The process takes its name from the interchange of the terms. The Predicate-term becomes the Subject-term, and the Subject-term the Predicate-term.

When propositions are analysed into relations of inclusion or exclusion between terms, the assertion of any such relation between one term and another, implies a Converse relation between the second term and the first. The statement of this implied assertion is technically known as the CONVERSE of the original proposition, which may be called the _Convertend_.

Three modes of Conversion are commonly recognised:--(_a_) SIMPLE CONVERSION; (_b_) CONVERSION _per accidens_ or by limitation; (_c_) CONVERSION BY CONTRAPOSITION.

(_a_) E and I can be simply converted, only the terms being interchanged, and Quantity and Quality remaining the same.

If S is wholly excluded from P, P must be wholly excluded from S. If Some S is contained in P, then Some P must be contained in S.

(_b_) A cannot be simply converted. To know that All S is contained in P, gives you no information about that portion of P which is outside S. It only enables you to assert that Some P is S; that portion of P, namely, which coincides with S.

O cannot be converted either simply or _per accidens_. Some S is not P does not enable you to make any converse assertion about P. All P may be S, or No P may be S, or Some P may be not S. All the three following diagrams are compatible with Some S being excluded from P.

[Illustration:

Concentric circles of S and P - P in centre

S in one circle and P in another circle.

S and P each in a circle, overlapping circle.

(_c_) Another mode of Conversion, known by mediaeval logicians following Boethius as _Conversio per contra positionem terminorum_, is useful in some syllogistic manipulations. This Converse is obtained by substituting for the predicate term its Contrapositive or Contradictory, not-P, making the consequent change of Quality, and simply converting. Thus All S is P is converted into the equivalent No not-P is S.[6]

Some have called it "Conversion by Negation," but "negation"

is manifestly too wide and common a word to be thus arbitrarily restricted to the process of substituting for one term its opposite.

Others (and this has some mediaeval usage in its favour, though not the most intelligent) would call the form All not-P is not-S (the Obverse or Permutation of No not-P is S), the Converse by Contraposition. This is to conform to an imaginary rule that in Conversion the Converse must be of the same Quality with the Convertend. But the essence of Conversion is the interchange of Subject and Predicate: the Quality is not in the definition except by a bungle: it is an accident. No not-P is S, and Some not-P is S are the forms used in Syllogism, and therefore specially named. Unless a form had a use, it was left unnamed, like the Subalternate forms of Syllogism: Nomen habent nullum: nec, si bene colligis, usum.

TABLE OF CONTRAPOSITIVE CONVERSES.

Con. Con.

All S is P No not-P is S No S is P Some not-P is S Some S is not P Some not-P is S Some S is P None.

When not-P is substituted for P, Some S is P becomes Some S is not not-P, and this form is inconvertible.

OTHER FORMS OF IMMEDIATE INFERENCE.

I have already spoken of the Immediate Inferences based on the rules of Contradictory and Contrary Opposition (see p. 145 - Part III, Ch.

II).

Another process was observed by Thomson, and named _Immediate Inference by Added Determinants_. If it is granted that "A negro is a fellow-creature," it follows that "A negro in suffering is a fellow-creature in suffering". But that this does not follow for every attribute[7] is manifest if you take another case:--"A tortoise is an animal: therefore, a fast tortoise is a fast animal". The form, indeed, holds in cases not worth specifying: and is a mere handle for quibbling. It could not be erected into a general rule unless it were true that whatever distinguishes a species within a class, will equally distinguish it in every class in which the first is included.

MODAL CONSEQUENCE has also been named among the forms of Immediate Inference. By this is meant the inference of the lower degrees of certainty from the higher. Thus _must be_ is said to imply _may be_; and _None can be_ to imply _None is_.

