In The Five Connectives you learned five symbols. Now we pin down exactly what each one means. The meaning of a connective is a rule: given the truth values of the parts, it tells you the truth value of the whole. Logicians call such a rule a truth function.
Key idea
Each connective is a truth function. $\neg p$ flips the value of $p$. $p \land q$ is true only when both are true. $p \lor q$ is false only when both are false. $p \to q$ is false only when $p$ is true and $q$ is false. $p \leftrightarrow q$ is true when $p$ and $q$ have the same value.
| $p$ | $q$ | $p \land q$ | $p \lor q$ | $p \to q$ | $p \leftrightarrow q$ |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
Why it works
Why is a conditional true when its antecedent is false? Think of $p \to q$ as a promise: "If you finish your chores, you can go to the movies." The promise is broken in one situation only: you finish your chores and are not allowed to go. If you did not finish them, nothing was promised, so the promise was not broken. Logic counts a claim that was not broken as true. This reading is called the material conditional, and it is the one every later rule assumes.
Worked examples
Worked example: A false antecedent
Let $p$ be F and $q$ be T. What is the value of $p \to q$?
- The only row that makes a conditional F has $p$ T and $q$ F.
- Here $p$ is F, so the conditional is not broken.
Answer: T
Worked example: Inclusive or
Let $p$ and $q$ both be T. What is $p \lor q$?
- A disjunction is F only when both disjuncts are F.
- Here both are T.
Answer: T
Worked example: The biconditional
Let $p$ be F and $q$ be F. What is $p \leftrightarrow q$?
- A biconditional is T when both sides match.
- Both are F, so they match.
Answer: T
Common mistake
Calling $p \to q$ false, or "not applicable", when $p$ is false. In logic every formula is true or false in every row, and a conditional with a false antecedent is true. A check: the only F in the conditional column is the row T, F.
- Negation: flip.
- Conjunction: one F makes it F.
- Disjunction: one T makes it T.
- Conditional: F only for T → F.
- Biconditional: T when the values match.
Because each connective is a truth function, the value of any formula, however long, is settled by the values of its letters. That is the fact behind every truth table in this unit, starting with Computing a Truth Value.