Example 1

Test whether ("x)(Fx ® Gx) is logically valid.

If it were then its negation would be unsatisfiable, and a tree constructed from it would have all branches closed.

      '      1.    Ø("x)(Fx ® Gx)           

   a  '      2.    ($x)Ø(Fx ® Gx)                From 1 and an application of quantifier negation (QN)

       '      3.    Ø(Fa ® Ga)                      2, EI.

                                                             Note the introduction of a new constant, a.

                                                             That constant is marked beside 2 for reference.

              4.    Fa                                     3, the rule for negated conditionals.

              5.    ØGa                                  

This tree has been carried to completion and it does not close.

Therefore there is a model to satisfy the formula at 1.

We can construct a model using information in that open branch.

If Fa is T, and Ga is F, then

Fa ® Ga is F, so

("x)(Fx ® Gx) is F.