Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 3.2.3 Wrong Premises

Is it possible to use the logical reasoning engine that we’ve just described to derive a conclusion that is obviously false? Sure. All we have to do is to choose premises that don’t correspond to the world we’re trying to reason about.

Let’s return to the Wet Sidewalks example. We’ve given names to the following statements:

R: It’s raining.

W: The sidewalks are wet.

S: The sidewalks are slippery.

C: It is important to be careful.

Here are the premises that we’ve been using:

[1] \(R \rightarrow W\) If it’s raining then the sidewalks will be wet.

[2] \(W \rightarrow S\) If the sidewalks are wet, they will be slippery.

[3] \(S \rightarrow C \) If the sidewalks are slippery then it is important to be careful.

[4] \(R\) It’s raining.

We can easily use a truth table to prove any of the conclusions that we derived earlier using everyday logic (actually we used the logical rule modus ponens that we’ll soon define formally). Let’s prove that the sidewalks will be wet:

Success. The final column is all T’s.

But now let’s change our premises. In particular, let’s change the first one so that it asserts that, if it’s raining, the sidewalks will be dry. Let D stand for the assertion that the sidewalks are dry. Now we can easily prove that, if it’s raining the sidewalks will be dry:

Oops. We’ve just proved something that’s clearly nonsense. This is typically what happens if we don’t choose our premises wisely.

Big Idea

We must separate the validity of an argument from the reasonableness of its premises.