(1) is a psychological product which may be, and usually is, logically imperfect; that is to say, it needs to be amplified if we are fully to realise the meaning of the proposition. Such amplification cannot be regarded as constituting inference. For, in making any inference, our starting point must be the proposition considered in its logical character. The inference comes in when we pass from (2) to (3). The question, however, arises as to how far the amplification is to extend if our object is to stop short at (2). In other words, where does meaning end and implication begin?
It has been pointed out at an earlier stage that in assigning to given combinations of words their logical import there is a certain element of arbitrariness. There is often a similar element of arbitrariness in formulating the fundamental axioms of a science, as well as in framing definitions. Thus, in geometry we cannot do without some special axiom relating to parallel straight lines, but we have some choice as to what the axiom shall be. Hence what is an axiom in one system may be a theorem in another, and vice versâ. Similarly, whether Q is to be regarded as part of the meaning of P, or as an inference from P, may be relative to the interpretation 422 adopted of the schedule of propositions to which P belongs. Some illustrations of this point will be given shortly.
We have cited cases in which it appears clear that we have inference and not mere verbal transformation. But in most of these cases intermediate steps may be inserted; and if this is done to the fullest possible extent, the progress at each step may be so slight that it may not be at all easy to say wherein precisely the inference is to be found.
We must then proceed to consider the limiting cases in which there may be legitimate doubt as to whether we have inference or not. One of these cases is that of conversion. The question whether there is inference in conversion may be in itself, as Mr Bosanquet puts it, “a point of little interest” (Essentials of Logic, p. 141). Nevertheless, as a limiting case, it is not lightly to be put on one side when we are attempting to decide what fundamentally constitutes inference.
It appears to me that conversion is a process of inference if we are dealing with a schedule of propositions in which the predicative reading is adopted. In such a schedule the primary import of the various propositions involves a differentiation between subject and predicate, and to predicate P of S or to deny that P can be predicated of S is a different thing from predicating S of P or denying that S can be predicated of P. Moreover we may grasp the one relation without necessarily realising whether it does or does not involve the other. But in an equational system it is different. If two classes are affirmed to be identical it is merely a verbal question which is mentioned first, and we cannot consider that we have made any progress in thought when we merely alter the order in which they are named. It follows that we must consider that we have inference when we reduce a proposition expressed predicatively to the equational form.
In either schedule, contraposition (or a process analogous to contraposition) presents itself as an inference. In the one case, we have All S is P, therefore, Anything that is not P is not S ; in the other case, S = SP, therefore, Pʹ = PʹSʹ.
Suppose again that we have an existential schedule, and that we start from the proposition SPʹ = 0 [There is nothing that is 423 S and at the same time not P]. Here what corresponds to conversion is the passage to Either PS > 0 or S = 0 [There is something that is both P and S or else S is non-existent]; and, what corresponds to contraposition is the passage to PʹS = 0 [There is nothing that is not P and at the same time S]. Conversion, but not contraposition, now appears as a process of inference. It follows that there is inference when we pass to this schedule from either of the others, or vice versâ.
A further consequence to be drawn from the above considerations is that if propositions are given at random, inference may at the outset be required in order to adapt them to a given logical schedule, though as a rule this will not be necessary. This point has already been touched upon in section [48].
380. Syllogisms and Immediate Inferences.—In the above argument we have confined ourselves mainly to the consideration of immediate inferences. The same question in relation to the syllogism usually presents itself in a slightly different form, namely, whether every, syllogism involves a petitio principii ; and we shall discuss it in this form in the following [section]. In the meantime, we may observe that if there is no such thing as immediate inference properly so called, then the claims of the syllogism to contain inference become very hard to maintain. For by the aid of immediate inferences the premisses of a syllogism can be combined into a single proposition, and the conclusion can then be obtained as an immediate inference from the combination.[444]