We may select a few examples of this fatal dilemma.
1. In dealing with what it calls 'the meaning' of terms, propositions, etc., Formal Logic has always to choose between the meaning of the words and the meaning of the man. For it is clear that words which may be used ambiguously may on occasion leave no doubt as to their meaning, while conversely all may become 'ambiguous' in a context. If, therefore, the occasion is abstracted from, all forms must be treated verbally as ambiguous formulae, which may be used in different senses. If it is, nevertheless, attempted to deal with their actual meaning on any given occasion, what its maker meant the words to convey must be discovered, and the inquiry at once becomes 'psychological'—that is to say, 'extralogical.'
2. If judgments are not to be verbal ('propositions'), but real assertions which are actually meant, they must proceed from personal selections, and must have been chosen from among alternative formulations because of their superior value for their maker's purpose. But all this is plainly an affair of psychology. So inevitable is this that a truly formal Ideal of 'Logic' would exclude ail judgment whatever from the complete system of 'eternal' Truth. For from such a system no part could be rightly extracted to stand alone. Such a selection could be effected and justified only by the exigencies of a human thinker.
The impotent verbalism of the formal treatment of judgment appears in another way when the question is raised how a 'true' judgment is to be distinguished from a 'false.' For the logician, if his public will not accept either the relegation of this distinction to 'psychology' or the proper formal answer that all judgments are (formally) 'true' and even 'infallible,' can think of nothing better to say than that if the 'judgment' is not true it was not a 'true judgment,' but a false 'opinion' which may be abandoned to 'psychology.'[[G]] Apparently he is not concerned to help men to discriminate between 'judgments' and 'opinions,' or even to show that true 'judgments' do in fact occur.
3. Inference involves Formal Logic in a host of difficulties.
(a) If it is not to be a verbal manipulation of phrases whose coming together is not inquired into, it must be a connected train of thought. But such a connection of thoughts cannot be conceived or understood without reference to the purpose of a reasoner, who selects what he requires from the totality of 'truths.' If, then, 'Logic' has merely to contemplate this eternal and immutable system of truth in its integrity, and forbids all selection from it for a merely human purpose, how can it either justify, or even understand, the drawing of any inference whatever?
(b) Formal Logic clearly will not quail before the charge of uselessness. But on its own principles it ought to be consistent. But by this test also, when it is rigorously judged by it, it fails completely. Its inconsistencies are many and incurable. It cannot even be consistent in its theory of the simplest fundamentals. It is found upon some occasions to define judgment as that which may be either true or false; and upon others as that which is 'true' (formally)—i.e., it cannot decide whether or not to ignore the existence of error.
(c) The Formal view of inference regards it as a 'paradox.' An inference is required on the one hand to supply fresh information, and on the other to follow rigorously from its premisses; it must, in a word, exhibit both novelty and necessity. It would seem, however, that if our inference genuinely had imparted new knowledge, the event must be merely psychological; for how can any process or event perturb, or add to, the completed totality of truth in itself? On the other hand, if the 'necessity' of the operation be taken seriously, the 'inference' becomes illusory; for if the conclusion inferred is already contained in the premisses, what sense is there in the purely verbal process of drawing it out?
(d) Most glaringly inadequate of all, however, is the Formal doctrine of 'Proof' contained in its theory of the Syllogism. A Formal or verbal syllogism depends essentially on the ability of its Middle Term to connect the terms in its conclusion. If, however, the Middle Term has not the same meaning in the two premisses, the syllogism breaks in two, and no 'valid' conclusion can be reached. Now, whether in fact any particular Middle Term bears the same meaning in any actual reasoning Formal Logic has debarred itself from inquiring, by deciding that actual meaning was 'psychological.' It has to be content, therefore, with an identity in the word employed for its Middle, But this evidence may always fail; for when two premisses which are (in general) 'true' are brought together for the purpose of drawing a particular conclusion, a glaring falsehood may result. E.g., it would in general be granted that 'iron sinks in water,' yet it does not follow that because 'this ship is iron' it will 'sink in water,' Hence syllogistic 'proof' seems quite devoid of the 'cogency' it claimed. After a conclusion has been 'demonstrated' it has still to come true in fact. This flaw in the Syllogism was first pointed out by Mr. Alfred Sidgwick.
(e) The formal Syllogism, moreover, conceals another formal flaw. An infinite regress lurks in its bosom. For if its premisses are disputed, they must in turn be 'proved.' Four fresh premisses are needed, and if these again are challenged, the number of true premisses needed to prove the first conclusion goes on doubling at every step ad infinitum. The only way to stop the process that occurred to logicians was an appeal to the 'self-evident' truth of 'intuitions'; but this has been shown to be argumentatively worthless. From this difficulty the pragmatist alone escapes, by assuming his premisses provisionally and arguing forwards, in order to test them by their consequences. If the deduced conclusion can be verified in fact, the premisses grow more assured. Thus every real inference is an experiment, and 'proof' is an affair of continuous trial and verification—not an infinite falling back upon an elusive 'certainty,' but an infinite reaching forwards towards a fuller consummation.