Show+that+a+formula+is+valid+iff+T=where+Tis+an+abbreviation+for+aninstance+p+V-p+ofLEM.

To show that a formula is valid if and only if T, where T is an abbreviation for an instance p V ~p of LEM, we can use the following steps:

1. Assume that the formula is valid.
2. By definition, a formula is valid if it is true in all interpretations.
3. Let p be a propositional variable.
4. Consider the interpretation I such that I(p) = true.
5. Since the formula is valid, it must be true in I.
6. Therefore, the formula (p V ~p) is true in I.
7. Since I(p) = true, we have (p V ~p) = true.
8. Now consider the interpretation J such that J(p) = false.
9. Again, since the formula is valid, it must be true in J.
10. Therefore, the formula (p V ~p) is true in J.
11. Since J(p) = false, we have (~p) = true.
12. Therefore, (p V ~p) = true in both cases where p is true and where p is false.
13. Thus, (p V ~p) is a tautology and T is true.
14. Conversely, assume that T is true.
15. Then (p V ~p) is a tautology.
16. Let I be an arbitrary interpretation.
17. If I(p) = true, then (p V ~p) = true by the tautology.
18. If I(p) = false, then (~p) = true by the tautology.
19. Therefore, (p V ~p) is true in all interpretations and the formula is valid.