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النشر الإلكتروني

VI.

IMMEDIATE INFERENCE.

CHAPTER XX.

IMMEDIATE INFERENCE.

Formal Inference and its Logical Principle.

FORMAL Inference is reasoning from accepted statements or premisses to conclusions implied in them. The Principle of Formal Inference may be called the Law of Formal Validity. It may be formulated as follows:

If a given proposition or set of propositions is accepted, then the further propositions which are implied in what is thus admitted, these, and these only, must also be accepted; and the further propositions which are in contradiction with any one of the admitted propositions, or with any one of their implications, these, and these only, must be rejected.

The Law of Formal Validity is no new addition to the Laws of Thought. It simply interprets the Laws of Formal Identity and Non-contradiction as formulated in relation to Inference.

Identity, in its relation to Inference, is a matter of Implication. If one statement is implied in another, the two must belong to one and the same identical system. This systematic intimacy between them constitutes their logical Identity. The Law of Formal Identity in its relation to Inference may be formulated follows:

as

If a given proposition or set of propositions is accepted, then the further propositions which are implied in what is thus admitted, these, and these only, must also be accepted.

So the Law of Non-contradiction in its relation to Inference may be formulated thus :

If a given proposition or set of propositions is accepted, then the further propositions which are in contradiction with any one of the admitted propositions, or with any propositions implied by them, these, and these only, must be rejected.

The Law of Formal Inference is sometimes stated in the form of the concise though negative precept 'Not to go beyond the premisses.' We may connect this injunction with the Law of Formal Validity by showing that in going beyond the premisses we necessarily contradict certain propositions implied by these premisses. Thus, from the accepted statement 'All S's are P's' we might conclude that 'All P's are S's.' In drawing this conclusion we should be going beyond the accepted premiss, though we should not be contradicting it. But though we do not contradict the premiss 'All S's are P's' when we accept the conclusion 'All P's are S's' as if this were an inference from it, we do thereby contradict certain implications of that premiss. Thus, when we posit 'All S's are P's '-i.e., 'All S's are some P's 'we expressly don't state anything about all the P's. Hence no statement dealing with all the P's is implied in the original statement. Consequently, no statement dealing with all the P's can be disimplicated from the original statement. But the incorrectly drawn conclusion tells us that at least one statement dealing with all the P's can be disimplicated from the original statement. Thus we see that the proposition "All P's are S's" is implied in the original statement "All S's are P's"" contradicts an implication of the accepted premiss, and must be rejected as inconsistent with it. (It is not, of course, the proposition 'All P's are S's' which contradicts an implication of the accepted premiss. Were this the case, we should be compelled, after accepting 'All S's are P's,' to reject 'All P's are S's' as inconsistent with it.)

We see, then, that, in going beyond the premiss, we have disregarded the Law of Validity, and fallen, at one point at least, into meaningless self-contradiction.

The most simple expression of the Principle of Formal Identity considered as a principle of Inference, an expression but one degree removed from the blank formula of Tautology-' If A is accepted, then A is accepted '-is provided by the First Law of Subalternation, the law which states that if the universal proposition is accepted, the particular proposition of the same quality must be accepted also.

The attempt to prove this law is instructive, as it serves to bring out the fact that the Principle of Formal Identity cannot be proved by means of the Principle of Non-contradiction.

Let us suppose that in accepting A we do not thereby logically bind ourselves to accept I. On this supposition, when we accept A we disable ourselves from rejecting the contradictory of I. But this disability involves us in a logical inconsistency, since A and E are contraries.

This proof, however, presupposes the truth of the law it is endeavouring to prove; for it assumes a law of Contrary Opposition which can be proved only by the help of the very law of Subalternation which we are considering. We have therefore committed the fallacy of reasoning in a circle. Hence the Principle of Identity, in this its simplest form, cannot be proved by means of the Principle of Non-contradiction.

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