Immediate Inference

The next thing to observe in connection with these four types of proposition is what is called the distribution of terms in each of them. A term is said to be distributed in a proposition when reference is made explicitly to the whole range or class of objects, etc., which the term denotes; otherwise it is said to be undistrib uted. Thus in SeP (no S whatever is any P whatever) both S and P are distributed. In SaP (every S is P but nothing is stated about every P) S is distributed, but P is not. In SiP (some S's are P, nothing definite is said about all S's or all P's) neither S nor P is distributed. In SoP (some S's are not any P whatsoever) S is not distributed, but P is. Briefly, universal propositions (A,E) distribute their subjects and negative propositions (E,0) distrib ute their predicates.

It is a general rule of all formal reasoning that no term may be distributed in the conclusion unless it is distributed in the premise or premises. For, otherwise, the conclusion would go beyond the evidence. A great many cases of invalid inference are due to a breach of this rule; and some of the most important rules of valid inference are meant to be safeguards against such a breach. Hence the importance of becoming familiar with these few points about the distribution of terms in propositions. There is no objection at all to not distributing in the conclusion a term that is distributed in the premises. Just as the maxim against extravagance does not demand that people should spend the whole of their income, but only that they should keep within it, 50 this rule only requires that people should not go beyond the evi dence; it does not require them to exhaust the whole of it when it is unnecessary.

Opposition and Eduction.

The doctrine of immediate in ference, as already remarked above, is concerned with the impli cations of single or isolated propositions, as distinguished from the joint implications of two or more propositions taken together. It has two main parts. One part, known as the doctrine of oppo sition, deals with the implications of each type of proposition in relation to others having precisely the same subject and the same predicate, but differing in quality or quantity. The other part, known as the doctrine of eductions, deals with the impli cations of each type of proposition in relation to others having not the same subject and the same predicate but the same terms or their contradictories. In other words, "opposition" is con cerned with the relations between SaP, SeP, SiP, SoP, whereas "eduction" is concerned with the relations between propositions of the types S-P, P-S, S-P, These tionships look more complicated than they really are, and can be determined with comparative ease with the aid of the laws of contradiction and excluded middle, and the rule relating to the distribution of terms just explained.

Opposition.

Given SaP, SeP and SoP must both be false, otherwise, all or some S's would both be P and not be P, which would violate the law of contradiction; but SiP must be true, as the some S's must be included in "every S" of SaP. Again,

given SeP, SaP and SiP must be false, otherwise all or some S's would both not be and be P, which is against the law of contradic tion; the SoP must be true, because its "some S's" are included in all the S's of which P is denied in SeP. Next, SiP excludes SeP, otherwise "some S's" would both be and not be P, in violation of the law of contradiction; but it throws no light on SaP, SoP, either of which (though not both at once of course) might be true or false, when SiP is true, without offending against any law of thought or rule of inference. Lastly, SoP excludes SaP, other wise "some S's" would both not be and be P, against the law of contradiction; but it has no implication in relation to SeP or SiP, either of which (though not both) might be true or false when SoP is true.

To complete the study of the interrelations between these propositional types, let us consider next the implications of the rejection of any one of them. So far we have only considered the implications of the acceptance of any one of them. To reject SaP is to maintain that not every S is P, that is, that it is wrong to say of one or more S's at least that they are P. But if so, then by the law of excluded middle it must be right to say of the one or more S's in question that they are not P, or SoP. Thus the rejection or falsity of SaP implies the acceptance or truth of SoP. On the other hand, SiP and SeP are not affected by it ; either (though not both of course) might be true or false, if SaP be untrue. Similarly, to reject SeP is to maintain that it is wrong to say of every S that it is not P, that is, that there are one or more S's of which it should not be said that they are not P. If so, then by the law of excluded middle it must be right to say of the one or more S's in question that they are P, or SiP. Thus the falsity of SeP implies the truth of SiP. But it does not affect SoP or SaP, either of which (but not both) might be true or false when SeP is untrue. Next, to reject SiP is to maintain that it is incorrect to say of even one S that it is P (for some S's are P means "one S at least is P"). If so, then by the law of excluded middle it must be right to say of every S that it is not P, or SeP, which, as has already been seen, implies SoP. Thus the falsity of SiP implies the truth of both SeP and SoP, each of which, as was already shown above, excludes the truth of SaP, by the law of contradiction. Similarly, the falsity of SoP implies the truth of both SaP and SiP. For to reject SoP is to maintain that it cannot be said of even one S that it is not P. Hence, by the law of excluded middle, it must be said of every S that it is P or SaP, which implies SiP, and each of these implies the falsity of SeP.