Dr. Bain includes also _Material Obversion_, the analogue of _Formal Obversion_ applied to a Subject. Thus Peace is beneficial to commerce, implies that War is injurious to commerce. Dr. Bain calls this Material Obversion because it cannot be practised safely without reference to the matter of the proposition. We shall recur to the subject in another chapter.

[Footnote 1: I purposely chose disputable propositions to emphasise the fact that Formal Logic has no concern with the truth, but only with the interdependence of its propositions.]

[Footnote 2: Mark Duncan, _Inst. Log._, ii. 5, 1612.]

[Footnote 3: There can be no doubt that in their doctrine of aequipollents, the Schoolmen were trying to make plain a real difficulty in interpretation, the interpretation of the force of negatives. Their results would have been more obviously useful if they had seen their way to generalising them.

Perhaps too they wasted their strength in applying it to the artificial syllogistic forms, which men do not ordinarily encounter except in the manipulation of syllogisms. Their results might have been generalised as follows:--

(1) A "not" placed before the sign of Quantity contradicts the whole proposition. Not "All S is P," not "No S is P,"

not "Some S is P," not "Some S is not P," are equivalent respectively to contradictories of the propositions thus negatived.

(2) A "not" placed after the sign of Quantity affects the copula, and amounts to inverting its Quality, thus denying the predicate term of the same quantity of the subject term of which it was originally affirmed, and _vice versa_.

All S is "not" P = No S is P.

No S is "not" P = All S is P.

Some S is "not" P = Some S is not P.

Some S is "not" not P = Some S is P.

(3) If a "not" is placed before as well as after, the resulting forms are obviously equivalent (under Rule 1) to the assertion of the contradictories of the forms on the right (in the illustration of Rule 2).

Not All S is "not" P = No S is P = Some S is P.

Not No S is "not" P = All S is P = Some S is not P.

Not Some S is "not" P = Some S is not P = All S is P.

Not Some S is "not" not P = Some S is P = No S is P.

[Footnote 4: _Formal_ to distinguish it from what he called the _Material Obverse_, about which more presently.]

[Footnote 5: The mediaeval word for the opposite of a term, the word Contradictory being confined to the propositional form.]

[Footnote 6: It is to be regretted that a practice has recently crept in of calling this form, for shortness, the Contrapositive simply. By long-established usage, dating from Boethius, the word Contrapositive is a technical name for a terminal form, not-A, and it is still wanted for this use.

There is no reason why the propositional form should not be called the Converse by Contraposition, or the Contrapositive Converse, in accordance with traditional usage.]

[Footnote 7: _Cf._ Stock, part iii. c. vii.; Bain, _Deduction_, p. 109.]

CHAPTER IV.

THE COUNTER-IMPLICATION OF PROPOSITIONS.

In discussing the Axioms of Dialectic, I indicated that the propositions of common speech have a certain negative implication, though this does not depend upon any of the so-called Laws of Thought, Identity, Contradiction, and Excluded Middle. Since, however, the counter-implicate is an important guide in the interpretation of propositions, it is desirable to recognise it among the modes of Immediate Inference.

I propose, then, first, to show that people do ordinarily infer at once to a counter-sense; second, to explain briefly the Law of Thought on which such an inference is justified; and, third, how this law may be applied in the interpretation of propositions, with a view to making subject and predicate more definite.

Every affirmation about anything is an implicit negation about something else. Every say is a gainsay. That people ordinarily act upon this as a rule of interpretation a little observation is sufficient to show: and we find also that those who object to having their utterances interpreted by this rule often shelter themselves under the name of Logic.

Suppose, for example, that a friend remarks, when the conversation turns on children, that John is a good boy, the natural inference is that the speaker has in his mind another child who is not a good boy.

Such an inference would at once be drawn by any actual hearer, and the speaker would protest in vain that he said nothing about anybody but John. Suppose there are two candidates for a school appointment, A and B, and that stress is laid upon the fact that A is an excellent teacher. A's advocate would at once be understood to mean that B was not equally excellent as a teacher.