Here then we have one form in which the point mainly at issue in regard to the nature of inference presents itself. Is it possible for two judgments to be different quâ judgments, although from the objective standpoint one of them states nothing that is not also stated by the other? Or, to put the question differently, can two judgments (or sets of judgments) be distinct as judgments although they are not logically independent, that is, although self-evident relations exist between them such that the truth of one of them involves the truth of the other?
I am ready to admit that it is no easy matter to draw a hard and fast line determining where mere verbal novelty ends and subjective novelty begins. Before attempting to deal with this difficulty, however, I will endeavour to shew that there undoubtedly are cases in which we have progress in thought without reaching anything that is objectively new.
Mill, after giving examples of so-called immediate inferences, says, “In all these cases there is not really any inference; there is in the conclusion no new truth, nothing but what was already asserted in the premisses, and obvious to whoever apprehends them” (Logic, ii. 1, § 2). Now it is certainly the case that in any formal inference the conclusion is implicitly contained in the premisses, and affirms no absolutely new fact. But it is one thing to say that a conclusion is virtually contained in certain premisses, and quite another to say that it is obvious to whoever apprehends the premisses. The identification of these two positions is one of the unfortunate consequences of taking simple conversion as the type of all immediate inference, and a single syllogism in Barbara as the type of all mediate formal inference. It may be difficult for anyone to apprehend that no S is P without at once apprehending that no P is S, or to apprehend the premisses of a syllogism in Barbara without at once apprehending the conclusion also. These cases will need discussion; but just now we are more concerned to point out 420 that there are other formal inferences against which any similar charge of obviousness cannot be brought.
All the theorems of geometry are virtually contained in certain axioms and postulates, and if we can exhaustively enumerate the axioms there is in a sense no new geometrical fact left for us to assert. Yet no one would say that the whole of geometry is at once obvious to anyone who has clearly apprehended the axioms. We shall, however, deal with syllogistic inference more in detail in a later [section]. For the present we will in the main confine ourselves to immediate inferences.
In order to shew that the conclusion of an immediate inference is not always immediately obvious to anyone who clearly apprehends the given premiss, it may be pointed out that it is Euclid’s practice to give independent proofs of contrapositives.[442] For example, the second part of Euclid I. 29 is the contrapositive of axiom 12. But it is impossible to suppose that if Euclid had regarded I. 29 as not really distinct from axiom 12, but merely as a repetition of that axiom in other words, he would have given an elaborate proof of it. The following are two other fairly simple examples of immediate inferences: Where B is absent, either A and C are both present or A and D are both absent, therefore, Where C is absent, either B is present or D is absent ; Where A is present, either B and C are both present, or C is present without D, or C is present without F, or H is present, therefore, Where C is absent, we never find H absent, A being present.
[442] See [note 4] on page 136.
In such cases as these, and they are comparatively simple ones of their kind, it cannot be maintained that the conclusion is at once obvious when the premiss is given. As a matter of fact, mistakes are not unfrequently made in immediate inferences of a still simpler and more elementary character.
379. The Direct Import and the Implications of a Proposition.—At this point a question may fairly be raised as to how we determine what is the explicit force of a given proposition, assuming the proposition to be clearly understood and fully grasped by the mind. This question is by no means easy 421 to answer, and the difficulty which it presents is the source of the doubt which sometimes arises when we attempt to draw the line between immediate inferences and mere verbal transformations.
If immediate inferences are possible, we must be able to discriminate between the direct logical import (or meaning) of a proposition and its logical implications; and it must be possible to grasp fully the meaning without at the same time necessarily realising all the implications.[443] We may begin by distinguishing between (1) the content of the judgment actually present to our mind when we utter or accept a proposition in ordinary discourse or in ordinary reading; (2) the content of the judgment which on reflection we are able to regard as constituting the full logical meaning of the proposition; (3) the content of this judgment together with the content of other judgments which it logically implies.