therefore symbol — Definition, Formula & Examples
The therefore symbol (∴) is three dots arranged in a triangle that means 'therefore' or 'it follows that.' It appears before a conclusion that logically follows from previous statements.
The symbol ∴, composed of three dots in an upward-pointing triangular arrangement, is a logical shorthand placed before a statement that is a direct consequence of the preceding premises or reasoning.
How It Works
You place ∴ at the beginning of a line to signal that what follows is a logical conclusion drawn from the statements above it. It is most commonly used in mathematical proofs, logical arguments, and syllogisms. The symbol saves space by replacing the word 'therefore' in written work. Its companion symbol ∵ (three dots in a downward triangle) means 'because' and introduces a reason rather than a conclusion.
Example
Problem: Use the therefore symbol to write a short geometric proof that the angles of a triangle with two 60° angles must have a third angle of 60°.
Premise 1: The sum of interior angles in any triangle is 180°.
Premise 2: Two of the angles each measure 60°.
Conclusion: Substitute and solve, then state the conclusion using ∴.
Answer: ∴ The triangle is equilateral, with all three angles equal to 60°.
Why It Matters
You will encounter ∴ frequently in geometry proofs, formal logic courses, and standardized tests. Learning to read and write it correctly helps you follow proof-based arguments and present your own reasoning concisely.
Common Mistakes
Mistake: Confusing ∴ (therefore) with ∵ (because).
Correction: Remember that ∴ has two dots on the bottom and one on top (pointing up toward the conclusion). The symbol ∵ is flipped — two dots on top, one on the bottom — and introduces a reason, not a conclusion.