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Implies (Logic Symbol) — Definition, Formula & Examples

Implies (→) is the logical symbol that connects two statements in an "if…then" relationship. Writing P→QP \rightarrow Q means "if PP is true, then QQ is true."

The material conditional P→QP \rightarrow Q is a logical connective that is false only when the antecedent PP is true and the consequent QQ is false; it is true in all other cases.

Key Formula

P→Q  ≡  ¬P∨QP \rightarrow Q \;\equiv\; \lnot P \lor Q
Where:
  • PP = The hypothesis (antecedent) — the "if" part
  • QQ = The conclusion (consequent) — the "then" part
  • ¬\lnot = Logical negation (NOT)
  • ∨\lor = Logical disjunction (OR)

How It Works

Read P→QP \rightarrow Q as "if PP, then QQ." The statement PP before the arrow is called the hypothesis (or antecedent), and QQ after the arrow is called the conclusion (or consequent). The implication is considered false in exactly one situation: when PP is true but QQ is false. If PP is false, the entire implication is automatically true regardless of QQ — this is called being "vacuously true." Common notations include P⇒QP \Rightarrow Q, P→QP \rightarrow Q, and P⊃QP \supset Q, which all mean the same thing in standard propositional logic.

Example

Problem: Let P = "It is raining" and Q = "The ground is wet." Determine the truth value of P → Q when P is true and Q is false.
Step 1: Identify the antecedent and consequent.
P=true,Q=falseP = \text{true},\quad Q = \text{false}
Step 2: Apply the definition: an implication is false only when P is true and Q is false.
P→Q=true→false=falseP \rightarrow Q = \text{true} \rightarrow \text{false} = \text{false}
Step 3: Verify using the equivalent form: not-P or Q.
¬P∨Q=false∨false=false\lnot P \lor Q = \text{false} \lor \text{false} = \text{false}
Answer: The statement "If it is raining, then the ground is wet" is false when it is raining but the ground is not wet.

Why It Matters

Nearly every theorem in mathematics is stated as an implication: "If these conditions hold, then this result follows." Understanding the implies symbol is essential for writing and reading proofs in geometry, discrete math, and any college-level math course.

Common Mistakes

Mistake: Assuming P → Q means the same thing as Q → P (confusing an implication with its converse).
Correction: "If it rains, the ground is wet" does not mean "if the ground is wet, it rained." The converse Q → P is a separate statement with its own truth value.