Example 5

Why is TUI important? Consider a test for the validity of ($x)(Fx ® ("x)Fx)

      ' 1.    Ø($x)(Fx ® ("x)Fx)

 a   \ 2.    ("x)Ø(Fx ® ("x)Fx)                1, QN

      ' 3.           Ø(Fa ® ("x)Fx)                2, UI

        4.               Fa                                 3

      ' 5.               Ø("x)Fx                        3

 b   \ 6.                ($x)ØFx                        5, QN

        7.                      ØFb                        6, EI

The tree has not closed yet. If we think we have finished we can construct a model for which Fa = T and Fb = F. If we expand the formula being tested in this universe of two elements it becomes:

        (Fa ® (Fa Ù Fb)) Ú (Fb ® (Fa Ù Fb))

which is true in the universe.

        (T  ®  (T  Ù  F)) Ú (F   ® (T  Ù  F))

        (T  ®        F     ) Ú (F   ®       F     )

              F                   Ú        T

                                    T

So the model isn't a counterexample.

The problem is 2. has a formula that is not TUI. Make it so now:

        8.           Ø(Fb ® ("x)Fx)                2, UI

        9.               Fb                                 8

       10.             Ø("x)Fx                          8

       11.              X                                    7, 9