Truth tables
A statement built from smaller ones is true in some cases and false in others. How do you list every case and see which is which?
Where you are 3 inside
What this area is about
Whether P \land Q is true depends only on whether P and Q are true, not on what they say. So its whole behavior fits in a short list: one row for each way the letters can come out, 2^n rows for n letters. That list is a truth table, and this area is about writing them and reading them.
How these fit together
The connective tables. Each connective’s table is its definition written out case by case: two rows for not, four for the rest. The tables for not, and, or and if and only if are taught with the connectives themselves, in statements and negation, and and or and the biconditional. The one with its own star here is the truth table for the conditional, whose false-hypothesis rows surprise almost everyone.
The procedure. Building a truth table for a compound statement joins those small tables into one for any statement, one column at a time, from the inside out.
Two special kinds of statement. Tautologies and contradictions: a tautology is true on every row, a contradiction false on every row. Most statements are neither.
A path through it
The tables for not, and and or come first, since building needs them. Then the conditional’s table, then the building method, then tautologies and contradictions.
Finish here and you can settle a claim about statements by checking every case. Testing an argument with a truth table does exactly that, and logical equivalence asks when two tables match row for row.