Example 2

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

As before, construct a tree from the negation of that formula.

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

   a  '      2.    ($x)Ø(Fx ® Fx)                1, QN

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

              4.    Fa                                    3, the rule for negated conditionals.

              5.    ØFa                                  

              6.    X                                     4, 5

This tree has been carried to completion and it no open branches remain.

Therefore the negation of the formula we were testing could not be satisfied.

Therefore the formula is logically valid.