Question

Using predicate logic, prove the argument using the precicate symbols shown.There is some movie star who is richer than everyone. Anyone who is richer than anyone else pays more taxes than anyone else does. Therefore, there is a movie star who pays more taxes than anyone. M(x), R(x,y), T(x,y

Answer 1

This statement can be written using predicate logic thus;

(there exists x)(for all y)[M(x) ∧ R(x,y)] ∧ (for all x)(for all y)[R(x,y) → T(x,y)] → (there exists x)(for all y)[M(x) ∧ T(x,y)]

Proof:

1. (there exists x)(for all y)[M(x) /\ R(x,y)] HYP
2. (for all x)(for all y)[R(x,y)-> T(x,y)] HYP
3. (for all y)[M(s) /\ R(s,y)] EI,1
4. M(s) /\ R(s,y) UI,3
5. (for all y)[R(s,y)-> T(s,y)] UI,2
6. R(s,y)-> T(s,y) UI,5
7. R(s,y) Simp,4
8. T(s,y) MP,6,7
9. M(s) Simp,4
10. M(s) /\ T(s,y) Conj,7,9
11. (for all y)[M(s) /\ T(s,y)] UG,10
12. (there exists x)(for all y)[M(x) /\ T(x,y)] EG,11