UNDERSTANDING TRUTH TABLES

Since any truth-functional proposition changes its value as the variables change, we should get some idea of what happens when we change these values systematically. Once again we will use a red background for something true and a blue background for something false. This time we will use example statements (or propositions) that would be represented by letters such as "P" or "Q."

First we have a conjunction, true only when both parts are true.

logic is easy and everyone studies
T T T
T F F
F F T
F F F
P & Q
T T T
T F F
F F T
F F F

Next we will see what happens with (inclusive) disjunction, when both parts must be false in order for the proposition itself to be false.

(either) logic is easy or everyone studies
T T T
T T F
F T T
F F F
P v Q
T T T
T T F
F T T
F F F

We then have the relationship of implication, when a proposition expressing a condition is false only at the time it has a true antecedent and a false consequent.

logic is easy only if everyone studies
T T T
T F F
F T T
F T F
P -> Q
T T T
T F F
F T T
F T F

Finally we have the relationship of equivalence or mutual implication (the opposite of mutual exclusion), when the only way a proposition can be false is for it to have different truth values for its parts.

logic is easy if and only if everyone studies
T T T
T F F
F F T
F T F
P <-> Q
T T T
T F F
F F T
F T F
Next, we go on to see how to use a truth table to decide the validity of an argument form. Click here to go ahead, or click to go back to the explanation of truth-functional propositions.