Are Philosophical Problems Soluble?

Edward Buckner writes, 

In my PhD thesis I argued that philosophical problems cannot be resolved. I think you still take the same view. My thinking today is that while the problems exist in some sense, they cannot be coherently stated in logical form. I.e. “The riddle does not exist. If a question can be put at all, then it can also be answered.”

I do indeed consider the central problems of philosophy to be insoluble. But I don't agree that the problems cannot be coherently stated in logical form. And I don't agree that a problem to be genuine must be soluble.  Consider the following antilogism:

1. All genuine problems are soluble.

2. No problem of philosophy is soluble.

3. Some problems of philosophy are genuine.

The above inconsistent triad is a clear and coherent presentation in logical form of a philosophical problem, namely, the meta-problem of whether only soluble problems are genuine.  The problem is obviously genuine (as opposed to pseudo), but not obviously soluble.  Hence it is reasonably held to be insoluble. 

If you disagree, tell me which of the three propositions you will reject, and why it must be rejected. For example, you might tell me that (3) is to be rejected and its negation accepted. The negation of (3) is:

    ~3. No problems of philosophy are genuine.

Now prove (~3).  You won't be able to do it.