Implications between the  A000629(4) = 150 statements with 4 variables in first-order logic

The formulas are represented by ordered set partitions. Black stands for the universal quantifier, red for the existential one.

The basic structure of this Graph is like the tesseract shown on the right.
The eight little cubes correspond to tesseract vertices with rank 1 and 3, the six rhombic dodecahedra to those with rank 2.

Examples:

matrix	abbreviation	formula
	a3 e2 a4 e1	${\displaystyle \forall y~\exists x~\forall z~\exists w~Pwxyz}$
	e1 a3 e24	${\displaystyle \exists w~\forall y~\exists x~\exists z~Pwxyz}$
	a1 e3 a24	${\displaystyle \forall w~\exists y~\forall x~\forall z~Pwxyz}$
	e3 a124	${\displaystyle \exists y~\forall w~\forall x~\forall z~Pwxyz}$