Formulas in predicate logic

4 places
Examples:

Pairs
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.







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

k =  1        2        3        4        5        6        7        8              sum = 2 * A083355(n) n 1          2                                                                                 2 2          2        6                                                                        8 3           2       18       26                                                              46 4           2       42      156      150                                                    350 5           2       90      650     1500     1082                                          3324 6           2      186     2340     9750    16230     9366                                37874 7           2      378     7826    52500   151480   196686    94586                      503458 8           2      762    25116   255150  1136100  2491356  2648408  1091670            7648564

Preferential arrangements of set partitions
A formula with an n-place predicate is PA of an n-set together with a Boolean value:

E.g. this PA corresponds to this  and this  formula.