Implies symbol (→)
→ means “implies” or “if … then” in logic. p → q, read “if p then q”, is a conditional: it is false only when p is true and q is false.
Easily confused with
- If and only if symbol (↔)→ runs one way; ↔ runs both ways.
- Reaction arrows (→, ⇌, ↑, ↓)The same arrow is a reaction arrow in chemistry.
- Subset symbols (⊆, ⊂)⊆ means “is a subset of”: A ⊆ B says that every element of A is also in B. ⊂ usually means “is a proper subset of”: A is inside B and B has at least one element A doesn’t.
How to read it
OpenStax’s Contemporary Mathematics calls p → q a conditional or implication; p is the hypothesis (antecedent) and q the conclusion (consequent). It compares the conditional to a one-way promise, broken only when the hypothesis is true and the conclusion is false. So p → q counts as true whenever p is false, whatever q is. Oscar Levin’s Discrete Mathematics stresses that reading P → Q as “P implies Q” does not claim any causal link between P and Q. Swapping the two sides gives the converse, q → p, which Levin says is not equivalent to the original; the contrapositive, ¬q → ¬p, is.
Also searched as: →, ⇒, ⊃, arrow in logic, conditional symbol, if then symbol, implication symbol.
→, ⇒ or ⊃
Three signs are in use. forall x: Calgary gives the arrow → and the horseshoe ⊃ as the two common symbols for the conditional of logic. Mathematicians writing a proof often use the double arrow ⇒ or the long ⟹ between steps; NIST’s Digital Library of Mathematical Functions lists ⟹ as “implies”. The horseshoe ⊃ looks like the superset sign of set theory, which is a different symbol.
Other arrows
The same → is used elsewhere with other meanings: Unicode notes it is the total function arrow of the Z specification notation. In chemistry the arrow is a reaction arrow.
Read next
- Next in Logic symbolsIf and only if symbol (↔)↔ means “if and only if” in logic. p ↔ q is a biconditional: it is true when p and q are both true or both false, and false when one is true and the other false.
- ∴ “therefore” also marks a conclusion.Therefore sign (∴)The therefore sign (∴), three dots in a triangle with one on top, means “therefore”. It comes before a conclusion that follows from the lines above it.
More logic symbols
The same shape elsewhere
Browse symbols drawn with arrows.
Sources
- Contemporary Mathematics, 2.2 Compound Statements, OpenStax (Rice University), Compound statements
- Discrete Mathematics, 0.2 Mathematical Statements, Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin, University of Northern Colorado (Open Math Books), Implications
- forall x: Calgary. An Introduction to Formal Logic, P. D. Magnus, Tim Button, Robert Trueman and Richard Zach, forall x: Calgary (Fall 2025 edition), Open Logic Project, University of Calgary, Sec. 5.4 Conditional; Appendix A.2 Alternative symbols
- Earliest Uses of Symbols of Set Theory and Logic, Jeff Miller, Earliest Uses of Various Mathematical Symbols, MacTutor History of Mathematics, University of St Andrews, Implication
- NIST Digital Library of Mathematical Functions: Introduction (Common Notations and Definitions), U.S. National Institute of Standards and Technology, Common Notations and Definitions
- Arrows: Range 2190–21FF (code chart), The Unicode Standard, Version 18.0, Unicode Consortium, Arrows, 2192 and 21D2
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research
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