Subsection 3.4.1 Introduction
In the last section, we looked at identities: ways of transforming a single logical statement into another (presumably more useful) one.
But proofs (in fact, more generally, arguments) require that we reason with multiple statements to see what new conclusions we can draw from an entire set of premises.
Recall the Wet Sidewalks example. We gave the following names to statements:
R: It’s raining.
W: The sidewalks are wet.
S: The sidewalks are slippery.
C: It is important to be careful.
We supplied the following premises:
[1] R W If it’s raining then the sidewalks will be wet.
[2] W S If the sidewalks are wet, they will be slippery.
[3] S C If the sidewalks are slippery then it is important to be careful.
[4] R It’s raining.
And then we reasoned as follows:
If it’s raining then the sidewalks will be wet. But it is raining. So the sidewalks will be wet.
If the sidewalks are wet, they will be slippery. But they are wet. So they are slippery.
If the sidewalks are slippery then it is important to be careful. But they are slippery. So it is important to be careful.
So we then have:
[5] C It is important to be careful.
In this section, we’ll formalize the inference (reasoning) rules that will enable us to make arguments (generate proofs) of this sort. We’ll start with modus ponens , the one we used informally in the Wet Sidewalks example.