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By the law of duality we have
and substituting this value in the second side of (2) we have
B = BCA Bca.
But for A in the above we may substitute its expression in (1), getting
and striking out one of these alternatives which is contradictory we finally obtain
B = Bca.
The meaning of this formula is that a planet is a planet not self-luminous, and not a sun, which only differs from the Aristotelian conclusion in being more full and precise.
43. The syllogism Camenes may be illustrated by the following example:
(1) All monarchs are human beings,
(2) No human beings are infallible, (3) No infallible beings, therefore, are monarchs.
This is proved by considering that every infallible being is either a monarch or not a monarch; but if a monarch, then by (1) he is a human being, and by (2) is not infallible, which is impossible; therefore, no infallible being is a monarch.
Or in symbols, taking
A = monarch,
B = human being,
the premises are
A = AB
(2) B = Bc.
Now by the law of duality
Substituting for A its value as derived from both the premises, we have
and, striking out the contradictory term,
C = aC.
44. By the indirect method we can obtain and prove the truth of the contra-positive of the ordinary proposition A is B, or
(1) A = AB.
What we require is the description of the term not-B or b; and by the law of duality this is, in the first place, either A or not-A:
Substituting for A in (2) its value as given in (1) we obtain
b = ABbab.
But the term ABb breaks the law of non-contradiction (p. 46), so that we have left only
b = ab,
or whatever is not-B is also not-A.
from the premise
A = metal,
All metals are elements
we conclude that all substances which are not-elements are not metals; which is proved at once by the consideration, symbolically expressed above, that if they were metals they would be elements, or at once elements and not-elements, which is impossible.
45. It is the peculiar character of this method of indirect inference that it is capable of solving and explaining, in the most complete manner, arguments of any degree of complexity. It furnishes, in fact, a complete solution of the problem first propounded and obscurely solved by Dr. Boole:
Given any number of propositions involving any number of distinct terms, required the description
of any of those terms or any combination of those terms as expressed in the other terms, under condition of the premises remaining true.
This method always commences by developing all the possible combinations of the terms involved according to the law of duality. Thus, if there are three terms, represented by A, B, C, then the possible combinations in which A can present itself will not exceed four, as follows:
(1) A = ABC ABC AbC Abc.
If we have any premises or statements concerning the nature of A, B, and C, that is, the combinations in which they can present themselves, we proceed to inquire how many of the above combinations are consistent with the premises. Thus, if A is never found with B, but B is always found with C, the two first of the combinations become contradictory, and we have
or, A is never found with B, but may or may not be found with C.
This conclusion may be proved symbolically by expressing the premises thus:
A = Ab
B = BC,
and then substituting the values of A and B wherever they occur on the second side of (1).
46. As a simple example of the process, let us take the following premises, and investigate the consequences which flow from them.1
"From A follows B, and from C follows D; but B and D are inconsistent with each other."
The possible combinations in which A, B, C, and D may present themselves are sixteen in number, as follows:
Each of these combinations is to be compared with the premises in order to ascertain whether it is possible under the condition of those premises. This comparison will really consist in substituting for each letter its description as given in the
1 See De Morgan's "Formal Logic," p. 123.