| A beautiful gem is not a rare stone. | (3) |
Our symbolic method gives only true conclusions; for if we take
A = gem
B = rare stone
C = beautiful stone,
the proposition (1) is of the form
| A | = B ꖌ C | |
| hence | AB | = B ꖌ BC |
| and | AC | = BC ꖌ C; |
but these inferences are not equivalent to the false ones (2) and (3).
We can readily represent disjunctive reasoning by the modus ponendo tollens, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of
Water is either salt or fresh,
and take
A = Water B = salt C = fresh,