CHAPTER V.
THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS.
125. The use of Diagrams in Logic.—In representing propositions by geometrical diagrams, our object is not that we may have a new set of symbols, but that the relation between the subject and predicate of a proposition may be exhibited by means of a sensible representation, the signification of which is clear at a glance. Hence the first requirement that ought to be satisfied by any diagrammatic scheme is that the interpretation of the diagrams should be intuitively obvious, as soon as the principle upon which they are based has been explained.[165]
[165] Hamilton’s “geometric scheme,” which he himself describes as “easy, simple, compendious, all-sufficient, consistent, manifest, precise, complete” (Logic, II. p. 475), fails to satisfy this condition. He represents an affirmative copula by a horizontal tapering line (
), the broad end of which is towards the subject. Negation is marked by a perpendicular line crossing the horizontal one (
). A colon (:) placed at either end of the copula indicates that the corresponding term is distributed; a comma (,) that it is undistributed. Thus, for All S is P we have,—
S :
, P ;
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:—
The force of the different propositional forms is to exclude one or more of these possibilities.
All S is P limits us to one of the two α, β ;
Some S is P to one of the four α, β, γ, δ ;
No S is P to ε ;
Some S is not P to one of the three γ, δ, ε.
It will be observed that there is great want of symmetry in the number of circles corresponding to the different propositional forms; also that there is an apparent inequality in the amount of information given by A and by E, and again by I and by O. We shall find that these anomalies disappear when account is taken of negative terms.
It is most misleading to attempt to represent All S is P by a single pair of circles, thus
or Some S is P by a single pair, thus
159 for in each case the proposition really leaves us with other alternatives. This method of employing the diagrams has, however, been adopted by a good many logicians who have used them, including Sir William Hamilton (Logic, I. p. 255), and Professor Jevons (Elementary Lessons in Logic, pp. 72 to 75); and the attempt at such simplification has brought their use into undeserved disrepute. Thus, Dr Venn remarks, “The common practice, adopted in so many manuals, of appealing to these diagrams—Eulerian diagrams as they are often called—seems to me very questionable. The old four propositions A, E, I, O, do not exactly correspond to the five diagrams, and consequently none of the moods in the syllogism can in strict propriety be represented by these diagrams” (Symbolic Logic, pp. 15, 16; compare also pp. 424, 425). This criticism, while perfectly sound as regards the use of Euler’s circles by Hamilton and Jevons, loses most of its force if the diagrams are employed with due precautions. It is true that the diagrams become somewhat cumbrous in relation to the syllogism; but the logical force of propositions and the logical relations between propositions can in many respects be well illustrated by their aid. Thus, they may be employed:—
(1) To illustrate the distribution of the predicate in a proposition. In the case of each of the four fundamental propositions we may shade the part of the predicate concerning which information is given us.
We then have,—
We see that with A and I, only part of P is in some of the cases shaded; whereas with E and O, the whole of P is in every case shaded; and it is thus made clear that negative propositions distribute, while affirmative propositions do not distribute, their predicates.
(2) To illustrate the opposition of propositions. Comparing two contradictory propositions, e.g., A and O, we see that they have no case in common, but that between them they exhaust all possible cases. Hence the truth, that two contradictory propositions cannot be true together but that one of them must be true, is brought home to us under a new aspect. Again, comparing two subaltern propositions, e.g., A and I, we notice that the former gives us all the information given by the latter and something more, since it still further limits the possibilities. The other relations involved in the doctrine of opposition may be illustrated similarly.
(3) To illustrate the conversion of propositions. Thus it is made clear by the diagrams how it is that A admits only of conversion per accidens. All S is P limits us to one or other of the following,—
What then do we know of P? In the first case we have All P is S, in the second Some P is S ; and since we are ignorant as to which of the two cases holds good, we can only state what is common to them both, namely, Some P is S.
Again, it is made clear how it is that O is inconvertible. Some S is not P limits us to one or other of the following,—
161 What then do we know concerning P? The three cases give us respectively,—(i) All P is S ; (ii) Some P is S and Some P is not S ; (iii) No P is S. But (i) and (iii) are inconsistent with one another. Hence nothing can be affirmed of P that is true in all three cases indifferently.
