Logic: Deductive and Inductive Part 7

Now, any relation between two terms may be viewed from either side--_A: B_ or _B: A_. It is in both cases the same fact; but, with the altered point of view, it may present a different character. For example, in the Immediate Inference--_A > B_ ? _B < a_--a="" diminishing="" turns="" into="" an="" increasing="" ratio,="" whilst="" the="" fact="" predicated="" remains="" the="" same.="" given,="" then,="" a="" relation="" between="" two="" terms="" as="" viewed="" from="" one="" to="" the="" other,="" the="" same="" relation="" viewed="" from="" the="" other="" to="" the="" one="" may="" be="" called="" the="" reciprocal.="" in="" the="" cases="" of="" equality,="" co-existence="" and="" simultaneity,="" the="" given="" relation="" and="" its="" reciprocal="" are="" not="" only="" the="" same="" fact,="" but="" they="" also="" have="" the="" same="" character:="" in="" the="" cases="" of="" greater="" and="" less="" and="" sequence,="" the="" character="">

We may, then, state the following rule for the conversion of propositions in which the whole relation explicitly stated is taken as the copula: Transpose the terms, and for the given relation subst.i.tute its reciprocal. Thus--

_A is the cause of B ? B is the effect of A._

The rule a.s.sumes that the reciprocal of a given relation is definitely known; and so far as this is true it may be extended to more concrete relations--

_A is a genus of B ? B is a species of A A is the father of B ? B is a child of A._

But not every relational expression has only one definite reciprocal. If we are told that _A is the brother of B_, we can only infer that _B is either the brother or the sister of A_. A list of all reciprocal relations is a desideratum of Logic.

-- 5. Obversion (otherwise called Permutation or aequipollence) is Immediate Inference by changing the quality of the given proposition and subst.i.tuting for its predicate the contradictory term. The given proposition is called the 'obvertend,' and the inference from it the 'obverse.' Thus the obvertend being--_Some philosophers are consistent reasoners_, the obverse will be--_Some philosophers are not inconsistent reasoners_.

The legitimacy of this mode of reasoning follows, in the case of affirmative propositions, from the principle of Contradiction, that if any term be affirmed of a subject, the contradictory term may be denied (chap. vi. -- 3). To obvert affirmative propositions, then, the rule is--Insert the negative sign, and for the predicate subst.i.tute its contradictory term.

A. _All S is P ? No S is not-P All men are fallible ? No men are infallible._

I. _Some S is P ? some S is not-P Some philosophers are consistent ? Some philosophers are not inconsistent._

In agreement with this mode of inference, we have the rule of modern English grammar, that 'two negatives make an affirmative.'

Again, by the principle of Excluded Middle, if any term be denied of a subject, its contradictory may be affirmed: to obvert negative propositions, then, the rule is--Remove the negative sign, and for the predicate subst.i.tute its contradictory term.

E. _No S is P ? All S is not-P No matter is destructible ? All matter is indestructible._

O. _Some S is not P ? Some S is not-P Some ideals are not attainable ? Some ideals are unattainable._

Thus, by obversion, each of the four propositions retains its quant.i.ty but changes its quality: A. to E., I. to O., E. to A., O. to I. And all the obverses are infinite propositions, the affirmative infinites having the sense of negatives, and the negative infinites having the sense of affirmatives.

Again, having obtained the obverse of a given proposition, it may be desirable to recover the obvertend; or it may at any time be requisite to change a given infinite proposition into the corresponding direct affirmative or negative; and in such cases the process is still obversion. Thus, if _No S is not-P_ be given us to recover the obvertend or to find the corresponding affirmative; the proposition being formally negative, we apply the rule for obverting negatives: 'Remove the negative sign, and for the predicate subst.i.tute its contradictory.' This yields the affirmative _All S is P_. Similarly, to obtain the obvertend of _All S is not-P_, apply the rule for obverting Affirmatives; and this yields _No S is P_.

-- 6. Contrariety.--We have seen in chap. iv. -- 8, that contrary terms are such that no two of them are predicable in the same way of the same subject, whilst perhaps neither may be predicable of it. Similarly, Contrary Propositions may be defined as those of which no two are ever both true together, whilst perhaps neither may be true; or, in other words, both may be false. This is the relation between A. and E. when concerned with the same matter: as A.--_All men are wise_; E.--_No men are wise_. Such propositions cannot both be true; but they may both be false, for some men may be wise and some not. They cannot both be true; for, by the principle of Contradiction, if _wise_ may be affirmed of _All men, not-wise_ must be denied; but _All men are not-wise_ is the obverse of _No men are wise_, which therefore may also be denied.

At the same time we cannot apply to A. and E. the principle of Excluded Middle, so as to show that one of them must be true of the same matter.

For if we deny that _All men are wise_, we do not necessarily deny the attribute 'wise' of each and every man: to say that _Not all are wise_ may mean no more than that _Some are not_. This gives a proposition in the form of O.; which, as we have seen, does not imply its subalternans, E.

If, however, two Singular Propositions, having the same matter, but differing in quality, are to be treated as universals, and therefore as A. and E., they are, nevertheless, contradictory and not merely contrary; for one of them must be false and the other true.

