logic
How do I prove a formula without a premise?
The question looks like this
⊢ (P→Q)
I have started by making the assumption NOT(P→Q) to get a contradiction, and have no idea where to go from here.
If anyone could offer me some guidance, or even direct me towards some similar questions(preferably with answers) that would also be much appreciated.
Any help is appreciated and thanks in advance.
Simple questions often have complex answers, or no answers at all.
If you were to ask me why the sky was blue, to give a complete answer I would have to describe the heliocentric model, the earth's atmosphere, and the electromagnetic spectrum.
If you asked me how my computer connected to the internet, the answer would take a considerable amount of time to explain.
If you asked my whether there was life on other planets, I wouldn't be able to give an answer; we just don't know.
Often, it is impossible to give a simple answer to a simple question. This is true in any field you might encounter, ranging from the sciences to the humanities. And it is true in mathematics,
Mathematics allows us to formalize our questions within an axiomatic structure. It lets us ask our questions more precisely. But it in no way guarantees that the simplicity of the question would translate into simplicity of the answer.
Some other simple problems which cannot be answered simply:
The Four color theorem states that any map made up of continuous regions can be colored with 4 colors such that each region gets 1 color and no two adjacent regions get the same color. The proof is quite complex, requiring the use of a computer.
Scheinerman's conjecture is the conjecture that "every planar graph is the intersection graph of a set of line segments in the plane" (sourced from http://en.wikipedia.org/wiki/Scheinerman%27s_conjecture). However, the proof was only completed in 2009, and is fairly difficult.
Layout: You want the first line to be true when $H,M$ and $B$ are all true. So your statement $F$ must look like $(H\land M\land B)\lor \text{Something}$.
But you also want it to be true when $H,M$ are true and $B$ is false, equivalently, when $H, M$ and $\neg B$ is true. So, using the information in the paragraph above, $F$ must look like $(H\land M\land B)\lor (H\land M\land \neg B)\lor \text{Something else}$.
Proceed in this fashion to find $F$.
Edit: Firstly note that we need only apply this technique for the true lines, because by exactly pinpointing the true lines, the falsehoods will be determined.
So inspecting the true lines you can find:
$$(H\land M\land B)\lor (H\land M\land \neg B)\lor (H\land \neg M\land B)\lor (\neg H\land M\land B),$$
which has the expected truth table: 
Best Answer
If you want to prove $P \rightarrow Q$ then you should assume $P$ and try to deduce $Q$ in some way.
In the comment you way that you want to prove $⊢ ∃x(Px → ∀y Py)$, however this is only true if we have a constantsymbol already inthe language we are studying, since else we could have the empty model as a counter example.
Here is one strategy to prove that $⊢ ∃x(Px → ∀y Py)$:
This may possibly be done in a simpler way. I hope that these "strategy hints" are enough for you to figure it out.