Some Other Logical Forms
Disjunctive Syllogism
PvQ
~P
Therefore, Q
Example, “We will go out to eat or I’ll cook steaks. I don’t have time to cook steaks. So
we will go out to eat.
Hypothetical Syllogism
P
Q
Q
R
Therefore, P
R
If Dole wins then Republicans will win back the senate. If the Republicans win back the
senate, conservative judges will be appointed. Thus, if Dole wins, conservative judges
will be appointed.
Constructive Dilemma
(P
Q) & (R
PvR
Therefore, Q v S
S)
If it rains, then we will meet at my house and if Betty calls, we will meet at your house.
Either it rains or Betty calls. Therefore either we meet at my house or we meet at your
house.
Material Implication
P
Q is logically equivalent to ~P v Q
Translations
“Unless” for example “we will have a picnic unless it rains”
Translates to “either we have a picnic or it rains” or to “if we do not have a picnic, then it
rained” or “if it did not rain, then we had a picnic”.
PvR
or if ~P
R or ~R
P
Reductio ad absurdum (indirect proof)
Suppose that you want to prove something but you don’t have a direct argument for it.
You can proceed in the following way. Let’s suppose that what I want to prove is NOT
TRUE. From that assumption we can derive an absurdity or a contradiction. Therefore,
what I am trying to prove must be true.
I want to prove P
Let’s assume ~P
~P plus other premises that you accept and logically valid inferences leads to (X & ~X),
it follows P must be true. This resembles modus tollens. If P then Q, ~Q; therefore ~P.