-- 7. Contradiction is a relation between two propositions a.n.a.logous to that between contradictory terms (one of which being affirmed of a subject the other is denied)--such, namely, that one of them is false and the other true. This is the case with the forms A. and O., and E.

and I., in the same matter. If it be true that _All men are wise_, it is false that _Some men are not wise_ (equivalent by obversion to _Some men are not-wise_); or else, since the 'Some men' are included in the 'All men,' we should be predicating of the same men that they are both 'wise' and 'not-wise'; which would violate the principle of Contradiction. Similarly, _No men are wise_, being by obversion equivalent to _All men are not-wise_, is incompatible with _Some men are wise_, by the same principle of Contradiction.

But, again, if it be false that _All men are wise_, it is always true that _Some are not wise_; for though in denying that 'wise' is a predicate of 'All men' we do not deny it of each and every man, yet we deny it of 'Some men.' Of 'Some men,' therefore, by the principle of Excluded Middle, 'not-wise' is to be affirmed; and _Some men are not-wise_, is by obversion equivalent to _Some men are not wise_.

Similarly, if it be false that _No men are wise_, which by obversion is equivalent to _All men are not-wise_, then it is true at least that _Some men are wise_.

By extending and enforcing the doctrine of relative terms, certain other inferences are implied in the contrary and contradictory relations of propositions. We have seen in chap. iv. that the contradictory of a given term includes all its contraries: 'not-blue,' for example, includes red and yellow. Hence, since _The sky is blue_ becomes by obversion, _The sky is not not-blue_, we may also infer _The sky is not red_, etc. From the truth, then, of any proposition predicating a given term, we may infer the falsity of all propositions predicating the contrary terms in the same relation. But, on the other hand, from the falsity of a proposition predicating a given term, we cannot infer the truth of the predication of any particular contrary term. If it be false that _The sky is red_, we cannot formally infer, that _The sky is blue_ (_cf._ chap. iv. -- 8).

-- 8. Sub-contrariety is the relation of two propositions, concerning the same matter that may both be true but are never both false. This is the case with I. and O. If it be true that _Some men are wise_, it may also be true that _Some (other) men are not wise_. This follows from the maxim in chap. vi. -- 6, not to go beyond the evidence.

For if it be true that _Some men are wise_, it may indeed be true that _All are_ (this being the subalternans): and if _All are_, it is (by contradiction) false that _Some are not_; but as we are only told that _Some men are_, it is illicit to infer the falsity of _Some are not_, which could only be justified by evidence concerning _All men_.

But if it be false that _Some men are wise_, it is true that _Some men are not wise_; for, by contradiction, if _Some men are wise_ is false, _No men are wise_ is true; and, therefore, by subalternation, _Some men are not wise_ is true.

-- 9. The Square of Opposition.--By their relations of Subalternation, Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O.

(having the same matter) are said to stand in Opposition: and Logicians represent these relations by a square having A. I. E. O. at its corners:

A. Contraries E.

S Co s S u nt e u b ra i b a di r a l ct o l t ct o t e di r e r ra i r n nt e n s Co s s

I. Sub-contraries O.

As an aid to the memory, this diagram is useful; but as an attempt to represent the logical relations of propositions, it is misleading. For, standing at corners of the same square, A. and E., A. and I., E. and O., and I. and O., seem to be couples bearing the same relation to one another; whereas we have seen that their relations are entirely different. The following traditional summary of their relations in respect of truth and falsity is much more to the purpose:

(1) If A. is true, I. is true, E. is false, O. is false.

(2) If A. is false, I. is unknown, E. is unknown, O. is true.

(3) If I. is true, A. is unknown, E. is false, O. is unknown.

(4) If I. is false, A. is false, E. is true, O. is true.

(5) If E. is true, A. is false, I. is false, O. is true.

(6) If E. is false, A. is unknown, I. is true, O. is unknown.

(7) If O. is true, A. is false, I. is unknown, E. is unknown.

(8) If O. is false, A. is true, I. is true, E. is false.

Where, however, as in cases 2, 3, 6, 7, alleging either the falsity of universals or the truth of particulars, it follows that two of the three Opposites are unknown, we may conclude further that one of them must be true and the other false, because the two unknown are always Contradictories.

-- 10. Secondary modes of Immediate Inference are obtained by applying the process of Conversion or Obversion to the results already obtained by the other process. The best known secondary form of Immediate Inference is the Contrapositive, and this is the converse of the obverse of a given proposition. Thus:

DATUM. OBVERSE. CONTRAPOSITIVE.

A. _All S is P_ ? _No S is not-P_ ? _No not-P is S_ I. _Some S is P_ ? _Some S is not not-P_ ? (none) E. _No S is P_ ? _All S is not-P_ ? _Some not-P is S_ O. _Some S is not P_ ? _Some S is not-P_ ? _Some not-P is S_

There is no contrapositive of I., because the obverse of I. is in the form of O., and we have seen that O. cannot be converted. O., however, has a contrapositive (_Some not-P is S_); and this is sometimes given instead of the converse, and called the 'converse by negation.'

Contraposition needs no justification by the Laws of Thought, as it is nothing but a compounding of conversion with obversion, both of which processes have already been justified. I give a table opposite of the other ways of compounding these primary modes of Immediate Inference.

A I E O --------------------------------------------------------------------------------

1 All A is B Some A is B No A is B Some A is not B

Obverse 2 No A is b Some A is not b All A is b Some A is b

Converse 3 Some B is A Some B is A No B is A