Chapter Three
3.6.4 Enthymemes
premise is identical to the antecedent of the second.
If we assume that both P→Q and Q→R are true, we can eliminate all of the possibilities where either one of them is false.
For instance, if I make the claim, (P1) If it gets below freezing outside, I can make ice out there.
And I also make the claim that, (P2) If I can make ice, my soft drinks will be deliciously refreshing.
Then I can conclude that, (C) If it gets below freezing outside, my soft drinks will be deliciously refreshing.
Essentially, we are demonstrating the transitive property of conditional statements. That is, if we have two conditional statements where the consequent of one is identical to the antecedent of another, we can eliminate them and mash the rest of the two premises together to get a conclusion that is definitely true.
P
Q
R
T
T
T
T
T
T
T
F
T
F
T
F
T
F
T
T
F
F
F
T
F
T
T
T
T
P→Q
Q→R
F
T
F
T
F
F
F
T
T
T
F
F
F
T
T
Now let’s take the values for P and R that are left over, and see what the values for P→R looks like. There are four possible combinations of P and R left, after we have taken into account the truth of our premises:
This argument takes the general form (P1) If P, then Q (P2) If Q, then R (C) If P, then R
Rendered symbolically: (P1) P→Q (P2) Q→R (C) P→R
The truth table proof of this argument now has to take into account three terms. Therefore when we make the table, we must account for all of the possible truth values of P, Q, and R, for a total of 8 combinations. Then we can fill in the truth values for the conditional statements acting as our premises:
P
R
P→R
T
T
T
F
T
T
F
T
T
F
F
T
Now it looks like no matter what leftover values of P and R we might choose, if P→Q and Q→R are true, P→R is definitely going to be true. But this could all be made clearer by taking a few examples. We can apply the hypothetical syllogism to categorical thinking: (P1) If Socrates is a man, Socrates is an animal. (P2) If Socrates is an animal, Socrates is a substance.
P
Q
R
T
T
T
T
T
T
T
F
T
F
T
F
T
F
T
T
F
F
F
T
F
T
T
T
T
(P1) If I set the house on fire, it will burn down.
F
T
F
T
F
(P2) If the house burns down, I’ll collect insurance
F
F
T
T
T
money.
F
F
F
T
T
P→Q
Q→R
(C) If Socrates is a man, Socrates is a substance.
We could also apply the hypothetical syllogism to causal relations:
(C) If I set the house on fire, I’ll collect insurance money.
59