173 Comparing the above with the five ordinary Eulerian diagrams (which may be designated α, β &c. as in section [126]), it will be seen that (i) corresponds to α; (ii) to β; (iii) to γ; (iv) and (v) represent the two cases now yielded by δ; (vi) and (vii) the two yielded by ε.
Our seven diagrams might also be arrived at as follows:—Every part of the universe must be either S or Sʹ, and also P or Pʹ ; and hence the mutually exclusive combinations SP, SPʹ, SʹP, SʹPʹ must between them exhaust the universe. The case in which these combinations are all to be found is represented by diagram (iv); if one but one only is absent we obviously have four cases which are represented respectively by (ii), (iii), (v), and (vi); if only two are to be found it will be seen that we are limited to the cases represented by (i) and (vii) or we should not fulfil the condition that neither S nor Sʹ, P nor Pʹ, is to be altogether non-existent; for the same reason the universe cannot contain less than two of the four combinations. We thus have the seven cases represented by the diagrams, and these are shewn to exhaust the possibilities.
174 The four traditional propositions are related to the new scheme as follows:—
A limits us to (i) or (ii);
I to (i), (ii), (iii), (iv), or (v);
E to (vi) or (vii);
O to (iii), (iv), (v), (vi), or (vii).
Working out the further question how each diagram taken by itself is to be expressed propositionally we get the following results:
(i) SaP and SʹaPʹ ;
(ii) SaP and SʹoPʹ ;
(iii) SʹaPʹ and SoP ;
(iv) SoP, SoPʹ, SʹoP, and SʹoPʹ ;
(v) SʹaP and SoPʹ ;
(vi) SaPʹ and SʹoP ;
(vii) SaPʹ and SʹaP.