(4) To illustrate the more complicated forms of immediate inference. Taking, for example, the proposition All S is P, we may ask, What does this enable us to assert about not-P and not-S respectively? We have one or other of these cases,—
As regards not-P, these yield respectively (i) No not-P is S ; (ii) No not-P is S. And thus we obtain the contrapositive of the given proposition.
As regards not-S, we have (i) No not-S is P, (ii) Some not-S is P and some not-S is not P.[168] Hence in either case we may infer Some not-S is not P.
[168] It is assumed in the use of Euler’s diagrams that S and P both exist in the universe of discourse, while neither of them exhausts that universe. This assumption is the same as that upon which our treatment of immediate inferences in the preceding chapter has been based.
E, I, O may be dealt with similarly.
(5) To illustrate the joint force of a pair of complementary or contra-complementary or sub-complementary propositions (compare section [100]). Thus, the pair of complementary propositions, SaP and PaS, taken together, limit us to
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.
We may first shew how Lambert’s lines may be used in such a manner as to be precisely analogous to Euler’s circles. 164 Thus, the four fundamental propositions may be represented as follows:—
These diagrams occupy less space than Euler’s circles. But they seem also to be less intuitively clear and less suggestive. The different cases too are less markedly distinct from one another. It is probable that one would in consequence be more liable to error in employing them.
The different cases may, however, be combined by the use of dotted lines so as to yield but a single diagram for each proposition much more satisfactorily than in Euler’s scheme. Thus, All S is P may be represented by the diagram
where the dotted line indicates that we are uncertain as to whether there is or is not any P which is not S. We obviously get two cases according as we strike out the dots or fill them in, and these are the two cases previously shewn to be compatible with an A proposition.
Again, Some S is P may be represented by the diagram
and here we get the four cases previously given for an I 165 proposition by (a) filling in the dots to the left and striking out those to the right, (b) filling in all the dots, (c) striking them all out, (d) filling in those to the right and striking out those to the left.
Two complete schemes of diagrams may be constructed on this plan, in one of which no part of any S line, and in the other no part of any P line, is dotted.[171] These two schemes are given below to the left and right respectively of the propositional forms themselves.
[171] It is important to give both these schemes as it will be found that neither one of them will by itself suffice when this method is used for illustrating the syllogism. For obvious reasons the E diagram is the same in both schemes.
It must be understood that the two diagrams given above in the cases of A, I, and O are alternative in the sense that we may select which we please to represent our proposition; but either represents it completely.
We shall find later on that for the purpose of illustrating the syllogistic moods, Lambert’s method is a good deal less cumbrous than Euler’s.[172] An adaptation of Lambert’s diagrams in which the contradictories of S and P are introduced as well 166 as S and P themselves will be given in section [131]. This more elaborated scheme will be found useful for illustrating the various processes of immediate inference.
[172] Dr Venn (Symbolic Logic, p. 432) remarks, “As a whole Lambert’s scheme seems to me distinctly inferior to the scheme of Euler, and has in consequence been very little employed by other logicians.” The criticism offered in support of this statement is directed chiefly against Lambert’s own representation of the particular affirmative proposition, namely,
This diagram certainly seems as appropriate to O as it does to I; but the modification introduced in the text, and indeed suggested by Dr Venn himself, is not open to a similar objection.
