We can show whether a propositional string is valid by working with the idea that in an invalid situation it would be consistent to have true premises and a false conclusion. To understand this better, use the drill TT lines on your disk. It will produce a 16-line truth table that will allow up to four variables (P,Q,R,S). You will be asked to indicate the number of premises (up to four) that you are using, then you will input your premises and your conclusion. Work with the two following patterns:
PQC, P |- Q and PQC, Q |- P (Use the drill before going on.)
Now think about what you saw. The first truth table did not have any line ending in 0 (for false) after the computer converted your original pattern into a single string expressing a conditional. It was a tautology (an expression that is always true). The second truth table did have lines ending in 0--what we would call bad lines in the sense that they should not appear in a valid pattern.

Doing complete truth tables is tedious for humans if not for a computer. We can accomplish the same goal by working in reverse to see if a bad line is even possible. For instance, in the two examples we can take these steps [click on to see the sequence]:

Go ahead to falsifying interpretations.

Go back to the starting page.