In mathematics, an open sentence (usually an equation or inequality) is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined (and hence the sentences are no longer regarded as "open"). These possible replacement values are assumed to range over a subset of either the real or complex numbers, depending on the equation or inequality under consideration (in applications, real numbers are usually associated also with measurement units). The replacement values which produce a true equation or inequality are called solutions of the equation or inequality, and are said to "satisfy" them.
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:
Example 4 is an identity. Examples 1, 3, and 4 are equations, while example 2 is an inequality. Example 5 is a contradiction.
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:
Here, the phrase "for all" implicitly requires a universe of discourse to specify which mathematical objects are "all" the possibilities for x.
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
which says that f(x) * f(x) = f(x) for every value of x. If the universe of discourse consists of all functions from the real line R to itself, then the solutions for f are all functions whose only values are one and zero. But if the universe of discourse consists of all continuous functions from R to itself, then the solutions for f are only the constant functions with value one or zero.
| It has been suggested that this article or section be merged with Predicate variable. (Discuss) |
Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. The notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written {x | P(x)}, is just a collection of all the objects for which P is sensible and true.
For instance, {x | x is a positive integer less than 4} is the set {1,2,3}.
Thus, an element of {x | P(x)} is an object t for which the statement P(t) is true. Such a sentence P(x) is called a Predicate. P(x) is also called a propositional function, because each choice of x produces a proposition P(x) that is either true or false.
In formal semantics a predicate is an expression of the semantic type of sets. An equivalent formulation is that they are thought of as indicator functions of sets, i.e. functions from an entity to a truth value.
In first-order logic, a predicate can take the role as either a property or a relation between entities.
The following explanation is from, http://www.cs.odu.edu/~toida/nerzic/content/logic/pred_logic/predicate/pred_intro.html
To cope with deficiencies of propositional logic we introduce two new features: predicates and quantifiers. A predicate is a verb phrase template that describes a property of objects, or a relationship among objects represented by the variables.
For example, the sentences "The car Jane is driving is blue", "The sky is blue", and "The cover of this book is blue" come from the template "is blue" by placing an appropriate noun/noun phrase in front of it. The phrase "is blue" is a predicate and it describes the property of being blue. Predicates are often given a name. For example any of "is_blue", "Blue" or "B" can be used to represent the predicate "is blue" among others. If we adopt B as the name for the predicate "is_blue", sentences that assert an object is blue can be represented as "B(x)", where x represents an arbitrary object. B(x) reads as "x is blue".
Similarly the sentences "Mary gives the book to John", "Jane gives a loaf of bread to Mary", and "John gives a lecture to Mary" are obtained by substituting an appropriate object for variables x, y, and z in the sentence "x gives y to z". The template "... gives ... to ..." is a predicate and it describes a relationship among three objects. This predicate can be represented by Give( x, y, z ) or G( x, y, z ), for example.
Note: The sentence "Mary gives the book to John" can also be represented by another predicate such as "gives a book to". Thus if we use B( x, y ) to denote this predicate, "Mary gives the book to John" becomes B( Mary, John ). In that case, the other sentences, "Jane gives a loaf of bread to Mary", and "John gives a lecture to Mary", must be expressed with other predicates.
In mathematics, a predicate is either a relation or the boolean-valued function that amounts to the characteristic function or the indicator function of such a relation.
A function P: X→ {true, false} is called a predicate on X. When P is a predicate on X, we sometimes say P is a property of X.
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  | Predicate -- from Wolfram MathWorld - Predicate -- from Wolfram MathWorld |
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