## Frankl, My Dear : 2

Supplied by the cache of definitions from Post 1, I can return to the passage from (2) that seemed to jog a bit of memory and see if what I imagined I saw there makes any sense.

Let us use ${J_{i}(f)}$ to denote those ${x}$ such that

$\displaystyle f(x) \neq f(x^{i}).$

Obviously the following is true:

$\displaystyle \frac{|J_{i}(f)|}{2^{n}} = I_{i}(f).$

There must be a better notation than ${J_{i}}$ — we are open to suggestions. Any?

I responded to their query about a better notation for ${J_{i}}$ in a series of comments along the following lines:

I think ${J_{i}(f)}$ is just the set of places where the partial differential of ${f}$ with respect to ${x_i}$ is ${1}.$

See Tables A9 and A10 in my article on Differential Logic and Dynamic Systems.

For example, look at ${f_8 (x, y),}$ which is just the logical conjunction ${xy}.$

In this case, ${\partial_x f = y}.$

This means that crossing the boundary of ${x}$ will change the value of ${f}$ exactly in those places where ${y}$ is true.

The See a Number, Make a Set principle leads to the following observation, that an arbitrary set of cells in a venn diagram or an arbitrary set of vertices in a ${k}$-cube is described by a proposition or a boolean function as its fiber of ${1},$ so the above types of differential operators take us from propositions to propositions, in other words, they stay within the same general datatype.

To be continued …

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