the premises, there remain only
The expression for not-opaque things consists of the three combinations containing d, thus
| d | = AB cd ꖌ A b C d ꖌ abcd , | |
|---|---|---|
| or | d | = A d (B c ꖌ b C) ꖌ abcd . |
In ordinary language, what is not-opaque is either metal which is gold, and then not-silver, or silver and then not-gold, or else it is not-metal and neither gold nor silver.
Fourth Example.
A good example for the illustration of the Indirect Method is to be found in De Morgan’s Formal Logic (p. 123), the premises being substantially as follows:—
From A follows B, and from C follows D; but B and D are inconsistent with each other; therefore A and C are inconsistent.
The meaning no doubt is that where A is, B will be found, or that every A is a B, and similarly every C is a D; but B and D cannot occur together. The premises therefore appear to be of the forms
| A = AB, | (1) |
|---|---|
| C = CD, | (2) |
| B = B d . | (3) |
On examining the series of sixteen combinations, only five are found to be consistent with the above conditions, namely,
In these combinations the only A which appears is joined to c, and similarly C is joined to a, or A is inconsistent with C.
Fifth Example.
A more complex argument, also given by De Morgan,7 contains five terms, and is as stated below, except that