Are predicates like $P(x)$ functions of $x$? Is it a function which can take values 'true' and 'false' depending on $x$? Then is $P(x)\to Q(x)$ equivalent to writing $P(x)=Q(x)$ for all $x$? Am I understanding this right?
EDIT- So I was supposed to prove that James is mortal using logic given that:


*

*James is a man

*All men are mortal


I did this:
$P(x)$: x is a man
$Q(x)$: x is mortal
$P(James)$
$\forall x(P(x)\rightarrow Q(x))$
$P(James)\rightarrow Q(James)$
$Q(James)$
Is this proof correct? In my proof, there are two lines which just say $P(James)$ and $Q(James)$. I took $P(James)$ as simply a shorthand for 'James is a man', and similarly for $Q(James)$. I don't see how the functional interpretation of $P(x)$ is equivalent to this shorthand interpretation of $P(x)$
 A: 
Is it a function which can take values 'true' and 'false' depending on $x$? Then is $P(x)\to Q(x)$ equivalent to writing $P(x)=Q(x)$ for all $x$? Am I understanding this right?

Yes and no.
Yes, $P(x)\to Q(x)$ is a function that maps a term, $x$, to truth values.
However, no.   In classical predicate logic, $P(x)\to Q(x)$ evaluates as true whenever $Q(x)$ evaluates as true or $P(x)$ evaluates as false.   So the predicates need not be equal.   Indeed if you consider Truth to be greater than Falsity, the comparison is more $P(x)\leqslant Q(x)$ ; $P(x)$ implies $Q(x)$ exactly when $Q(x)$ is at least as true as $P(x)$.
A: I don't know that I perfectly understand your question. If this doesn't answer it, please let me know and I'll update my answer. 

Yes, you can look at predicates as being functions from your domain of discourse to truth values. 
In general $P(x)$ might be something like $x>5$. This is not true or false, but becomes true or false once we fix a value of $x$. However, $\exists x. P(x)$ is no longer a "function" -- it is true, since $6$ (among other things) will witness the existence of such an $x$. 
However, $P(x) \to Q(x)$ is not saying that $P(x) = Q(x)$. Instead it says that $\lnot P(x) \lor Q(x)$. Said another way, it says $\{ x ~|~ P(x)\} \subseteq \{ x~|~ Q(x)\}$. 
Remember, though, $P(x) \to Q(x)$ still depends on $x$. Saying that $x>5 \to x>7$ is false when $x=6$ but it's true when $x=8$. However, once we quantify, and write $\forall x. (x> 5 \to x>7)$ then we get something which is either true or false (in this instance, it is false).
To get $P(x) = Q(x)$ you want $P(x) \leftrightarrow Q(x)$, which is an abbreviation for $(P(x) \to Q(x)) \land (Q(x) \to P(x))$. This says, following the above, that $\{ x ~|~ P(x)\} = \{ x~|~ Q(x)\}$. 
Edit:
In your example, we might take $P(x)$ to be "$x$ is a man" and $Q(x)$ to be "$x$ is mortal". 
Then if we assume $P(\text{James})$ and $\forall x. P(x) \to Q(x)$, what can we say?
Well, since we know for ALL $x$, $P(x) \to Q(x)$, we know that, in particular for James, $P(\text {James}) \to Q(\text{James})$. Further, since we know $P(\text{James})$, we can conclude $Q(\text{James})$. That is, "James is mortal". 
There is a lot of mathematics which is cantered around formalizing this argument. It is called "proof theory", and might be interesting for you to read about. At a high level, though, what I have just said is how it works. 

I hope this helps ^_^
