Question from Intermediate Logic, Lesson 7, Exercise 7a: 1-6:
Is this exercise to help students visually see a tautology vs. a self contradiction? Or do they need to write out truth tables to find the truth-value?
The purpose of this exercise is to train students to understand the truth-functionality of compound propositions; that is, so they can see how the truth value of the proposition depends on the truth value of the component parts and the logical operators. Part of that is recognizing tautologies. But yes, I have students create truth tables to solve some of these.
Note that the number of rows in the truth table only depends on the number of variables (with unknown truth value). This means that the truth table for #3, for example, must be four rows long (two variables, P and Q), but #6 need only be two rows long (one variable, P). The other truth values will be rows of T (for A, B) or rows of F (for X, Y). For example, a truth table for #2 would start like this:
(P ⊃ P) ⊃ ~A
T T FT
F F FT
You would then finish the truth values under the conditionals. If done correctly, the final column (under the second horseshoe) is FF.