Let L be a language of the first-order logic. Assume that the language L has the following sets of nonlogical symbols:

1. C is the set of constant symbols of L. (These are nullary function symbols.)

2. P is the set of predicate symbols of L, and for each P in P, alpha(P) is the arity of P. The symbols in P are also called relation symbols of the language L.

3. F is the set of function symbols of L, and for each f in F, alpha(f) is the arity of f. The symbols in F are also called operation symbols of the language L.

4.  forall is the universal quantifier symbol of L.

A structure for L is a tuple A=(A,(c^A)_(c in C), (P^A)_(P in P), (f^A)_(f in F)) where A is a set (called the underlying set of A), and the following hold:

1. For each c in C, c^A in A,

2. For each P in P, P^A subset= A^(alpha(P)),

3. For each f in F, f^A:A^(alpha(f))->A.

See also

First-Order Logic, Lattice

This entry contributed by Matt Insall (author's link)

Explore with Wolfram|Alpha


Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.

Referenced on Wolfram|Alpha


Cite this as:

Insall, Matt. "Structure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications