Propositional function

In

), which leaves the statement undetermined. The sentence may contain several such variables (e.g. n variables, in which case the function takes n arguments).

Overview

As a mathematical function, A(x) or A(x₁, x₂, ..., x_n), the propositional function is abstracted from predicates or propositional forms. As an example, consider the predicate scheme, "x is hot". The substitution of any entity for x will produce a specific proposition that can be described as either true or false, even though "x is hot" on its own has no value as either a true or false statement. However, when a value is assigned to x, such as lava, the function then has the value true; while one assigns to x a value like ice, the function then has the value false.

Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):

"...it has become necessary to take propositional function as a primitive notion.

Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types.^[1]

A Propositional Function, or a predicate, in a variable x is an open formula p(x) involving x that becomes a proposition when one gives x a definite value from the set of values it can take.

According to

.

See also

• Propositional formula
• Boolean-valued function
• Formula (logic)
• Sentence (logic)
• Truth function
• Open sentence

References