In mathematical logic, an open sentence is a sentence which contains variables. Unlike an ordinary sentence, which contains constants, open sentences do not express propositions; they are neither true nor false. Hence, the open sentence:
(1) x is a number
Has no truth-value. An open sentence is said to be satisfied by any object(s) such that if it is written in place of the variable(s), it will form a sentence expressing a true proposition. Hence, "5" satisfies (1). Any sentence which resembles an open sentence in form is said to be a substitution instance of that sentence. Hence, "5 is a number" is a substitution instance of (1).
Mathematicians have not adopted that nomenclature, but refer instead to equations, inequalities with free variables, etc.
Such replacements are known as solutions to the sentence. An identity is an open sentence for which every number is a solution.
Examples of open sentences include:
Every open sentence must have (usually implicitly) a universe of discourse describing which numbers are under consideration as solutions. For instance, one might consider all real numbers or only integers. For example, in example 2 above, 1.6 is a solution if the universe of discourse is all real numbers, but not if the universe of discourse is only integers. In that case, only the integers greater than 3/2 are solutions: 2, 3, 4, and so on. On the other hand, if the universe of discourse consists of all complex numbers, then example 2 doesn't even make sense (although the other examples do). An identity is only required to hold for the numbers in its universe of discourse.
This same universe of discourse can be used to describe the solutions to the open sentence in symbolic logic using universal quantification. For example, the solution to example 2 above can be specified as:
The idea can even be generalised to situations where the variables don't refer to numbers at all, as in a functional equation. For example of this, consider
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