128. Dr Venn’s Diagrams.—In the diagrammatic scheme employed by Dr Venn (Symbolic Logic, chapter 5) the diagram
does not itself represent any proposition, but the framework into which propositions may be fitted. Denoting not-S by Sʹ and what is both S and P by SP, &c., it is clear that everything must be contained in one or other of the four classes SP, SPʹ, SʹP, SʹPʹ ; and the above diagram shews four compartments (one being that which lies outside both the circles) corresponding to these four classes. Every universal proposition denies the existence of one or more of such classes, and it may therefore be diagrammatically represented by shading out the corresponding compartment or compartments. Thus, All S is P, which denies the existence of SPʹ, is represented by
No S is P by
167 With three terms we have three circles and eight compartments, thus,—
All S is P or Q is represented by
All S is P and Q by
It is in cases involving three or more terms that the advantage of this scheme over the Eulerian scheme is most manifest. The diagrams are not, however, quite so well adapted to the case of particular propositions. Dr Venn (in Mind, 1883, pp. 599, 600) suggests that we might draw a bar across the compartment declared to be saved by a particular proposition;[173] thus, Some S is P would be represented by drawing a bar across the SP compartment. This plan can be worked out satisfactorily; but in representing a combination of propositions in this way special care is needed in the interpretation of the diagrams. For example, if we have the diagram for three terms S, P, Q, and are given Some S is P, 168 we do not know that both the compartments SPQ, SPQʹ, are to be saved, and in a case like this a bar drawn across the SP compartment is in some danger of misinterpretation.
[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.
130. Euler’s diagrams and the class relations between S, not-S, P, not-P.—In Euler s diagrams, as ordinarily given, there is no explicit recognition of not-S and not-P; but it is of course understood that whatever part of the universe lies outside S is not-S, and similarly for P, and it may be thought that no further account of negative terms need be taken. Further consideration, however, will shew that this is not the case; and, assuming that S, not-S, P, not-P all represent existing classes, we shall find that seven, not five, determinate class relations between them are possible.
Taking the diagrams given in section [126], the above assumption clearly requires that in the cases of α, β, and γ, there should be some part of the universe lying outside both the circles, since otherwise either not-S or not-P or both of them would no longer be contained in the universe. But in the cases of δ and ε it is different. S, not-S, P, not-P are now all of them represented within the circles; and in each of these cases, therefore, the pair of circles may or may not between them exhaust the universe.
Our results may also be expressed by saying that in the cases of α, β, and γ, there must be something which is both not-S and not-P; whereas in the cases of δ and ε, there may or may not be something which is both not-S and not-P. Euler’s circles, as ordinarily used, are no doubt a little apt to lead us to overlook the latter of these alternatives. If, indeed, there were always part of the universe outside the circles, every proposition, whether its form were A, E, I, or O, would have an inverse and the same inverse, namely, Some not-S is not-P ; also, every proposition, including I, would have a contrapositive. These are erroneous results against which we have to be on our guard in the use of Euler’s fivefold scheme.
We find then that the explicit recognition of not-S and not-P practically leaves α, β, and γ unaffected, but causes δ and ε each to subdivide into two cases according as there is or is not anything that is both not-S and not-P; and the 171 Eulerian fivefold division has accordingly to give place to a sevenfold division.
In the diagrammatic representation of these seven relations, the entire universe of discourse may be indicated by a larger circle in which the ordinary Eulerian diagrams (with some slight necessary modifications) are included. We shall then have the following scheme:—
172 It may be useful to repeat these diagrams with an explicit indication in regard to each subdivision of the universe as to whether it is S or not-S, P or not-P.[175] The scheme will then appear as follows:—
[175] We might also represent the universe of discourse by a long rectangle divided into compartments, shewing which of the four possible combinations SP, SPʹ, SʹP, SʹPʹ are to be found. This plan will give the following which precisely correspond, as numbered, with those in the text:—
| (i) | SP | SʹPʹ |
| (ii) | SP | SʹP | SʹPʹ |
| (iii) | SP | SPʹ | SʹPʹ |
| (iv) | SP | SPʹ | SʹP | SʹPʹ |
| (v) | SP | SPʹ | SʹP |
| (vi) | SPʹ | SʹP | SʹPʹ |
| (vii) | SPʹ | SʹP |
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.
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).
EXERCISES.
132. Illustrate by means of the Eulerian diagrams (1) the relation between A and E, (2) the relation between I and O, (3) the conversion of I, (4) the contraposition of O, (5) the inversion of E. [K.]
133. A denies that none but X are Y ; B denies that none but Y are X. Which of the five class relations between X and Y (1) must they agree in rejecting, (2) may they agree in accepting? [C.]
134. Take all the ordinary propositions connecting any two terms, combine them in pairs so far as is possible without contradiction, and represent each combination diagrammatically. [J.]