Super Complicated Logical Proof Question? :/?

Question

At least, its super complicated to me. And super confusing...

X -> N
~X -> P
(M ^ N) ->
~Q
M
___________
therefore P

If you don't feel like showing me every step, I'd be grateful to know where to start :)


THANKS SO MUCH~!

Woops @_@ sorry

Q's supposed to be after that suspending arrow

My apologies ^_^;

Answer 1

Something's amiss here, seeing as Q only appears once among these statements. Are you absolutely sure that the question is specified correctly?

Ok now that it's been fixed:

Start with any truth values you're given: M and ~Q. How can you take advantage of this? Look at the statements that involve them.

(M ^ N) -> Q is equivalent to
~(M ^ N) v Q, or, by De Morgan's Law,
~M v ~N v Q.

You know M is true and Q is false, so for the statement to be true, you must have ~N.

Now find out where N is involved: X -> N, which is equivalent to ~X v N. But N is false, so you must have ~X.

Now what consequence does ~X have? Go to the statement ~X -> P, which is equivalent to ~(~X) v P, or X v P. But X is false, so P must be true.

Note the frequent use of this equivalence: P -> Q is equivalent to ~P v Q. It's a lifesaver :)

Answer 2

The way that I would prove this would be to use the tableau method, where you consider the set of the premises and the negation of the conclusion. The aim of the game is to show that the premises and the negation of the conclusion are inconsistent-that is, that there is no possible situation in which all of the premises are true and the negation of the conclusion true. So start off like this:

(x->n)
(~x->p)
((m^n)->q)
~q
m
~p

Now, go on from here and apply the tableau rules to each formula, and close each branch when you get a formula and its negation together in the same branch. The tableau will close (I've checked it, but I can't input it into this space-I've tried but unfortunately I can't put the lines in and it looks confusing).

If you're stuck, this is a great website: