By an “unmeaning proposition” in the above argument we have understood a so-called proposition which has no meaning for the person who utters it or for anyone else. To a given individual a statement made by someone else may be unmeaning because he does not understand the force of the terms employed; but this in no way affects the principle that the statement will as a matter of fact be either true or false.
Whilst, however, every judgment must be either true or false, it is quite possible that unsuitable questions may be put, the correct answers to which will be negative, but will be felt to be barren and insignificant because anyone who understands the meaning of the terms employed will recognise at once that the predicate cannot in any intelligible sense be attributed to the subject.[459]
Is virtue circular? This question is felt to be absurd; but it is not unmeaning. By saying that anything is circular we mean that it has some figure and that its figure is circular. If, therefore, the question of circularity is raised in connexion with something that is immaterial, and therefore has no figure at all, the answer must be in the negative.[460]
[460] Compare Bradley, Principles of Logic, p. 145. Mr Bradley puts the question, “When a predicate is really known not to be ‘one which can in any intelligible sense be attributed to the subject,’ is not that itself ground enough for denial?”
This point may perhaps hardly seem worth raising. It helps, however, to explain how Mill is led to his denial of the universal applicability of the law of excluded middle. In his criticism of Hamilton’s doctrine of noumena the question is raised whether matter in itself has a minimum of divisibility or is infinitely divisible. Mill’s answer is that although we appear here to have contradictory alternatives, both may have to be rejected, since divisibility may not be predicable at all of matter in itself. In other words, the proposition that matter in itself has a minimum of divisibility is neither true nor false, but unmeaning.
It is to be observed, however, that “having a minimum of 463 divisibility” and “being infinitely divisible” are not contradictories except within the sphere of the divisible. If a wider point of view be taken, the contradictory of “having a minimum of divisibility” must be expressed simply in the form “not having a minimum of divisibility,” the latter including the case of “infinite divisibility,” and also that of “the absolute inapplicability of the attribute of divisibility.”
420. Are the Laws of Thought also Laws of Things?—On the view taken of the laws of thought in the preceding pages, the question whether these laws are also laws of things must be regarded as somewhat misleading. We have described the laws as postulates which are fundamental in all valid thought and reasoning, and we have regarded them as concerned essentially with judgments. Our results may be very briefly summarised as follows.
The truth affirmed in any judgment, when fully expressed, is independent of time and context. It is accordingly not open to us to accept a judgment at one stage of an argument or course of reasoning and reject it at another. This unambiguity of the fact of judgment is declared by the law of identity, and again by the law of contradiction, the one looking at the question from the positive, and, the other from the negative, point of view. Again, all judgment involves both affirmation and denial; and the force of any judgment is not fully grasped by us until we realise clearly what it denies as well as what it affirms. The law of contradiction, in conjunction with the law of excluded middle, has the function of making explicit what we mean by denial. The three laws may be expressed by these formulae: I affirm what I affirm, and deny what I deny ; If I make any affirmation, I thereby deny its contradictory ; If I make any denial, I thereby affirm its contradictory.
It follows that we cannot make any progress in material knowledge except in subordination to these laws. But at the same time they do not directly advance our knowledge of things. They are distinctly laws relating to judgments, and not directly to the things about which we judge.