Example 4

For the argument:

              P1          ("x)(Sx ® Px)

              P2          ("x)(Rx ® Px) .

              C            ("x)(Sx ® Rx)

Construct the tree:

a    \ 1.    ("x)(Sx ® Px)                                P1

a    \ 2.    ("x)(Sx ® Px)                                P2

     '  3.    Ø("x)(Sx ® Rx)                              NC

a   '  4.    ($x)Ø(Sx ® Rx)                               3, QN

     '  5.          Ø(Sa ® Ra)                               4, EI

        6.              Sa                                          5

        7.              ØRa                                       5

     '  8.             (Sa ® Pa)                               1, UI

     '  9.             (Ra ® Pa)                               2, UI

                            /     \                              

       10.           ØSa        Pa                              8

       11.             X                                            6, 10

                                    /     \                              

       12.                   ØRa        Pa                      9

This tree will not close (no matter how many further applications of UI are invoked.)

Tracking up the open branches indicates a model may be constructed with Ra = F, Pa = Sa = T.