Re: Timothy Chow • Shifting Paradigms?
I can’t remember when I first started playing with Gödel codings of graph-theoretic structures, which arose in logical and computational settings, but I remember being egged on in that direction by Martin Gardner’s 1976 column on Catalan numbers, planted plane trees, polygon dissections, etc. Codings being injections from a combinatorial species 
The most interesting bijection I found was one between positive integers 
An interesting tangent to the main subject, but one that I had some ready thoughts on.
can be verified by establishing the corresponding equation in matrices.
and 
are two 1-dimensional matrices over the same index set 
then 
if and only if 
for every 
Thus, a routine way to check the validity of 
is to check whether the following equation holds for arbitrary 
in ordinary arithmetic holds when the values 
are restricted to the boolean domain 
Interpreted as a logical statement, the law of exponents ![\begin{matrix} (a \Leftarrow b) \Leftarrow c & = & a \Leftarrow b \land c \\[8pt] c \Rightarrow (b \Rightarrow a) & = & c \land b \Rightarrow a \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%28a+%5CLeftarrow+b%29+%5CLeftarrow+c+%26+%3D+%26+a+%5CLeftarrow+b+%5Cland+c++%5C%5C%5B8pt%5D++c+%5CRightarrow+%28b+%5CRightarrow+a%29+%26+%3D+%26+c+%5Cland+b+%5CRightarrow+a++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=1&c=20201002)
will denote whatever stands to every woman in the relation of servant of every lover of hers; and 
will denote whatever is a servant of everything that is lover of a woman. So that
![\begin{array}{*{15}{c}} X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i\ & \} \\[6pt] L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} \\[6pt] S & = & \{ & b\!:\!a, & b\!:\!c, & d\!:\!c, & d\!:\!d, & d\!:\!e, & f\!:\!e, & f\!:\!f, & f\!:\!g, & h\!:\!g, & h\!:\!i\ & \} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++X+%26+%3D+%26+%5C%7B+%26+a%2C+%26+b%2C+%26+c%2C+%26+d%2C+%26+e%2C+%26+f%2C+%26+g%2C+%26+h%2C+%26+i%5C+%26+%5C%7D++%5C%5C%5B6pt%5D++L+%26+%3D+%26+%5C%7B+%26+b%5C%21%3A%5C%21a%2C+%26+b%5C%21%3A%5C%21c%2C+%26+c%5C%21%3A%5C%21b%2C+%26+c%5C%21%3A%5C%21d%2C+%26+e%5C%21%3A%5C%21d%2C+%26+e%5C%21%3A%5C%21e%2C+%26+e%5C%21%3A%5C%21f%2C+%26+g%5C%21%3A%5C%21f%2C+%26+g%5C%21%3A%5C%21h%2C+%26+h%5C%21%3A%5C%21g%2C+%26+h%5C%21%3A%5C%21i+%26+%5C%7D++%5C%5C%5B6pt%5D++S+%26+%3D+%26+%5C%7B+%26+b%5C%21%3A%5C%21a%2C+%26+b%5C%21%3A%5C%21c%2C+%26+d%5C%21%3A%5C%21c%2C+%26+d%5C%21%3A%5C%21d%2C+%26+d%5C%21%3A%5C%21e%2C+%26+f%5C%21%3A%5C%21e%2C+%26+f%5C%21%3A%5C%21f%2C+%26+f%5C%21%3A%5C%21g%2C+%26+h%5C%21%3A%5C%21g%2C+%26+h%5C%21%3A%5C%21i%5C+%26+%5C%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
and 
if and only if 
But the vacuous inclusions, that is, the cases where 
have the effect of adding non‑intuitive links to the mix.
if and only if there exists a 
such that 
and 
for “lover of every woman”.
at an argument 
![\begin{array}{*{15}{c}} X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \} \\[6pt] W & = & \{ & d, & f & \} \\[6pt] L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B15%7D%7Bc%7D%7D++X+%26+%3D+%26+%5C%7B+%26+a%2C+%26+b%2C+%26+c%2C+%26+d%2C+%26+e%2C+%26+f%2C+%26+g%2C+%26+h%2C+%26+i+%26+%5C%7D++%5C%5C%5B6pt%5D++W+%26+%3D+%26+%5C%7B+%26+d%2C+%26+f+%26+%5C%7D++%5C%5C%5B6pt%5D++L+%26+%3D+%26+%5C%7B+%26+b%5C%21%3A%5C%21a%2C+%26+b%5C%21%3A%5C%21c%2C+%26+c%5C%21%3A%5C%21b%2C+%26+c%5C%21%3A%5C%21d%2C+%26+e%5C%21%3A%5C%21d%2C+%26+e%5C%21%3A%5C%21e%2C+%26+e%5C%21%3A%5C%21f%2C+%26+g%5C%21%3A%5C%21f%2C+%26+g%5C%21%3A%5C%21h%2C+%26+h%5C%21%3A%5C%21g%2C+%26+h%5C%21%3A%5C%21i+%26+%5C%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
within 
within 
by means of the relative term 
which conveys the same information.
as follows.
otherwise 
otherwise 
for each 
is 
then the product 
is 
otherwise it is 
goes to 
such that 
and 
but 
If there is no such 
the only case producing a non-zero result is 
That portion of the work can be sketched as follows.
Inquiry Into Inquiry 