Some of these modes exemplify different special rules and theorems of logical writers, of which a few are subjoined.

Fig. 1. A A A and AA I are the only modes to which the dictum de omni directly applies-"Whatever is said of a class may be said of a contained part of the class."

Fig. 1. AUA is a formula into which a "perfect induction " might fall, where we affirm something of a whole class, because we have found it true of all the individuals or species which the class contains, Thus

Leibniz gives the formula "Cui singula insunt, etiam ex ipsis constitutum inest."

Fig. 1. EAE and EIO are the only modes to which the dictum de nullo applies. "What is denied of a class must be denied of any part of the class."

EUE and UEE in all figures.

"Si duorum quæ sunt

eadem inter se unum diversum sit a tertio, etiam alterum ab eo erit diversum." Leibniz.

Fig. 1 and 11. U A A. “Quod inest uni coincidentium, etiam alteri inest."

Leibniz.

M = P

All S is M

.. All S is P

UUU in all figures. "Quæ sunt eadem uni tertio, eadem sunt inter se."

§ 101. A mode of Notation.

To be able to represent to the eye by figures the relation which subsists in thought between concep

tions, tends so greatly to facilitate logical analysis, that many attempts have been made to attain it. Of two important schemes, that of Euler and that of Sir W. Hamilton, an account will be given hereafter. The scheme now to be explained is that which Lambert makes use of, in his Neues Organon.

A distributed term is marked by a horizontal line, with the letter S, P, or M attached, to denote that it is the subject, predicate, or middle term of the syllogism.

P

An undistributed term is marked, not by a definite line, but by a row of dots, to show its indefiniteness, thus

S.

These are the two forms of quantity in which separate conceptions may occur. But when two conceptions are joined in a judgment, another power as to quantity must be represented also. Let the judgment be, "All plants are organized," and let the lower line represent the subject and the upper the predicate; will this representation convey the whole truth?

P.....
S-

In one point it is inadequate, that the term

nized" is not wholly indefinite. We mean indeed by it, only some organized things; but then one part of it is made definite by affirming it of plants. We do not know how many, or what, individuals, come into the conception "Some organized things". by itself; but when it occurs in this judgment, we are certain of some individuals in it, viz. those which are "all plants." This we are able to express by a line partly definite, partly undetermined, thus

P....

S

Every affirmative judgment may be represented by a line drawn under another, the lower being always the subject. Negative judgments, which express that one conception cannot be contained under another, are represented by two lines drawn apart from each other, the predicate being a little higher than the subject, thus

S

P

But in a syllogism there are three terms, so that we require three lines to represent their relations; and the diagram thus drawn will supply some important illustrations of the nature of inference. Suppose the premisses are, "All matter undergoes change, and the

diamond is a kind of matter," the relations of the

three terms may be thus exhibited.

P..
M

S

From this notation, besides the two premisses given,

1. All M is P

2. All S is M

we may, by reading downwards, gather that

3. Some P is M, and

4. Some M is S

which are in fact immediate inferences by conversion from each of the premisses respectively. But further, from knowing that M stands under P, and S under M, we have learnt that S stands also under P, and this we may express, leaving M altogether out of our statement,

5. All S is P

6. Some P is S

the former being the proper conclusion from our premisses, and the latter the converse of the conclusion. Where one premiss is negative, and by the canon

of syllogism one only can be of that quality, the nota

Finally, every universal judgment of substitution, or U, may be expressed by two equal lines

66

Р
S

But when such a judgment expresses a logical division, as Organised beings are either plants, brutes, or men," the divided character of the predicate may be expressed by breaking up the line which represents it, thus

which would be read, "All S is either x y or z." The contrary process, of logical composition, which is used to express induction, as "Plants, brutes, and