[173] Dr Venn’s scheme differs from the schemes of Euler and Lambert, in that it is not based upon the assumption that our terms and their contradictories all represent existing classes. It involves, however, the doctrine that particulars are existentially affirmative, while universals are existentially negative.
129. Expression of the possible relations between any two classes by means of the propositional forms A, E, I, O.—Any information given with respect to two classes limits the possible relations between them to something less than the five à priori possibilities indicated diagrammatically by Euler’s circles as given at the beginning of section [126]. It will be useful to enquire how such information may in all cases be expressed by means of the propositional forms A, E, I, O.
The five relations may, as before, be designated respectively α, β, γ, δ, ε.[174] Information is given when the possibility of one or more of these is denied; in other words, when we are limited to one, two, three, or four of them. Let limitation to α, or β, the exclusion of γ, δ, ε be denoted by α, β ; limitation to α, β, or γ (i.e., the exclusion of δ and ε) by α, β, γ ; and so on.
[174] Thus, the classes being S and P, α denotes that S and P are wholly coincident; β that P contains S and more besides; β that S contains P and more besides; δ that S and P overlap each other, but that each includes something not included by the other; ε that S and P have nothing whatever in common.
In seeking to express our information by means of the four ordinary propositional forms, we find that sometimes a single proposition will suffice for our purpose; thus α, β is expressed by All S is P. Sometimes we require a combination of propositions; thus α is expressed by the pair of complementary propositions All S is P and all P is S, (since all S is P excludes γ, δ, ε, and all P is S further excludes β). Some other cases are more complicated; thus the fact that we are limited to α or δ cannot be expressed more simply than by saying, Either All S is P and all P is S, or else Some S is P, some S is not P, and some P is not S.
Let A = All S is P, A1 = All P is S, and similarly for the other propositions. Also let AA1 = All S is P and all P is S, &c. Then the following is a scheme for all possible cases:— 169
| Limitation to | denoted by | Limitation to | denoted by |
| α | AA1 | α, β, γ | A or A1 |
| β | AO1 | α, β, δ | A or IO1 |
| γ | A1O | α, β, ε | A or E |
| δ | IOO1 | α, γ, δ | A1 or IO |
| ε | E | α, γ, ε | A1 or E |
| α, β | A | α, δ, ε | AA1 or OO1 |
| α, γ | A1 | β, γ, δ | IO or IO1 |
| α, δ | AA1 or IOO1 | β, γ, ε | AO1 or A1O or E |
| α, ε | AA1 or E | β, δ, ε | O1 |
| β, γ | AO1 or A1O | γ, δ, ε | O |
| β, δ | IO1 | α, β, γ, δ | I |
| β, ε | AO1 or E | α, β, γ, ε | A or A1 or E |
| γ, δ | IO | α, β, δ, ε | A or O1 |
| γ, ε | A1O or E | α, γ, δ, ε | A1 or O |
| δ, ε | OO1 | β, γ, δ, ε | O or O1 |
It will be found that any combinations of propositions other than those given above either involve contradictions or redundancies, or else give no information because all the five relations that are à priori possible still remain possible.
For example, AI is clearly redundant; AO is self-contradictory; A or A1O is redundant (since the same information is given by A or A1); A or O gives no information (since it excludes no possible case). The student is recommended to test other combinations similarly. It must be remembered that I1 = I, and E1 = E.
170 It should be noticed that if we read the first column downwards and the second column upwards we get pairs of contradictories.