Constructing a Proof

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.

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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:

1. (R v F) ~ P Premise
2. R / ~ P Premise / Conclusion
3. R v F 2 Add
4. ~ P 1, 3 MP
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.

Proof

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