In the second diagram, the four rectangles at each of the corners represent the possible unary predicates on a domain of two individuals.
Specifically, if we consider a predicate P as a subset of a domain D = {c, d}, then either:

P = {}, drawn as an empty rectangle

P = {c}, drawn as a rectangle with a dot on the left (for c)

P = {d}, drawn as a rectangle with a dot on the right (for d)

P = {c,d}, drawn as a rectangle with two dots.

The rectangles are colored red iff the predicate they represent validates the formula near which they are placed.
For example only P = {c,d} validates the formula forall x: P(x) at the top left corner, hence only the rectangle with the two dots is colored red.
Thus, the squares together with their coloring represents a subset of the set of unary predicates P on a domain with two individuals.

In the first diagram, the subsets are summarised in a different graphical notation, called a sketch.
There, each square represents a specific subset of the set of unary predicates on a domain of arbitrary size.
This subset is the extension of the formula near which it is placed, and with that, it is a canonical representation of that formula.

Formulas with n-place predicates can be broken down in T_{(n-1)} formulas with 2-place predicates.
These triangles (or vectors) with up to 8 different entries are a convenient way to determine whether one formula implies another one.

The image captions in this section are the abbreviated formulas and the pseudo-octal strings.

Among the following four formulas - visualized in the different ways used here - the left one implies a1 e2 a3, and the two on the right are implied by it.

The number of formulas with n place predicates and n different variables is A000629(n) = 2 * OrderedBell(n).
These formulas form the lattices shown above.

The number of formulas with n place predicates and k different variables is 2 * A232598 = 2 * Stirling2(n,k) * OrderedBell(k):