Predicate Logic, Howard Pospesel, Prentice Hall. 2003, Chpt 11, ex. 20

Symbolize the following argument and show that it is valid.

Logical equivalence is mutual entailment. Every statement entails any logical truth. Therefore, all logical truths are logically equivalent.

Qxy - x is logically equivallent to y

Nxy - x entails y

Tx - x is a logical truth

1. (x)(y)(Qxy ≡ Nxy)
2. (x)(y)(Ty ⊃Nxy)
3. ∴ (x)(y)(Ty ⊃Qxy)
4. * Ty / ACP
5. * (y)(Ty ⊃Nxy) / 3UI x/x
6. * Ty ⊃Nxy / 5UI y/y
7. * Nxy / 6,7MP
8. * (y)(Qxy ≡ Nxy) / 1UI x/x
9. * Qxy ≡ Nxy / 1UI y/y
10. * (Qxy ⊃Nxy) • (Nxy ⊃Qxy) / 9BE
11. * Nxy ⊃Qxy / 10Simp.
12. * Qxy / 7,11MP
13. Ty ⊃Qxy / 4-12CP
14. (y)(Ty ⊃Qxy) / 13UG
15. (x)(y)(Ty ⊃Qxy) / 14UG