrow \neg ((q \uparrow q) \uparrow q) & | (c)\\ & = & p \uparrow (((q \uparrow q) \uparrow q) \uparrow ((q \uparrow q) \uparrow q)) .\\ \end{array}\end{split}\]

Replacing NAND expressions with different language constructs

(5)¶\[\begin{split}\begin{array}{ll} p\quad \mbox{is incompatible with}\quad q , & (f) \\ p\quad \mbox{is mutually exclusive with}\quad q , & (g) \\ \mbox{the fact that}\quad p\quad \mbox{cannot coexist with the fact that}\quad q , & (h) \\ p\quad \mbox{conflicts with}\quad q , & (i) \end{array}\end{split}\]

for each parenthesized level, assigned to the deepest innermost level first, formula (4) translates to

(6)¶\[\begin{split}\begin{array}{l} \hphantom{quad}\mbox{[the cock is crowing on the heap]}\\ \mbox{conflicts with (}\\ \hphantom{quad}\mbox{the fact that (}\\ \hphantom{quad}\hphantom{quad}\hphantom{quad}\mbox{( [the weather is changing] is incompatible with [the weather is changing] )}\\ \hphantom{quad}\hphantom{quad}\mbox{is mutually exclusive with}\\ \hphantom{quad}\hphantom{quad}\hphantom{quad}\mbox{[the weather is changing])}\\ \hphantom{quad}\mbox{cannot coexist with the fact that (}\\ \hphantom{quad}\hphantom{quad}\hphantom{quad}\mbox{( [the weather is changing] is incompatible with [the weather is changing] )}\\ \hphantom{quad}\hphantom{quad}\mbox{is mutually exclusive with}\\ \hphantom{quad}\hphantom{quad}\hphantom{quad}\mbox{[the weather is changing])))} . \end{array}\end{split}\]

Although parentheses are kept as structural support, even providing indentation for clarification, the statement’s translated form (6) still makes the issue at hand utterly confusing, if not completely incomprehensible for a human brain.

So, while the human brain is able to produce such fabulous and wondrous things, it is not capable of understanding such fabulous and wondrous things, which would have been truely fabulous and wondrous. But this is not the case. Therefore humans are basically on the same level as monkeys randomly typing on typewriters, producing all of Shakespeare’s works eventually \(\blacksquare\).

To the unsuspecting eye the phrasing of the NAND statement (6) may appear as a complete denial – even asserting ultimate falsehood – although it is in fact – just like its original form (1) – always true.

References

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