Editing First-order logic (section)

===Validity, satisfiability, and logical consequence===
{{see also|Satisfiability}}
If a sentence φ evaluates to ''true'' under a given interpretation ''M'', one says that ''M'' ''satisfies'' φ; this is denoted<ref>It seems that symbol <math>\vDash</math> was introduced by Kleene, see footnote 30 in Dover's 2002 reprint of his book Mathematical Logic, John Wiley and Sons, 1967.</ref> <math>M \vDash \varphi</math>. A sentence is ''satisfiable'' if there is some interpretation under which it is true. This is a bit different from the symbol <math>\vDash</math> from model theory, where <math>M\vDash\phi</math> denotes satisfiability in a model, i.e. "there is a suitable assignment of values in <math>M</math>'s domain to variable symbols of <math>\phi</math>".<ref>F. R. Drake, ''Set theory: An introduction to large cardinals'' (1974)</ref>

Satisfiability of formulas with free variables is more complicated, because an interpretation on its own does not determine the truth value of such a formula. The most common convention is that a formula φ with free variables <math>x_1</math>, ..., <math>x_n</math> is said to be satisfied by an interpretation if the formula φ remains true regardless which individuals from the domain of discourse are assigned to its free variables <math>x_1</math>, ..., <math>x_n</math>. This has the same effect as saying that a formula φ is satisfied if and only if its [[universal closure]] <math>\forall x_1 \dots \forall x_n \phi (x_1, \dots, x_n)</math> is satisfied.

A formula is ''logically valid'' (or simply ''valid'') if it is true in every interpretation.<ref>Rogers, R. L., ''Mathematical Logic and Formalized Theories: A Survey of Basic Concepts and Results'' (Amsterdam/London: [[Elsevier#Imprints|North-Holland Publishing Company]], 1971), [https://books.google.com/books?id=fJziBQAAQBAJ&pg=PA39 p. 39].</ref> These formulas play a role similar to [[tautology (logic)|tautologies]] in propositional logic.

A formula φ is a ''logical consequence'' of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.