The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill 2004, 10.4E, 12(b)

We are required to symbolise the following argument and show that it is valid. Despite the name used in the agrument, here abbreviated to lower case 'g', there is nothing that is cock-eyed about the argument.

Groucho Marx doesn't stay in any hotel that is willing to have him as a guest. Any hotel not willing to have Groucho Marx as a guest doesn't. Therefore, Groucho Marx doesn't stay in any hotel.

1. (x)[(Hx • Wxg) ⊃¬ Sgx]
2. (x)[(Hx • ¬ Wxg) ] ⊃¬ Sgx]
3. ∴(x)(Hx ⊃¬ Sgx]
4. * Hx ......... ACP
5. * (Hx • Wxg) ⊃¬ Sgx ......... 1UI x/x
6. * Hx ⊃(Wxg ⊃¬ Sgx) ......... 5Exp.
7. * Wxg ⊃¬ Sgx ......... 4,6MP
8. * (Hx • ¬ Wxg) ] ⊃¬ Sgx ......... 2UI x/x
9. * Hx ⊃ (¬ Wxg ⊃¬ Sgx) ......... 8Exp.
10. * ¬ Wxg ⊃¬ Sgx ......... 4,9MP
11. * Sgx ⊃Wxg ......... 10Contrap.
12. * Sgx ⊃¬ Sgx ......... 7,11HS
13. * ¬ Sgx ∨¬ Sgx ......... 12MI
14. * ¬ Sgx ......... 13Taut.
15. Hx ⊃¬ Sgx ......... 4-14CP
16. (x)(Hx ⊃¬ Sgx) ......... 15UG