Conditional Statement
A conditional statement is an "if-then" statement of the form "If P, then Q," which asserts that P being true forces Q to be true.
Formula
P \to Q \text{ (if P then Q)}
Definition
A conditional statement has the form "If $P$, then $Q$": $P$ is the hypothesis (the "if" part) and $Q$ is the conclusion (the "then" part), claiming that whenever $P$ is true, $Q$ must also be true. Formally, $P \to Q$ is false only when $P$ is true and $Q$ is false, with truth table TT=T, TF=F, FT=T, FF=T; theorems are often stated as conditionals, like "If $f$ is differentiable, then $f$ is continuous." In propositional logic, $P \to Q$ is equivalent to $\neg P \vee Q$, and universal statements $\forall x\, P(x) \to Q(x)$ are the standard form of mathematical theorems; the material conditional captures implication in classical logic but not in modal or intuitionistic logics, where "if $P$ then $Q$" carries a stronger, constructive meaning.
Example
"If it is raining, then the ground is wet": when it rains, the ground is guaranteed to be wet. "If $n$ is divisible by $4$, then $n$ is divisible by $2$" is true for all integers, since the hypothesis is stronger and the conclusion weaker. In intuitionistic logic (constructive math), $P \to Q$ requires a constructive proof converting any proof of $P$ into a proof of $Q$, unlike the truth-table definition that suffices in classical logic.
Key Insight
A conditional statement can be false only one way: if $P$ is true but $Q$ is false; if $P$ is false, the statement is considered true regardless of $Q$ (a vacuous truth, so "If pigs fly, then $2+2=5$" is technically true). The choice of conditional semantics (classical vs. intuitionistic vs. modal) fundamentally affects what theorems are provable, linking logic to the foundations of mathematics and computer science (type theory).