To see how these inference rules are used in constructing a
proof, let's start with the very simple argument:
If there is a recession this year or a foreign affairs
fiasco, then the president will not be re-elected.
There will be a recession.
The president will not be re-elected.
This is not an instance of modus ponens because the antecedent
of the first premise is a disjunction and the second premise
affirms only one disjunct.
To keep track of what we're doing in a proof, it helps to use a
special notation. A proof is a sequence of steps, so we number
each step, starting with the premises and ending with the
conclusion. In a separate column to the right, we make a note of
our justification for each step:
The first two lines state the premises, and after the second
premise, we make a note of the conclusion so that we know where
we're headed. The slash mark indicates that this is merely a note
to ourselves, and that we can't use ~ P in the proof itself
(that would be circular). Line #3 is our first inference from the
premises; the "2 Add" part tells us that the
statement R v F was derived from line #2 by means of addition.
Similarly, the "1, 3 MP" part in line #4 says that ~ P was derived
from lines #1 and #3 by means of modus ponens. Each line in a proof,
therefore, is either a premise or a statement that follows from
statements on preceding lines in accordance with one of the nine
basic inference forms.
|1. (R v F) ~ P
|2. R / ~ P
||Premise / Conclusion
|3. R v F
|4. ~ P
||1, 3 MP