“All metals except gold and silver are opaque; therefore what is not opaque is either gold or silver or is not-metal.”
There is more implied in this statement than is distinctly asserted, the full meaning being as follows:
| All metals not gold or silver are opaque, | (1) |
| Gold is not opaque but is a metal, | (2) |
| Silver is not opaque but is a metal, | (3) |
| Gold is not silver. | (4) |
Taking our letters thus—
| A = metal | C = silver |
| B = gold | D = opaque, |
we may state the premises in the forms
| Abc | = AbcD | (1) |
| B | = ABd | (2) |
| C | = ACd | (3) |
| B | = Bc. | (4) |
To obtain a complete solution of the question we take the sixteen combinations of A, B, C, D, and striking out those which are inconsistent with the premises, there remain only
ABcd
AbCd
AbcD
abcD
abcd.
The expression for not-opaque things consists of the three combinations containing d, thus