In other words, a propositional function is like an algorithm.

We may consider "the horse" to be the product of a propositional function.

Propositional functions are useful in set theory for the formation of sets.

Russell starts out by defining the "fundamental" notion of a propositional function.

Sometimes, P(x) is also called a propositional function, as each choice of x produces a proposition.

It expresses that a propositional function can be satisfied by every member of a domain of discourse.

They are propositional functions, functions whose values are truth values.

A system of logical connectives is functionally complete if and only if it can express all propositional functions.

(4) Assertion of a propositional function.

If a proposition's truth value is "truth" then the variable's value is said to satisfy the propositional function.

Finally, per Russell's definition, "a class [set] is all objects satisfying some propositional function" (p. 23).

A quantified propositional function is a statement; thus, like statements, quantified functions can be negated.

Russell has a very similar proof in Principles of Mathematics (1903, section 348), where he shows that that there are more propositional functions than objects.

We do not restrict the values taken by a computable function to be natural numbers; we may for instance have computable propositional functions."

In this case, the propositional function of interest is one that takes any proposition p and returns a proposition of the form p is true'.

The functions of propositions . . . are a particular case of propositional functions".

By this he meant that the class would designate only the elements that satisfied the propositional function (e.g. d and s) and nothing else.

Universal generalization concludes the propositional function must be universally true if it is true for any arbitrary element of the Universe of Discourse.

This new section eliminates the first edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'.

Substitution of one of these values for variable ŷ yields a proposition; this proposition is called a "value" of the propositional function.

The reason for the disappearance of the words "propositional function" e.g., in , and , is explained by together with further explanation of the terminology:

And since the logical subject was made up only of reference, tied together in strings by propositional functions, in logic there was no meaning except reference.

Propositional functions: Because his terminology is different from the contemporary, the reader may be confused by Russell's "propositional function".

Von Neumann avoided axiom schemas by formalizing the concept of "definite propositional function" with his functions, whose construction requires only finitely many axioms.

Fraenkel and Skolem formalized Zermelo's imprecise concept of "definite propositional function", which appears in his axiom of separation.