If you don't put any money into the soda-pop machine, and it gives you a bottle of soda anyway, do you have grounds for complaint? Has it violated the principle, "if you put money in, then a soda comes out"? I wouldn't think you have grounds for complaint. If the machine gives a soda to every passerby, then it is still obeying the principle that if one puts money in, one gets a soda out.
Similarly, the only grounds for complaint against $p\to q$ is the situation where $p$ is true, but $q$ is false. This is why the only F entry in the truth table occurs in this row.
If you imagine putting an F on the row to which you refer, the truth table becomes the same as what you would expect for $p\iff q$, but we don't expect that "if p, then q" has the same meaning as "p if and only if q".
From the point of view of formalism, after we set up a (first order) formal language, a (mathematical) statement is neither true nor false, but simply a (meaningless) string of symbols. Even if we agree that some of the statements are axioms, we are not really saying they are true, but simply playing the game of printing new strings of symbols from them under certain rules, and there are statements that could be printed (provable), and others that could not (unprovable). There is neither truth nor falsehood within the formal system. Proof by contradiction is just one of the rules used to generate new statements from old ones.
For example, neither Euclidean geometry nor non-Euclidean ones are "true" or "false". It's just that if you accept the axioms of one kind, there are inevitable consequences you must accept as well. Whether the axioms (hence theorems) match physical reality is beyond the scope of mathematical investigation.
To speak about truth, we need a "model" of the theory, that is a set (with some functions and relations) that fits the axiomatic setup, i.e., each statement makes sense, and each axiom is true. It's more or less clear that for any given model, a statement is either true or false. For example, $\exists x,y,z\in\mathbb N$ such that $x^n+y^n=z^n, xyz\not=0$ is either true or false for the (standard) model $\mathbb N$. The difference between a formal system and a model, is like the difference between the English word "rose" and rose -- the flower.
There is a fundamental theorem (or meta-theorem) in mathematical logic that says proof (including proof by contradiction) is "sound". This means that if you start with true statements about a model, everything that can be proved is also true. To be more specific, the tautology theorem claims all tautologies hold, and "Reductio ad absurdum" is amongst them. So, the closest thing to a proof of "reductio ad absurdum" is the one of the tautology theorem.
Most of us would accept this, probably because we (including Godel) do believe the existence of a Platonic universe that reflects our axiomatic reasoning. (Still, questions like whether CH or its negation hold are very much for debate.). There are other values of more constructive proofs, but philosophically to completely reject reductio ad absurdum, I guess it tries not to assume our reasoning is about some kind of (unspecified or unreachable) Platonic "model" or reality, but about the status of our knowledge itself. I personally believe even if a murder case was never solved, the existence of a criminal is not in doubt.
Neither "this is false" nor "this is unprovable" is a well-formed formula in some formal language, and statements of natural languages are too ambiguous to have definite truth values, even more innocent ones might still suffer: such as whether a 50 years old is “old”. However, given a well-formed statement within a formal system, whether it can be proved or not from the given axioms has objective truth value. But this is like you're talking about whether a printer can print certain symbols, and the printer itself is incapable of "talking" about this. However, Godel's genius idea is roughly that, if such a proof exists, it can be encoded using natural numbers, and "this is unprovable" itself can be made a first-order statement about natural numbers, that is a sentence within the vocabulary of the printer if it can “speak” about natural numbers.
Best Answer
You're not going to get anything like a logical paradox or anything because the way you are considering defining it is not self-contradictory. In fact, as @Prime Mover and you both point out, this would make it equivalent to $\Leftrightarrow$.
Nevertheless, this would lead to what would probably be considered an unsatisfactory operator. For example, "$x < 5 \Rightarrow x < 10$" would be false which is not what we want from an "if ... then ..." statement.