Re: Dick Lipton & Ken Regan • (1) • (2)
Here’s a few pages on differential logic, whose ideas I’ll be trying out in the present setting:
- Differential Logic : Introduction
- Differential Propositional Calculus
- Differential Logic and Dynamic Systems
I next need to look at the following “key lemma” from (2) and see if I can wrap my head, or at least my own formalism, around what it says.
Lemma. Let 
These sets have the following properties:
- The variable
on
- The variable
on
- The union
is equal to
- The function is always
- The function is always
- Finally
and
and
This may take a while …
to denote those 
such that
— we are open to suggestions. Any?
with respect to 
is 
which is just the logical conjunction 
is true.
-cube is described by a proposition or a boolean function as its fiber of 
so the above types of differential operators take us from propositions to propositions, in other words, they stay within the same general datatype.
to denote those 
such that
— we are open to suggestions. Any?
and let 
be a boolean function. The influence of the 
input of 
is defined by![\displaystyle I_{i}(f) = \mathrm{Prob}_{x}[ f(x) \neq f(x^{i}) ],](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+I_%7Bi%7D%28f%29+%3D+%5Cmathrm%7BProb%7D_%7Bx%7D%5B+f%28x%29+%5Cneq+f%28x%5E%7Bi%7D%29+%5D%2C&bg=ffffff&fg=000000&s=0&c=20201002)
is equal to
I like a bit better. Here is the formal definition, using 
instead of 
as the index of the indicated variable.
be defined for each integer ![{[1, k]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2C+k%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
by the following equation:
with a tilde as 
or with distinctive parentheses as 
Thus we may see the function ![\begin{matrix} \lnot_j (x_1, \ldots, x_j, \ldots, x_k) & = & x_1 x_2 \ldots x_{j-1} \tilde{x}_j x_{j+1} \ldots x_{k-1} x_k \\[4pt] & = & x_1 x_2 \ldots x_{j-1} \texttt{(} x_j \texttt{)} x_{j+1} \ldots x_{k-1} x_k \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%5Clnot_j+%28x_1%2C+%5Cldots%2C+x_j%2C+%5Cldots%2C+x_k%29++%26+%3D+%26++x_1+x_2+%5Cldots+x_%7Bj-1%7D+%5Ctilde%7Bx%7D_j+x_%7Bj%2B1%7D+%5Cldots+x_%7Bk-1%7D+x_k++%5C%5C%5B4pt%5D++%26+%3D+%26++x_1+x_2+%5Cldots+x_%7Bj-1%7D+%5Ctexttt%7B%28%7D+x_j+%5Ctexttt%7B%29%7D+x_%7Bj%2B1%7D+%5Cldots+x_%7Bk-1%7D+x_k++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
is a 
is false and all the rest of the 
are true. In other work, I used the above notations for 
in defining 
Inquiry Into Inquiry 