Open Sentence
An open sentence is a mathematical statement containing one or more variables that is neither true nor false until specific values are substituted for the variables.
Definition
An open sentence is a math statement with a variable that is not yet true or false; once you substitute a number, it becomes either true or false. It is called "open" because the question is still open, like an unanswered question waiting for the right value to close it, and the values that make it true form the solution set. The contrast is with a closed sentence, a statement that is simply true or false on its own, such as "$5 + 3 = 8$." In formal logic, an open sentence (or open formula) is a well-formed formula containing at least one free variable; it is not a proposition, which must have a fixed truth value, but becomes one upon binding all free variables with quantifiers or substituting values, so solving an equation is equivalent to finding a valuation that closes the formula to "true."
Example
$x + 3 = 10$ is an open sentence: it is true when $x = 7$ and false when $x = 4$. $2x - 1 > 5$ is an open inequality, true for all $x > 3$ and false otherwise, so its solution set is $\{x : x > 3\}$. The formula $P(x)$: $x^2 = 4$ is open; closing it by universal quantification gives "for all $x$, $x^2 = 4$" (false), while closing it by existential quantification gives "there exists $x$ such that $x^2 = 4$" (true, with $x = 2$ or $x = -2$).
Key Insight
The distinction between open and closed formulas is fundamental to predicate logic and model theory, and the notion of a "free variable" is what separates algebraic equations from logical propositions.