The most noteworthy reformation of recent years in logic is the discovery and development by Mr. Bertrand Russell of the fact that the paradoxes—of Burali-Forti, Russell, KÖnig, Richard, and others—which have appeared of late years in the mathematical theory of aggregates and have just been referred to, are of an entirely logical nature, and that their avoidance requires us to take account of a principle which has been hitherto unrecognized, and which renders invalid several well-known arguments in refutation of scepticism, agnosticism, and the statement of a man that he asserts nothing.

Dr. Whitehead and Mr. Russell say:[113] “The principle which enables us to avoid illegitimate totalities may be stated as follows: ‘Whatever involves all of a collection must not be one of the collection,’ or conversely: ‘If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.’ We shall call this the ‘vicious-circle principle,’ because it enables us to avoid the vicious circles involved in the assumption of illegitimate totalities. Arguments which are condemned by the vicious-circle principle will be called ‘vicious-circle fallacies.’ Such arguments, in certain circumstances, may lead to contradictions, but it often happens that the conclusions to which they lead are in fact true, though the arguments are fallacious. Take, for example, the law of excluded middle in the form ‘all propositions are true or false.’ If from this law we argue that, because the law of excluded middle is a proposition, therefore the law of excluded middle is true or false, we incur a vicious-circle fallacy. ‘All propositions’ must be in some way limited before it becomes a legitimate totality, and any limitation which makes it legitimate must make any statement about the totality fall outside the totality. Similarly the imaginary sceptic who asserts that he knows nothing and is refuted by being asked if he knows that he knows nothing, has asserted nonsense, and has been fallaciously refuted by an argument which involves a vicious-circle fallacy. In order that the sceptic’s assertion may become significant it is necessary to place some limitation upon the things of which he is asserting his ignorance; the proposition that he is ignorant of every member of this collection must not itself be one of the collection. Hence any significant scepticism is not open to the above form of refutation.”

In fact, the world of things falls into various sets of things of the same “type.” For every propositional function ?(x) there is a range of values of x for which ?(x) has a signification as a true or a false proposition. Until this theory was brought forward, there were occasionally discussions as to whether an object which did not belong to the range of a certain propositional function possessed the corresponding property or not. Thus, Jevons, in early days,[114] was of opinion that virtue is neither black nor not-black because it is not coloured, but rather later[115] he admitted that virtue is not triangular.[116]

CHAPTER XXXIX