Re: John Baez • Entropy and Information in Biological Systems
- To develop the concept of evolutionary games as “learning” processes in which information is gained over time.
A fund of ideas toward that end can be found in the work of C.S. Peirce on the themes of evolution, inquiry, and their interaction. Peirce stands out as one of the few pioneers in the study of scientific method who avoided the dead‑ends of naive deductivisim and naive inductivisim. He developed Aristotle’s concept of abductive reasoning in a way that anticipated later insights into the dynamics of paradigm shifts. A question worth exploring in that connection is whether abductive hypothesis formation is to scientific method what random mutation is to natural selection.
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.
will denote everything which is an 
for every individual of 
Thus 
or the exponentiation of 
is the repeated application of the multiplier 
for as many times as there are individuals under the term 
On Peirce’s interpretive rules, the repeated applications of the base term 
In Peirce’s notation the fact may be written as follows.
is a lover of 
that is a lover of 
that is a lover of 
In other words, a lover of every woman in the universe at hand is a lover of 
with 
collect the subset 
of elements which appear as the first components of the pairs in 
is a lover of every woman is to say 
if 
in the boolean domain 
are tabulated as follows.
All that remains is to see how operations like these on values in 
induce the corresponding operations on sets and terms.
are represented by the matrices 
and 
respectively, then the operation on terms which produces the term 
which produces the matrix 
In short, the involution operation on matrices must be defined in such a way that the following equation holds.
The value of the matrix 
is the monadic relation, or set, whose elements fall under the absolute term 
The elements of 
is the dyadic relation associated with the relative term 
is the dyadic relation associated with the relative term 
is the 1-dimensional matrix representation of the set 
is the 2-dimensional matrix representation of the relation 
is the 2-dimensional matrix representation of the relation 
and the relative term 
![\begin{array}{lll} u \star L & = & L_{u\,@\,1} \\[6pt] & = & \{ (u, x) \in L \} \\[6pt] & = & \text{ordered pairs in}~ L ~\text{with}~ u ~\text{in the 1st place}. \\[9pt] L \star v & = & L_{v\,@\,2} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{ordered pairs in}~ L ~\text{with}~ v ~\text{in the 2nd place}. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++u+%5Cstar+L++%26+%3D+%26+L_%7Bu%5C%2C%40%5C%2C1%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+%28u%2C+x%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bordered+pairs+in%7D%7E+L+%7E%5Ctext%7Bwith%7D%7E+u+%7E%5Ctext%7Bin+the+1st+place%7D.++%5C%5C%5B9pt%5D++L+%5Cstar+v++%26+%3D+%26+L_%7Bv%5C%2C%40%5C%2C2%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+%28x%2C+v%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bordered+pairs+in%7D%7E+L+%7E%5Ctext%7Bwith%7D%7E+v+%7E%5Ctext%7Bin+the+2nd+place%7D.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ x \in X : (u, x) \in L \} \\[6pt] & = & \text{loved by}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{lover of}~ v. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++u+%5Ccdot+L++%26+%3D+%26+%5Cmathrm%7Bproj%7D_2+%28u+%5Cstar+L%29++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+x+%5Cin+X+%3A+%28u%2C+x%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bloved+by%7D%7E+u.++%5C%5C%5B9pt%5D++L+%5Ccdot+v++%26+%3D+%26+%5Cmathrm%7Bproj%7D_1+%28L+%5Cstar+v%29++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+x+%5Cin+X+%3A+%28x%2C+v%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Blover+of%7D%7E+v.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![[\mathit{1}] ~=~ 1.](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7B1%7D%5D+%7E%3D%7E+1.&bg=ffffff&fg=000000&s=1&c=20201002)
denotes the dyadic identity relation 
while the antique figure 
denotes what Peirce otherwise defines as 
![[\mathit{1}] = \mathfrak{1},](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7B1%7D%5D+%3D+%5Cmathfrak%7B1%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002)
which does not make sense. Chronological Edition CE 2 gives the 1’s in different styles of italics but reading the equation as ![[\mathit{1}] = 1](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7B1%7D%5D+%3D+1&bg=ffffff&fg=000000&s=0&c=20201002)
makes better sense if the latter “1” is the numeral denoting the natural number 1 and not the absolute term “1” denoting the universe of discourse. The quantity ![[\mathit{1}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7B1%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
is defined as the average number of things related by the identity relation ![[\mathit{1}] = 1 \in \mathbb{N},](https://s0.wp.com/latex.php?latex=%5B%5Cmathit%7B1%7D%5D+%3D+1+%5Cin+%5Cmathbb%7BN%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002)
where 
is the set of non-negative integers 
in the non-negative integers 
we have the following.![v(\mathit{1}) ~=~ [\mathit{1}] ~=~ 1.](https://s0.wp.com/latex.php?latex=v%28%5Cmathit%7B1%7D%29+%7E%3D%7E+%5B%5Cmathit%7B1%7D%5D+%7E%3D%7E+1.&bg=ffffff&fg=000000&s=1&c=20201002)
has another one of the properties required of an arrow from logical terms in 
Inquiry Into Inquiry 