It follows that we must reject Jevons’s further statement that “a proposition of moderate complexity has an almost unlimited number of contradictory propositions, which are more or less in conflict with the original. The truth of any one or more of these contradictories establishes the falsity of the original, but the falsity of the original does not establish the truth of any one or more of its contradictories.”[111] No doubt a proposition which is complicated in form may yield an indefinite number of other non-equivalent propositions the truth of any one of which is inconsistent with its own. It will also be true that its contradictory can be expressed in more than one form. But these forms will necessarily be equivalent to one another, since it is impossible for a proposition to have two or more non-equivalent contradictories. This position may be formally established as follows. Let Q and R be both contradictories of P. They will be equivalent if it can 114 be shewn that if Q then R, and if R then Q. Since P and Q are contradictories, we have If Q then not P, and since P and R are contradictories we have If not P then R. Combining these two propositions we have the conclusion If Q then R. If R then Q follows similarly. Hence we have established the desired result.
[111] It must be admitted that it has not been uncommon for logicians to use the word contradict somewhat loosely. For example, in the Port Royal Logic, we find the following: “Except the wise man (said the Stoics) all men are truly fools. This may be contradicted (1) by maintaining that the wise man of the Stoics was a fool as well as other men; (2) by maintaining that there were others, besides their wise man, who were not fools; (3) by affirming that the wise man of the Stoics was a fool, and that other men were not” (p. 140). The affirmation of any one of these three propositions certainly renders it necessary to deny the truth of the given proposition, but no one of them is by itself the contradictory of the given proposition. The true contradictory is the alternative proposition: Either the wise man of the Stoics is a fool or some other men are not fools.
In connexion with the same point, Jevons raises another question, in regard to which his view is also open to criticism. He says, “But the question arises whether there is not confusion of ideas in the usual treatment of this ancient doctrine of opposition, and whether a contradictory of a proposition is not any proposition which involves the falsity of the original, but is not the sole condition of it. I apprehend that any assertion is false which is made without sufficient grounds. It is false to assert that the hidden side of the moon is covered with mountains, not because we can prove the contradictory, but because we know that the assertor must have made the assertion without evidence. If a person ignorant of mathematics were to assert that ‘all involutes are transcendental curves,’ he would be making a false assertion, because, whether they are so or not, he cannot know it.” We should, however, involve ourselves in hopeless confusion were we to consider the truth or falsity of a proposition to depend upon the knowledge of the person affirming it, so that the same proposition would be now true, now false. It will be observed further that on Jevons’s view both the propositions S is P and S is not P would be false to a person quite ignorant of the nature of S. This would mean that we could not pass from the falsity of a proposition to the truth of its contradictory; and such a result as this would render any progress in thought impossible.
81. Contrary Opposition.—Seeking to generalise the relation between A and E, we might naturally be led to characterize the contrary of a given proposition by saying that it goes beyond mere denial, and sets up a further assertion as far as possible removed from the original assertion; so that, whilst the contradictory of a proposition denies its entire truth, its contrary may be said to assert its entire falsehood. A pair of contraries as thus defined may be regarded as standing at the opposite 115 ends of a scale on which there are a number of intermediate positions.
On this definition, however, the notion of contrariety cannot very satisfactorily be extended much beyond the particular case contemplated in the ordinary square of opposition. For if we have a proposition which cannot itself be regarded as standing at one end of a scale, but only as occupying an intermediate position, such proposition cannot be regarded as forming one of a pair of contraries. Plurative and numerically definite propositions may be taken as illustrations.
Hence if it is desired to define contrariety so that the conception may be generally applicable, the idea of two propositions standing, as it were, furthest apart from each other must be given up, and any two propositions may be described as contraries if they are inconsistent with one another without at the same time exhausting all possibilities. Contraries must on this definition always admit of a mean, but they may not always be what we should speak of as diametrical opposites, and any given proposition is not limited to a single contrary, but may have an indefinite number of non-equivalent contraries. At the same time, it will be observed that this definition still suffices to identify A and E as a pair of contraries, and as the only pair in the traditional scheme of opposition.
82. The Opposition of Singular Propositions.—Taking the proposition Socrates is wise, its contradictory is Socrates is not wise ;[112] and so long as we keep to the same terms, we cannot go beyond this simple denial. The proposition has, therefore, no formal contrary.[113] This opposition of singulars has been called secondary opposition (Mansel’s Aldrich, p. 56).
[112] This must be regarded as the correct contradictory from the point of view reached in the present chapter. The question becomes a little more difficult when the existential interpretation of propositions is taken into account.
[113] We can obtain what may be called a material contrary of the given proposition by making use of the contrary of the predicate instead of its mere contradictory; thus, Socrates has not a grain of sense. This is spoken of as material contrariety because it necessitates the introduction of a fresh term that could not be formally obtained out of the given proposition. It should be added that the distinction between formal and material contrariety might also be applied in the case of general propositions.
116 If, however, there is secondary quantification in a proposition having a singular subject, then we may obtain the ordinary square of opposition. Thus, if our original proposition is Socrates is always (or in all respects) wise, it is contradicted by the statement that Socrates is sometimes (or in some respects) not wise, while it has for its contrary, Socrates is never (or in no respects) wise, and for its subaltern, Socrates is sometimes (or in some respects) wise. It may be said that when we thus regard Socrates as having different characteristics at different times or under different conditions, our subject is not strictly singular, since it is no longer a whole indivisible. This is in a sense true, and we might no doubt replace our proposition by one having for its subject “the judgments or the acts of Socrates.” But it does not appear that this resolution of the proposition is necessary for its logical treatment.