For All

The For All at U+2200 is a mathematical quantifier meaning “for every” or “for each.” It is part of the Mathematical Operators block and is used to state that a claim applies to all objects in a specified domain, not just to one example. Its main home is logic, set theory, and higher mathematics. A statement such as ∀x ∈ ℝ, x² ≥ 0 reads “for all real numbers x, x squared is nonnegative,” and a line like ∀ε > 0 signals that every positive epsilon must be considered. It often pairs with the existential quantifier, as in ∀x ∃y, meaning “for every x there exists a y.” Computer science uses the same symbol in formal specifications, type theory, theorem provers, database theory, and program verification. For example, an invariant may be written to say that ∀i in an array range, the stored value satisfies some condition. In plain prose it is usually replaced by words such as “all,” “every,” or “for each,” because the symbol is specialized notation. It is sometimes confused visually with an upside-down capital A, but in mathematical writing its position before a variable marks it as a quantifier.