For what is ordinarily known as the law of excluded middle, Jevons proposes the name law of duality.[458] This he does on the ground that the law in question asserts that at every step there are two possible alternatives, and hence gives to all the formulae of reasoning a dual character. The law of duality occupies an important position in Jevons’s system of formal logic, which is based on the repeated application of the principle of dichotomal division. It may, however, be questioned whether, as thus employed, the law of duality ought not to include the law of contradiction as well as the law of excluded middle. It is as important at each stage that the alternatives are exclusive as that they are exhaustive.
[458] Principles of Science, 1, § 3.
419. Grounds on which the absolute universality and necessity of the law of excluded middle have been denied.—The universal applicability of the law of excluded middle has been more frequently denied than that of either of the two laws previously discussed. The denial usually depends upon a confusion between contradictory opposition and contrary opposition. It is said, for example, that there is a mean between greater and less. This is true; but the law of excluded middle does not exclude the possibility of such a mean. That law does not tell us that a given quantity must be either greater or less than another given quantity; it only tells us that it must be either greater or not greater.
Closely connected with this is the case where our inability 461 (through lack of the requisite knowledge or power of discernment) to decide in favour of either of two contradictory alternatives is supposed to yield a third alternative; as, for example, where to the two alternatives “guilty” and “not guilty” is added the third alternative “not proven.” “Guilty” and “not guilty,” considered purely in relation to the supposed culprit, are true contradictories, and they admit of no mean. But “proved to be guilty” and “proved to be not guilty” are contraries, not contradictories; and it is here that the third alternative “not proven” comes in.
Some difficulty may also arise from ambiguity or uncertainty in the use of language. Thus it may perhaps be said that a prisoner may be neither “guilty” nor “not guilty,” but “partially guilty.” By “guilty,” however, we must understand either “entirely guilty” or “guilty in any degree”; and whichever of these meanings we adopt the difficulty is resolved.
We may deal similarly with the question whether an action occupying a finite interval of time for its completion has or has not taken place when it is actually proceeding; for example, whether a battle has or has not been fought when it is half through, or whether the sun has or has not risen when half its circumference is above the horizon.
The difficulties which arise in such cases as these are really verbal difficulties.
Other difficulties arising from uncertainty as to the precise range of application of terms are partly verbal and partly dependent upon our imperfect powers of discrimination. We may perhaps hesitate to say of a given colour whether it is “blue” or “green,” and therefore whether it is “blue” or “not blue.” If, however, by means of the spectrum or otherwise we are able to determine quite precisely what we mean by “blue,” the difficulty is obviated.
Mill remarks, on a different ground from any of the above, that the principle of excluded middle is not true unless with a large qualification. “A proposition must be either true or false, provided that the predicate be one which can in any intelligible sense be attributed to the subject. ‘Abracadabra is a second intention’ is neither true nor false. Between the true and the false there is a third possibility, the unmeaning” (Logic, ii. 7, § 5).
The reply to this is that the law of excluded middle applies only to propositions properly so-called, that is, to propositions regarded as the verbal expressions of judgments, a condition which clearly is 462 not satisfied by a sentence (falsely called a proposition) which is unmeaning. If we define a proposition as the verbal expression of a judgment, then an “unmeaning proposition”—a mere fortuitous jumble of words that conveys nothing to the mind—is in reality a contradiction in terms.