what the logician would call the point of view of extension, by saying that two parts of the same class are not necessarily identical. In these instances the observers' state of miud seemed to be not so much the consciousness of certain ideas that should have been but were not included in the argument, like the 'carrots and cabbages' in (i) or the insane people in (3), as a more vague and abstract mental process for which no more fitting descriptive term than 'conscious- ness of inequality' presents itself. Such a mental process must be of frequent occurrence in purely mathematical reasoning. It probably varies from a practically unanalyzable 'relational' process, through a more complex process associated with some vague ideas of the omitted factors, the other parts of the class which have not been taken into account, and thus by imperceptible gradations may pass into a process involving definite thought of the neglected ideas, such as we have discussed under (c). It is to be noted that the mere 'conscious- ness of inequality' would not of itself be a source of unpleasantness; the unpleasantness must arise from the fact that the terms ought not to be unequal. The thought of two sub-classes as not being identical with each other would not be disagreeable unless the argument de- manded their equality. Is it not probable that the unpleasantness in this case arises ultimately from the same source as it did in (c);that is, from the confusion and" division of attention that result when in- stead of passing smoothly from one term to the aext, attention is 'led aside' to consider the omitted factors? In the instances we are considering, the omitted factors are not definitely thought of as they are in the cases under (c), but even although they are represented only by an unanalyzable relational process, the affective tone which would accompany the clear and developed thought of them may be transferred to the relational 'consciousness of inequality.'
(e) Under this head we may consider the cases where unpleasant- ness arose because of a quite vague and indefinite sense of something wrong with the argument. The observer cannot or at least does not state that anything is wrong with a particular part of the reasoning; she does not 'place' the wrongness; it is only vaguely felt. "Some- thing is wrong with the 'therefore,' " said two observers in the case of syllogism (i). Two persons complained that (5) was 'unconvincing' and therefore unpleasant. Four gave 'incompleteness' without fur- ther specification as the source of unpleasantness in (i). It is, of course, impossible to be sure, with our untrained observers, that the sense of incompleteness was in every case perfectly vague; the omitted factors might have been thought of with some definiteness although the observer did not take the trouble to report the fact. But it seems probable that a sense of something lacking did some- times accompany the reading of the arguments without being at- tached to anything definite. The term most frequently used to de- scribe the vague sense of something wrong was 'confusion.' Five persons named this as the sole source of unpleasantness in the first syllogism; three in the second, four in the third, five in the fourth, and two in the fifth. 'Confusion' would seem to be one degree vaguer than a sense of 'something omitted.' One might enumerate the logi- cal sources of the unpleasantness of a fallacy in the following order, beginning with the most indefinite, the passage from each stage to the next being brought about by better attention and analysis: general sense of confusion, sense of something omitted, sense of a lack of equivalence between two of the terms, clear idea of the omitted factors. All of these processes are unpleasant for what is ultimately the same reason: a tendency to division of the attention instead of allowing it to pass smoothly from one term to the next.
