Question 582497
This one is a bit of a doozy. Feel free to ask about any piece of the proof.

<pre>

1.  B > [(O v ~O) > (T v U)]
2.  U > ~(G v ~G)                 /           B > T
-------------------------------------------------------
	3.  ~(B > T)                                       AIP
	4.  ~(~B v T)                             3        MI
	5.  ~~B & ~T                              4        DM
	6.  B & ~T                                5        DN 
	7.  ~T & B                                6        Comm
	8.  B                                     6        Simp
	9.  ~T                                    7        Simp
	10. (O v ~O) > (T v U)                    1,8      MP
	11. ~(O v ~O) v (T v U)                   10       MI
	12. (~O & ~~O) v (T v U)                  11       DM
	13. (~O & O) v (T v U)                    12       DN
	14. (T v U) v (~O & O)                    13       Comm
	15. ((T v U) v ~O) & ((T v U) v O)        14       Dist
	16. ((T v U) v O) & ((T v U) v ~O)        15       Comm
	17. (T v U) v O                           16       Simp
	18. T v (U v O)                           17       Assoc
	19. U v O                                 18,9     DS
	20. ~~U v O                               19       DN
	21. ~U > O                                20       MI
	22. (T v U) v ~O                          15       Simp
	23. T v (U v ~O)                          22       Assoc
	24. U v ~O                                23,9     DS
	25. ~O v U                                24       Comm
	26. O > U                                 25       MI
	27. ~U > U                                21,26    HS
	28. ~~U v U                               27       MI
	29. U v U                                 28       DN
	30. U                                     29       Taut
	31. ~(G v ~G)                             2,30     MP
	32. ~G & ~~G                              31       DM
	33. ~G & G                                32       DN
34.  B > T                                        3-33     IP
</pre>


=======================================================


Notes: To make things short and fit, I used the following abbreviations

AIP: Assumption for indirect proof. This is where you assume the complete opposite of the conclusion, and you try to find a contradiction. If a contradiction arises because you assumed that the opposite of the conclusion is true, then the conclusion must be true.



MI: Material Implication. This says that P > Q is the same as ~P v Q



DM: De Morgan's Law/Theorem: ~(P v Q) is the same as ~P & ~Q. Similarly, ~(P & Q) is the same as ~P v ~Q



DN: Double Negation: ~~P is the same as P



Comm: Commutation: P v Q is the same as Q v P. Also, P & Q is the same as Q & P



Assoc: Association: P v (Q v R) is the same as (P v Q) v R. In addition, P & (Q & R) is the same as (P & Q) & R



Simp: Simplification: P & Q can be simplified to just P (note: they are not equivalent)



Dist: Distribution: P v (Q & R) can be expressed as (P v Q) & (P v R). Also, P & (Q v R) can be written as (P & Q) v (P & R)



DS: Disjunctive Syllogism: If you have the statement P v Q and you know that ~P is the case, then you automatically can deduce Q (ex: I state that "I either had eggs or toast for breakfast" and I add in "I did NOT have eggs". So you can conclude that I must have had toast for breakfast)



HS: Hypothetical Syllogism: If you had P > Q and you have Q > R, then you can say that P > R. Ex: If it rains, then it is wet outside. If it is wet outside, then my grass will grow. Therefore, if it rains, then my grass will grow.



MP: Modus Ponens: If you have the conditional P > Q and you have P as a premise, then Q is the conclusion. Ex: If it rains, then it is wet outside. It rains. Therefore, it is wet outside.



Taut: Tautology: The basic idea that P v P is the same as P (ex: "I ate pizza or I ate pizza" is the same as "I ate pizza"). Similarly, P & P is the same as P as well (using the same reasoning)



IP: Indirect proof: This is the end of the proof by contradiction. If you start by assuming the opposite of the conclusion, do a bit of derivation, and you find a contradiction, then you have shown that the original conclusion must be valid. This is added to wrap things up and show that this conclusion was proven to be valid indirectly.