Similarly the pair of contra-complementary propositions, SaP and PoS, limit us to the relation marked β on page [158]; and the pair of contra-complementary propositions, SoP and 162 PaS, to γ ; while the pair of sub-complementary propositions, SoP and PoS, give us a choice between δ and ε.
The application of the diagrams to syllogistic reasonings will be considered in a subsequent [chapter].
With regard to all the above, it may be said that the use of the circles gives us nothing that could not easily have been obtained independently. This is of course true; but no one, who has had experience of the difficulty that is sometimes found by students in properly understanding the elementary principles of formal logic, and especially in dealing with immediate inferences, will despise any means of illustrating afresh the old truths, and presenting them under a new aspect.
The fact that we have not a single pair of circles corresponding to each fundamental form of proposition is fatal if we wish to illustrate any complicated train of reasoning in this way; but in indicating the real nature of the information given by the propositions themselves, it is rather an advantage than otherwise, inasmuch as it shews how limited in some cases this information actually is.[169]
[169] Dr Venn writes in criticism of Euler’s scheme, “A fourfold scheme of propositions will not very conveniently fit in with a fivefold scheme of diagrams… What the five diagrams are competent to illustrate is the actual relation of the classes, not our possibly imperfect knowledge of that relation” (Empirical Logic, p. 229). The reply to this criticism is that inasmuch as the fourfold scheme of propositions gives but an imperfect knowledge of the actual relation of the classes denoted by the terms, the Eulerian diagrams are specially valuable in making this clear and unmistakeable. By the aid of dotted lines it is indeed possible to represent each proposition by a single Eulerian figure; but the diagrams then become so much more difficult to interpret that the loss is considerably greater than the gain. The first and second of the following diagrams are borrowed from Ueberweg (Logic, § 71). In the case of O, Ueberweg’s diagram is rather complicated; and I have substituted a simpler one.
In the last of these diagrams we get the three cases yielded by an O proposition by (1) filling in the dotted line to the left and striking out the other, (2) filling in both dotted lines, (3) filling in the dotted line to the right and striking out the other. These three cases are respectively those marked γ, δ, ε, on page [158].
163 127. Lambert’s Diagrams.—A scheme of diagrams was employed by Lambert[170] in which horizontal straight lines take the place of Euler’s circles. The extension of a term is represented by a horizontal straight line, and so far as two such lines overlap it is indicated that the corresponding classes are coincident, while so far as they do not overlap these classes are shewn to be mutually exclusive. Both the absolute and the relative length of the lines is of course arbitrary and immaterial.
[170] Johann Heinrich Lambert was a German philosopher and mathematician who lived from 1728 to 1777. His Neues Organon was published at Leipzig in 1768. Lambert’s own diagrammatic scheme differs somewhat from both of those given in the text; but it very closely resembles the one in which portions of the lines are dotted. The modifications in the text have been introduced in order to obviate certain difficulties involved in Lambert’s own diagrams. See note [2] on page 165.