Conditional & Biconditional Statements
Conditional (\(\to\))
| \(p\) | \(q\) | \(p \to q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
By definition, \(p \to q \equiv \sim p \lor q\) — this lets any law of logical equivalence (see logical-equivalence-laws) be applied to conditionals.
\(p \to q\) can be phrased: “\(p\) implies \(q\)”, “if \(p\) then \(q\)”, “\(q\) if \(p\)”, “\(p\) is sufficient for \(q\)”, “\(p\) only if \(q\)”, “\(q\) is necessary for \(p\)”.
Warning: \(p \to q \not\equiv q \to p\).
Contrapositive, converse, inverse
| Name | Form | Equivalent to |
|---|---|---|
| Conditional | \(p \to q\) | Contrapositive |
| Contrapositive | \(\sim q \to \sim p\) | Conditional |
| Converse | \(q \to p\) | Inverse |
| Inverse | \(\sim p \to \sim q\) | Converse |
The conditional and its contrapositive are always logically equivalent. The converse and inverse are always logically equivalent to each other — but not to the original conditional.
Necessary and sufficient conditions
- \(p\) is a necessary condition for \(q\) ⟺ “if \(\sim p\) then \(\sim q\)” ⟺ “if \(q\) then \(p\)” ⟺ “\(q\) only if \(p\)”.
- \(p\) is a sufficient condition for \(q\) ⟺ “if \(p\) then \(q\)” ⟺ “\(q\) if \(p\)”.
Examples:
- Non-negative is necessary but not sufficient for being a perfect square (\(-9\) isn’t a square; \(7\) is non-negative but not a square).
- Divisible by 4 is sufficient but not necessary for being even (\(8\): divisible by 4 ⟹ even; \(10\): even but not divisible by 4).
Biconditional (\(\leftrightarrow\))
\[a \leftrightarrow b \equiv (a \to b) \land (b \to a)\]
“\(a\) iff \(b\)” ≡ “\(a\) only if \(b\), and \(a\) if \(b\)” ≡ “\(b\) is necessary and sufficient for \(a\)”.
Order of operations
- \(\sim\)
- \(\land, \lor\) (equal)
- \(\to, \leftrightarrow\) (equal)
Parenthesise explicitly rather than relying on left-to-right tie-breaking.
See also
- logical-connectives
- logical-equivalence-laws
- practice-problems — §3 Conditional Statements, with full worked solutions