-- 434. Inductive inferences are wholly extraneous to the science of formal logic, which deals only with formal, or necessary, inferences, that is to say with deductive inferences, whether immediate or mediate. These are called formal, because the truth of the consequent is apparent from the mere form of the antecedent, whatever be the nature of the matter, that is, whatever be the terms employed in the proposition or pair of propositions which const.i.tutes the antecedent. In deductive inference we never do more than vary the form of the truth from which we started. When from the proposition "Brutus was the founder of the Roman Republic," we elicit the consequence "The founder of the Roman Republic was Brutus," we certainly have nothing more in the consequent than was already contained in the antecedent; yet all deductive inferences may be reduced to ident.i.ties as palpable as this, the only difference being that in more complicated cases the consequent is contained in the antecedent along with a number of other things, whereas in this case the consequent is absolutely all that the antecedent contains.
-- 435. On the other hand, it is of the very essence of induction that there should be a process from the known to the unknown. Widely different as these two operations of the mind are, they are yet both included under the definition which we have given of inference, as the pa.s.sage of the mind from one or more propositions to another. It is necessary to point this out, because some logicians maintain that all inference must be from the known to the unknown, whereas others confine it to "the carrying out into the last proposition of what was virtually contained in the antecedent judgements."
-- 436. Another point of difference that has to be noticed between induction and deduction is that no inductive inference can ever attain more than a high degree of probability, whereas a deductive inference is certain, but its certainty is purely hypothetical.
-- 437. Without touching now on the metaphysical difficulty as to how we pa.s.s at all from the known to the unknown, let us grant that there is no fact better attested by experience than this--"That where the circ.u.mstances are precisely alike, like results follow." But then we never can be absolutely sure that the circ.u.mstances in any two cases are precisely alike. All the experience of all past ages in favour of the daily rising of the sun is not enough to render us theoretically certain that the sun will rise tomorrow We shall act indeed with a perfect reliance upon the a.s.sumption of the coming day-break; but, for all that, the time may arrive when the conditions of the universe shall have changed, and the sun will rise no more.
-- 438. On the other hand a deductive inference has all the certainty that can be imparted to it by the laws of thought, or, in other words, by the structure of our mental faculties; but this certainty is purely hypothetical. We may feel a.s.sured that if the premisses are true, the conclusion is true also. But for the truth of our premisses we have to fall back upon induction or upon intuition. It is not the province of deductive logic to discuss the material truth or falsity of the propositions upon which our reasonings are based. This task is left to inductive logic, the aim of which is to establish, if possible, a test of material truth and falsity.
-- 439. Thus while deduction is concerned only with the relative truth or falsity of propositions, induction is concerned with their actual truth or falsity. For this reason deductive logic has been termed the logic of consistency, not of truth.
-- 440. It is not quite accurate to say that in deduction we proceed from the more to the less general, still less to say, as is often said, that we proceed from the universal to the particular. For it may happen that the consequent is of precisely the same amount of generality as the antecedent. This is so, not only in most forms of immediate inference, but also in a syllogism which consists of singular propositions only, e.g.
The tallest man in Oxford is under eight feet.
So and so is the tallest man in Oxford.
.". So and so is under eight feet.
This form of inference has been named Traduction; but there is no essential difference between its laws and those of deduction.
-- 441. Subjoined is a cla.s.sification of inferences, which will serve as a map of the country we are now about to explore.
Inference ________________________|__________ | | Inductive Deductive _________________|_______________ | | Immediate Mediate ___________|__________ ______|______ | | | | Simple Compound Simple Complex ______|________________ | ______|_____________|_ | | | | | | | Opposition Conversion Permutation | Conjunctive Disjunctive Dilemma | _________|________ | | Conversion Conversion by by Negation position
CHAPTER II.
_Of Deductive Inferences._
$ 442. Deductive inferences are of two kinds--Immediate and Mediate.
-- 443. An immediate inference is so called because it is effected without the intervention of a middle term, which is required in mediate inference.
-- 444. But the distinction between the two might be conveyed with at least equal aptness in this way--
An immediate inference is the comparison of two propositions directly.
A mediate inference is the comparison of two propositions by means of a third.
-- 445. In that sense of the term inference in which it is confined to the consequent, it may be said that--
An immediate inference is one derived from a single proposition.
A mediate inference is one derived from two propositions conjointly.
-- 446. There are never more than two propositions in the antecedent of a deductive inference. Wherever we have a conclusion following from more than two propositions, there will be found to be more than one inference.
-- 447. There are three simple forms of immediate inference, namely Opposition, Conversion and Permutation.
-- 448. Besides these there are certain compound forms, in which permutation is combined with conversion.
CHAPTER III.
_Of Opposition._
-- 449. Opposition is an immediate inference grounded on the relation between propositions which have the same terms, but differ in quant.i.ty or in quality or in both.
-- 450. In order that there should be any formal opposition between two propositions, it is necessary that their terms should be the same. There can be no opposition between two such propositions as these--
(1) All angels have wings.
(2) No cows are carnivorous.
-- 451. If we are given a pair of terms, say A for subject and B for predicate, and allowed to affix such quant.i.ty and quality as we please, we can of course make up the four kinds of proposition recognised by logic, namely,
A. All A is B.
E. No A is B.
I. Some A is B.
O. Some A is not B.
-- 452. Now the problem of opposition is this: Given the truth or falsity of any one of the four propositions A, E, I, O, what can be ascertained with regard to the truth or falsity of the rest, the matter of them being supposed to be the same?
-- 453. The relations to one another of these four propositions are usually exhibited in the following scheme--
A ... . Contrary ... . E . . . .
Subaltern Contradictory Subaltern . . . .
I ... Sub-contrary ... O
-- 454. Contrary Opposition is between two universals which differ in quality.