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indicate anything which unites the qualities of both A and B, and then it follows that
AB = BA.
This principle of logical symbols has been fully explained by Dr. Boole in his "Laws of Thought " (pp. 29, 30), and also in my "Pure Logic" (p. 15); and its truth will be assumed here without further proof. It must be observed, however, that this property of logical symbols is true only of adjectives, or their equivalents, united to nouns, and not of words connected together by prepositions, or in other ways. Thus table of wood is not equivalent to wood of table; but if we treat the words of wood as equivalent to the adjective wooden, it is true that a table of wood is the same as a wooden table.
26. We may now proceed to consider the ordinary proposition of the form
A = AB,
which asserts the identity of the class A with a particular part of the class B, namely the part which has the properties of A. It may seem when stated in this way to be a truism, but it is not, because it really states in the form of an identity the inclusion of A in a wider class B. Aristotle happened to treat it in the latter aspect only, and the extreme incompleteness of his syllo
gistic system is due to this circumstance. It is only by treating the proposition as an identity that its relation to the other forms of reasoning becomes apparent.
27. One of the simplest and by far the most common form of argument in which the proposition of the above form occurs is the mood of the syllogism known by the name Barbara.
As an example, we may take the following:
(1) Iron is a metal,
(2) A metal is an element, therefore
The propositions thus expressed in the ordinary manner become, in a strictly logical form :
(1) Iron = metallic iron,
(2) Metal = elementary metal.
Now for metal or metallic in (1) we may substitute its equivalent in (2) and we obtain
(3) Iron = elementary, metal, iron;
which in the elliptical expression of ordinary conversation becomes Iron is an element, or Iron is some kind of element, the words an or some kind being indefinite substitutes for a more exact description.
The form of this mode of inference must be
stated in symbols on account of its great importance. If we take
A = iron,
B = metal,
C = element,
the premises are obviously,
A = AB,
(2) B = BC,
and substituting for B in (1) its description in (2) we have the conclusion
A = ABC,
which is the symbolic expression of (3).
28. The mood Darii, which is distinguished from Barbara in the doctrine of the syllogism by its particular minor premise and conclusion, cannot be considered an essentially different form, For if, instead of taking A in the previous example iron, we had taken it
A = some native minerals;
B and C remaining as before, we should then have the conclusion
A = ABC,
some native minerals are elements;
which affords an instance of the syllogism Darii exhibited in exactly the same form as Barbara.
29. The sorites or chain of syllogisms consists but in a series of premises of the same kind, allow-ing of repeated substitution. Let the premises be
(1) The honest man is truly wise, (2) The truly wise man is happy, (3) The happy man is contented, (4) The contented man is to be envied, the conclusion being—
(5) The honest man is to be envied.
Taking the letters A, B, C, D, and E to indicate respectively honest man, truly wise, happy, contented, and to be envied, the premises are represented thus:
A = AB,
(2) B = BC,
C = CD,
(4) D= DE,
and successive substitutions by (4) in (3), by (3) in (2), and by (2) in (1), give us
C = CDE,
B = BCDE,
A = ABCDE.
Or we may get exactly the same conclusion by substitution in a different order, thus:
A = AB = ABC = ABCD = ABCDE.
The ordinary statement of the conclusion in (5) is only an indefinite expression of the full description of A given in A = ABCDE.
30. All the affirmative moods of the syllogism may be represented with almost equal clearness and facility. As an example of Darapti in the third figure we may take
(1) Oxygen is an element,
(2) Oxygen is a gas,
(3) Some gas, therefore, is an element.
= oxygen, C = element,
the premises become
(1) B = BC,
Hence, by obvious substitution, either by (1) in (2) or by (2) in (1), we get
(3) BA = BC.
Precisely interpreted this means that gas which is oxygen is element which is oxygen; but when this full interpretation is unnecessary, we may substitute