Disjunctive Syllogism

(I may be completely misreading expressions on faces in class, but I have the impression that DS is tripping some of you up. Even if it is not, this brief post may be helpful.)

Here is the rule called Disjunctive Syllogism (DS)

p v q
~p
:. q

Let's show that DS is a valid rule:

REMEMBER: An argument is valid if, and only if, the truth of the premises guarantees the truth of the conclusion.

Here is a proof of DS in English:
If p v q is true, then either p is true or q is true (or both).
If ~p is true, then p is false.
So, q must be true.

We can show that DS is valid by using indirect proof:

1. p v q
2. ~p          / q

     3. ~q                             AIP
     4. ~p & ~q                    2, 3, CONJ
     5. ~(p v q).                    4, DM
     6. (p v q) & ~(p v q)     1, 5, CONJ
7. q                                     3-6, IP


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