Couldn't we have defined the material conditional differently? I've been mulling this over lately, and I can't seem to understand why exactly the material conditional wasn't defined in a completely different way. 

Obviously, the motivation behind our current definition of the material conditional is, amongst others, so that we don't end up with a definition that makes most restricted quantifiers impossible to work with. For example, take two triangles. A triangle is called equilateral if the lengths of its three sides are the same, whereas a triangle is defined to be isosceles if two of its three sides have the same length. We know then, by definition, that any equilateral triangle is also an isosceles triangle. 

With this in mind, we can set up the following implication where $P(T)$ and $Q(T)$ are open sentences over the domain S of all triangles:
$$P(T): T \; \textit{is equilateral.} \; \text{and} \; Q(T): T \; \textit{is isosceles.}$$

$$P(T) \implies Q(T)$$

This implication should then be required to hold true for all triangles $T$ that are equilateral, so we could write:

$$\forall \, T, \, P(T) \implies Q(T)$$

but we run into the problem of "what if T isn't equilateral?" If we choose to define the truth values of the material conditional any other way than we currently have, say by making the implication false whenever $P(T)$ is false, we'd end up with a false implication. But surely, our implication (which we know to be true by definition) can't be proven wrong by looking at triangles that the implication doesn't even deal with in the first place. Therefore, we must require this implication to be vacuously true.

However, couldn't all of this have been avoided by simply stating:

$$\forall \, T: \textit{T is equilateral}, \, P(T) \implies Q(T)$$

and assigning a false truth value to the material conditional whenever its antecedent is false? That way, we'd avoid situations like:

$$\forall x \in \mathbb{R}, \; x&gt;2 \implies x^2&gt;4$$
$$x=-1$$
$$\therefore (-1)^2&gt;4$$

...situations which we know to be untrue.

I'm probably not really getting the big picture here, but it seems to me that a simple specification of the properties of a quantified statement fix the issue of quantifier incompatibility. Could somebody enlighten me?

Cheers.
Your inference is wrong.

If we start with the reasonable law :


  $∀x \in \mathbb R,x &gt; 2 \to x^2 &gt; 4$


and we instantiate it with $x=-1$, we get :


  $-1 &gt; 2 \to (-1)^2 &gt; 4$.


But we cannot infer the false : $(-1)^2 &gt; 4$ because, in order to do it, we have to use modus ponens, and to do this we have to assert the false : $-1 &gt; 2$, and we have no reason to assert it.
Consider the statement: If it is raining, then it is cloudy.


  Symbolically: $Raining\implies Cloudy$


What does this mean? First, what it does not mean: It does not mean that rain somehow causes cloudiness, or that cloudiness somehow causes rain. Neither is the case. It means only that it is not both raining and not cloudy. 


  Symbolically: $\neg[Raining \land \neg Cloudy]$


There is no suggestion of causality -- of one thing causing another -- or of time-based data about a historical relationships between raining and cloudiness.

Everything about material implication -- the truth table for $\implies$, and the detachment and contra-positives rules -- all make sense when you understand this. Simply stated, the rule is:


  $A\implies B\equiv \neg[A\land \neg B]$




EDIT:

The Truth Table

$\begin{array}{c|c|c|c}
A&amp;B&amp;A\implies B\\
\hline
T&amp;T&amp;T\\
T&amp;F&amp;F\\
F&amp;T&amp;T\\
F&amp;F&amp;T
\end{array}$

The Intuition

In the first row, we have $A\implies B$ and $A$ being true. Of course, $B$ must also then be true, as indicated.

In the second row, we $A$ being true and $B$ being false. Then $A\implies B$ must be false, as indicated.

Note that when $A\implies B$ is true and $A$ is false, as you would expect, $B$ is indeterminate, i.e. $B$ could be either true or false, as indicated in the the last two rows. 

We also need $A\implies B$ to be true when $A$ is false to prove vacuous truth, an essential method of proof in mathematics.