Let A be a relational system, and let L be a language which is appropriate for A. Let phi be a well-formed formula of L, and let s be a valuation in A. Then A|=_sphi is written provided that one of the following holds:

1. phi is of the form x=y, for some variables x and y of L, and s maps x and y to the same element of the structure A.

2. phi is of the form Rx_1...x_n, for some n-ary predicate symbol R of the language L, and some variables x_1,...,x_n of L, and {s(x_1),...,s(x_n)} is a member of R^A.

3. phi is of the form (psi ^ gamma), for some formulas psi and gamma of L such that A|=_spsi and A|=_sgamma.

4. phi is of the form (( exists  x)psi), and there is an element a of A such that A|=_(s(x|a))psi.

In this case, A is said to satisfy phi with the valuation s.

See also

Łoś' Theorem

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

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Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1969.Enderton, H. E. A Mathematical Introduction to Logic. Boston, MA: Academic Press, 1972.

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

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