We can represent each categorical proposition in a tabular way, as follows:
| X | Y | |
|---|---|---|
| All X is Y | -1 | 1 |
| No X is Y | -1 | -1 |
| Some X is Y | 1 | 1 |
| Some X is not y | 1 | -1 |
Given this, we can represent syllogisms in a similar way.
| BARBARA | S | M | P |
|---|---|---|---|
| All M is P | 0 | -1 | 1 |
| All S is M | -1 | 1 | 0 |
| All S is P | -1 | 0 | 1 |
Notice that the premises add to the conclusion, All S is P. We can do the same for the other First Figure syllogisms:
| CELARENT | S | M | P |
|---|---|---|---|
| No M is P | 0 | -1 | -1 |
| All S is M | -1 | 1 | 0 |
| All S is P | -1 | 0 | -1 |
| DARII | S | M | P |
|---|---|---|---|
| All M is P | 0 | -1 | 1 |
| Some S is M | 1 | 1 | 0 |
| Some S is P | 1 | 0 | 1 |
| FERIO | S | M | P |
|---|---|---|---|
| No M is P | 0 | -1 | -1 |
| Some S is M | 1 | 1 | 0 |
| Some S is not P | 1 | 0 | -1 |
If we look at Second, Third, and Fourth Figure, we find that the C's all have the same pattern as Celarent, showing that they can be directly converted to Celarent in the First Figure. The D's and F's reduce to Darii and Ferio, for the most part; in fact, the only exceptions to this general pattern in the traditional figures are Bramantip/Baralipton/Bamalip (Fourth Figure), Darapti (Third Figure), Felapton (Third Figure), and Fesapo (Fourth Figure). These all have to involve subalternation in some way so as to get particular conclusions from universal premises. If we tabulate the the way we tabulated the First Figure, we find that the premises do not directly add to the conclusion. For instance, this is Bramantip:
| BRAMANTIP | S | M | P |
|---|---|---|---|
| All P is M | 0 | 1 | -1 |
| All M is S | 1 | -1 | 0 |
| Some S is P | 1 | 0 | 1 |
The P's do not add. But this is because there is a subalternation step. In Bramantip, this subalternation step is 'Some P is P', which gives us a double-dose of P. Thus:
| BRAMANTIP | S | M | P |
|---|---|---|---|
| All P is M | 0 | 1 | -1 |
| All M is S | 1 | -1 | 0 |
| Some P is P | 0 | 0 | 2 |
| Some S is P | 1 | 0 | 1 |
Bramantip, using 'Some P is P', is the weirdest of the valid syllogisms; Darapti, Felapton, and Fesapo use 'Some M is M' , because they all have -1 for both the M places in the premise, and therefore need something that can cancel out a -2 for M. The same method will work for subalternated moods that take ordinary syllogisms with universal conclusions that are then subalternated (Barbari, Celaront, Cesaro, etc.), except that in those cases the subalternation can be handled extramodally -- i.e., one way to do them is to reach the conclusion using the standard mood and figure and then add the subalternation premise to the conclusion to get the particular conclusion (for these, the subalternation premise is always 'Some S is S').
The premises adding to the conclusion is a necessary, not a sufficient, condition for validity of syllogism; the tables don't actually track figure (which requires considering order, not just value), so they only identify syllogisms that are invalid purely because of mood. For validity, syllogisms also need to be regular, i.e., universal conclusions have to come from all universal premises, and particular conclusions have to come from premises that have one and only one particular proposition (which may be the subalternation premise).
