Another Logical Proof Question?

Question

Thanks to the guys who helped me on the last one :)!!!
Just one more-I promise.

R ->B
~ (~R ^Q)
B -> P
Q
____________
therefore P

This time there's four lines :( So how do I solve it?


-Deepest gratitude

Answer 1

Again work backwards, with Q being true (#4)...

Logically ~(~R ^ Q) is the same as R v ~Q (#2)

But since Q is true, ~Q would be false. For the statement to be true, one of them must be true, so this requires R to be true.

Since R is true, this implies B is true (#1)

And since B is true, this implies P is true (#3)

Therefore P

Answer 2

Work on the second line a bit:
~(~R ^ Q) = R v ~Q = Q -> R

(Remember ~Q v R is equivalent to Q -> R)

Now you have:
(1) R -> B
(2) Q -> R
(3) B -> P
(4) Q

Chain together (4), (2), (1), (3) and you get P

Answer 3

Backwards,

Q is true.

Non (Non R and Q) is R or Non Q. Since Q is true, R is true and R or Non Q is true. R true implies B is true. Since B is true P is true.

Here we are.