part i.. If We Assign the Letters of the Alphabet in Succession, Thus,

If we assign the letters of the alphabet in succession, thus,

A = he B = rich C = absolutely mad D = weakness itself E = subjected to bad advice F = subjected to most unfavourable circumstances, the proposition will take the form

A = AB{C ꖌ D (E ꖌ F)},

and if we develop the alternatives, expressing some of the different cases which may happen, we obtain

A = ABC ꖌ AB*c*DEF ꖌ AB*c*DE*f* ꖌ AB*c*D*e*F.

The above gives the strict logical interpretation of the sentence, and the first alternative ABC is capable of development into eight cases, according as D, E and F are or are not present. Although from our knowledge of the matter, we may infer that weakness of character cannot be asserted of a person absolutely mad, there is no explicit statement to this effect.

*Inference by Disjunctive Propositions.*

Before we can make a free use of disjunctive propositions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms. In the first place, to combine a simple term with a disjunctive one, we must combine it with every alternative of the disjunctive term. A vegetable, for instance, is either a herb, a shrub, or a tree. Hence an exogenous vegetable is either an exogenous herb, or an exogenous shrub, or an exogenous tree. Symbolically stated, this process of combination is as follows,

A(B ꖌ C) = AB ꖌ AC.

Secondly, to combine two disjunctive terms with each other, combine each alternative of one with each alternative of the other. Since flowering plants are either exogens or endogens, and are at the same time either herbs, shrubs or trees, it follows that there are altogether six alternatives--namely, exogenous herbs, exogenous shrubs, exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. This process of combination is shown in the general form

(A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE.

It is hardly necessary to point out that, however numerous the terms combined, or the alternatives in those terms, we may effect the combination, provided each alternative is combined with each alternative of the other terms, as in the algebraic process of multiplication.

Some processes of deduction may be at once exhibited. We may always, for instance, unite the same qualifying term to each side of an identity even though one or both members of the identity be disjunctive. Thus let

A = B ꖌ C.

Now it is self-evident that

AD = AD,

and in one side of this identity we may for A substitute its equivalent B ꖌ C, obtaining

AD = BD ꖌ CD.

Since “a gaseous element is either hydrogen, or oxygen, or nitrogen, or chlorine, or fluorine,” it follows that “a free gaseous element is either free hydrogen, or free oxygen, or free nitrogen, or free chlorine, or free fluorine.”

This process of combination will lead to most useful inferences when the qualifying adjective combined with both sides of the proposition is a negative of one or more alternatives. Since chlorine is a coloured gas, we may infer that “a colourless gaseous element is either (colourless) hydrogen, oxygen, nitrogen, or fluorine.” The alternative chlorine disappears because colourless chlorine does not exist. Again, since “a tooth is either an incisor, canine, bicuspid, or molar,” it follows that “a not-incisor tooth is either canine, bicuspid, or molar.” The general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred. Now this result clearly follows from our process of substitution; for if we have the proposition

A = B ꖌ C ꖌ D,

and we insert this expression for A on one side of the self-evident identity

A*b* = A*b*,

we obtain A*b* = AB*b* ꖌ A*b*C ꖌ A*b*D;

and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction: there remains

A*b* = A*b*C ꖌ A*b*D.

Thus our system fully includes and explains that mood of the Disjunctive Syllogism technically called the *modus tollendo ponens*.

But the reader must carefully observe that the Disjunctive Syllogism of the mood *ponendo tollens*, which affirms one alternative, and thence infers the denial of the rest, cannot be held true in this system. If I say, indeed, that

Water is either salt or fresh water,

it seems evident that “water which is salt is not fresh.” But this inference really proceeds from our knowledge that water cannot be at once salt and fresh. This inconsistency of the alternatives, as I have fully shown, will not always hold. Thus, if I say

Gems are either rare stones or beautiful stones, (1)

it will obviously not follow that

A rare gem is not a beautiful stone, (2)

nor that

A beautiful gem is not a rare stone. (3)

Our symbolic method gives only true conclusions; for if we take

A = gem B = rare stone C = beautiful stone,

the proposition (1) is of the form

A = B ꖌ C hence AB = B ꖌ BC and AC = BC ꖌ C;

but these inferences are not equivalent to the false ones (2) and (3).

We can readily represent disjunctive reasoning by the *modus ponendo tollens*, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of

Water is either salt or fresh,

and take

A = Water B = salt C = fresh,

then the premise is apparently of the form

A = AB ꖌ AC;

but in reality there is an unexpressed condition that “what is salt is not fresh,” from which follows, by a process of inference to be afterwards described, that “what is fresh is not salt.” We have then, in letter-terms, the two propositions

B = B*c* C = *b*C.

If we substitute these descriptions in the original proposition, we obtain /* A = AB*c* ꖌ A*b*C; */

uniting B to each side we infer

AB = AB*c* ꖌ AB*b*C or AB = AB*c*;

that is,

Water which is salt is water salt and not fresh.

I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name *abscissio infiniti*. Take the case:

Red-coloured metal is either copper or gold (1) Copper is dissolved by nitric acid (2) This specimen is red-coloured metal (3) This specimen is not dissolved by nitric acid (4) Therefore, this specimen consists of gold (5)

Let us assign the letter-symbols thus--

A = this specimen B = red-coloured metal C = copper D = gold E = dissolved by nitric acid.

Assuming that the alternatives copper or gold are intended to be exclusive, as just explained in the case of fresh and salt water, the premises may be stated in the forms

B = BC*d* ꖌ B*c*D (1) C = CE (2) A = AB (3) A = A*e* (4)

Substituting for C in (1) by means of (2) we get

B = BC*d*E ꖌ B*c*D

From (3) and (4) we may infer likewise

A = AB*e*

and if in this we substitute for B its equivalent just stated, it follows that

A = ABC*d*E*e* ꖌ AB*c*D*e*

The first of the alternatives being contradictory the result is

A = AB*c*D*e*

which contains a full description of “this specimen,” as furnished in the premises, but by ellipsis asserts that it is gold. It will be observed that in the symbolic expression (1) I have explicitly stated what is certainly implied, that copper is not gold, and gold not copper, without which condition the inference would not hold good.