It will be observed that the new scheme is in itself more symmetrical than Euler’s, and also that it succeeds better in bringing out the symmetry of the fourfold schedule of propositions.[176] A and E give two alternatives each, I and O give five each; whereas with Euler’s scheme E gives only one alternative, A two, O three, I four, and it might, therefore, seem as if E afforded more definite and unambiguous information than A, and O than I, which is not really the case. Further, the problem of expressing each diagram propositionally yields a more symmetrical result than the corresponding problem in the case of Euler’s diagrams.
[176] We have seen that, similarly, in the case of immediate inferences symmetry can be gained only by the recognition of negative terms.
This sevenfold scheme of class relations should be compared with the sevenfold scheme of relations between propositions given in section [84].
131. Lambert’s diagram and the class-relations between S, not-S, P, not-P.—The following is a compact diagrammatic representation of the seven possible class-relations between S, not-S, P, not-P, based upon Lambert’s scheme. 175
In this scheme each line represents the entire universe of discourse, and the first line must be taken in connexion with each of the others in turn. Further explanation will be unnecessary for the student who clearly understands the Lambertian method.
On the same principle and with the aid of dotted lines the four fundamental propositional forms may be represented as follows:
176 In each case the full extent of a line represents the entire universe of discourse; any portion of a line that is dotted may be either S or Sʹ (or P or Pʹ, as the case may be).
This last scheme of diagrams is perhaps more useful than any of the others in shewing at a glance what immediate inferences are obtainable from each proposition by conversion, contraposition, and inversion (on the assumption that S, Sʹ, P, and Pʹ all represent existing classes). Thus, from the first diagram we can read off at a glance SaP, PiS, PʹaSʹ, SʹiPʹ ; from the second SeP, PeS, PʹoSʹ, SʹoPʹ ; from the third SiP and PiS ; and from the fourth SoP and PʹoSʹ. The last two diagrams are also seen at a glance to be indeterminate in respect to Pʹ and Sʹ, P and Sʹ, respectively (that is to say, I has no contrapositive and no inverse, O has no converse and no inverse).