Formulas in predicate logic

1 placeEdit

 
Implications between 1-place formulas, on the right the negations (To match the traditional square of opposition the subalterns point downwards here.)
 
All cases with 2 individuals



Rdrdo.svg 2 placesEdit


 
3 ordered partitions in Cayley graph
 
Pairs, nodes of the square
 
Sketches in Hasse diagram
 
6 matrices in Hasse diagram
 
Pairs in Hasse diagram


Rdrdo.svg 3 placesEdit

 
13 ordered partitions in Cayley graph
 
Pairs, nodes of the cube
 
Sketches in Hasse diagram
 
26 matrices in Hasse diagram
 
Pairs in Hasse diagram

4 placesEdit

Examples:

matrix abbreviation formula
  a3 e2 a4 e1  
  e1 a3 e24  
  a1 e3 a24  
  e3 a124  
 
75 ordered partitions in Cayley graph
 
150 matrices in Hasse diagram
 
Pairs, nodes of the tesseract

2-element subsets of a 4-element set:  


weight 1
Ordered partitions
 
1000
 
0100
 
0010
 
0001

PairsEdit

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.

 
a(12)3  =  20-0
 
a1 e2 a3  =  60-1
 
e12 a3  =  71-1
 
a1 e(23)  =  66-5
 
e2 a14 e3  =  460-71-3
 
Symbolic representation of pairs and the corresponding pseudo-octal digits
 
e(12)8 a(35)(46) e7  =  5111177-111177-02064-0264-064-64-7
 
e(128) a(356)4 e7  =  5111175-111175-02264-0064-264-64-7


Places and different variablesEdit

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):

      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


Rdr.svg Preferential arrangements of set partitionsEdit

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.