Are all philosophical questions unsolvable?

Questions do not have solutions, so your question needs rephrasing. It contains a category mistake. Perhaps one could say that questions are to answers as problems are to solutions. At any rate, it is questions have answers, and problems that have solutions. So we can put your question as this: 'Is no philosophical question answerable?'

Your question has a history starting in classical philosophy, as the question whether there are insolubilia, or unanswerable questions, such as whether the statement made by the Cretan Liar is true, or whether it is false. But in modern philosophy the question about insolubilia expanded from logical annoyances into the entire world of metaphysics and epistemology. Can we ever find reality and know it as it is? Kant famously thought not, that we are somehow imprisoned within our own conceptual schemes. This is a big deal. His arguments are complicated and take some study to understand, if they can be understood. There is a possible lesson here, which is that we should look carefully at the reasons given for saying that reality is unknowable, rather than wallowing in that conclusion.

Suppose that the answer to your question is affirmative, so that no philosophical question is answerable. Then that is the answer to your philosophical question. But then the answer is wrong, since at least that question, the one you asked, has an answer. We have just given it!

So then suppose the answer to the question is negative. It is that some philosophical questions are answerable. So at least one philosophical question has an answer.

Either way, at least one philosophical question (yours!) has an answer. What goes for questions also goes for problems. So there is at least one philosophical problem with a solution.

It is interesting to see that to get an answer to your question we do not have to look at the specific arguments as to why philosophical questions might be answerable or as to why they might be unanswerable. We have an argument that cuts through any other argument, and delivers a reassuring result, even if it is a narrow one. But it would be a good thing to explore the historical question, 'What reasons have been given in the history of philosophy for thinking that there are insolubilia, of the narrower logical sort or of the wider metpaphysical and epistemological sorts?'

It would also be good to think about whether the argument I have given can somehow be generalized to all philosophical questions. A good place to start might be Moritz Schlick's "Unanswerable Questions?" of 1935. Schlick's view is that there are no unanswerable questions of an empirical kind, because the answers to such questions would be unverifiable, and hence meaningless. So the questions, being the answers with a question mark at the end, would also be meaningless.

You might also want to ask why you ask the question, what is your own reason for thinking that all it might be the case that all philosophical questions are unanswerable.

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