and similarly for the other propositions.
Dr Venn rightly observes that this scheme is purely symbolical, and does not deserve to rank as a diagrammatic scheme at all. There is clearly nothing in the two ends of a wedge to suggest subjects and predicates, or in a colon and comma to suggest distribution and non-distribution” (Symbolic Logic, p. 432). Hamilton’s scheme may certainly be rejected as valueless. The schemes of Euler and Lambert belong to an altogether different category.
A second essential requirement is that the diagrams should be adequate; that is to say, they should give a complete, and 157 not a partial, representation of the relations which they are intended to indicate. Hamilton’s use of Euler’s diagrams, as described in the following section, will serve to illustrate the failure to satisfy this requirement.
In the third place, the diagrams should be capable of representing all the propositional forms recognised in the schedule of propositions which are to be illustrated, e.g., particulars as well as universal. One scheme of diagrams may, however, be better suited for one purpose, and another scheme for another purpose. It will be found that Dr Venn’s diagrams, to be described presently, are not quite so well adapted to the representation of particulars as of universals.
Lastly, it is advantageous that a diagrammatic scheme should be as little cumbrous as possible when it is desired to represent two or more propositions in combination with one another. This is the weak point of Euler’s method. A scheme of diagrams may, however, serve a very useful function in making clear the full force of individual propositions, even when it is not well adapted for the representation of combined propositions.
A further requirement is sometimes added, namely, that each propositional form should be represented by a single diagram, not by a set of alternative diagrams. This is, however, by no means essential. For if we adopt a schedule of propositions some of which yield only an indeterminate relation in respect of extension between the terms involved, it is important that this should be clearly brought out, and a set of alternative diagrams may be specially helpful for the purpose. This point will be illustrated, with reference to Euler’s diagrams, in the following section.[166]
[166] It must be borne in mind that in all the schemes described in this chapter the terms of the propositions which are represented diagrammatically are taken in extension, not in intension.
126. Euler’s Diagrams.—We may begin with the well-known scheme of diagrams, which was first expounded by the Swiss mathematician and logician, Leonhard Euler, and which is usually called after his name.[167]
[167] Euler lived from 1707 to 1783. His diagrammatic scheme is given in his Lettres à une Princesse d’Allemagne (Letters 102 to 105).
158 Representing the individuals included in any class, or denoted by any name, by a circle, it will be obvious that the five following diagrams represent all possible relations between any two classes:—