Justifying ‘No Problem of Philosophy is Soluble’

Earlier, I presented the following antilogism:

1. All genuine problems are soluble.
2. No problem of philosophy is soluble.
3. Some problems of philosophy are genuine.

I claimed that "(2) is a good induction based on two and one half millenia of philosophical experience." The inductive inference, which I am claiming is good, is not merely from 'No problem has been solved' to 'No problem will be solved'; but from the former to the modal 'No problem can  be solved.'  From a deductive point of view, this is of course doubly invalid.  I use 'valid' and 'invalid' only in connection with deductive arguments.  No inductive argument is valid.  No news there.

Peter Lupu's objection, which he elaborated as best he could after I stuffed him with L-tryptophan-rich turkey and fixin's, was along the following lines.   If the problems of philosophy are insoluble, then so is the problem of induction.  This is the problem of justifying induction, of showing it to be rational.  So if all the problems are insoluble, then we cannot ever know that inductive inference is rational.  But if we cannot ever know this, then we cannot ever know that the inductive inference to (2) is rational.  Peter concludes that this is fatal to my metaphilosophical argument which proceeds from (2) and (3) to the negation of (1).  What he is maintaining, I believe, is that my argument is not rationally acceptable, contrary to what I stated, because (2) is not rationally acceptable.

Perhaps Peter's objection can be given the following sharper formulation.

(2) is either true or false. If (2) is true, then (2) is not rationally justifiable, hence not rationally acceptable, in which case the argument one of whose premises it is is not rationally acceptable.  If, on the other hand, (2) is false, then the argument is unsound.  So  my metaphilosophical argument is either rationally unacceptable or unsound.  Ouch!

I concede that my position implies that we cannot know that the inductive inference to (2) is rationally justified. But it might be rationally justified nonetheless.  Induction can be a rational procedure even if we cannot know that it is or prove that it is.  Induction is not the same as the problem of induction.  If I am right, the latter is insoluble.  But surely failure to solve the problem of induction does not show that induction is not rationally justified.  Peter seems to be assuming the following principle:

If S comes to believe that p on the basis of some cognitive procedure CP, then S is rationally justified in believing that p on the basis of CP only if S has solved all the philosophical problems pertaining to CP.

I don't see why one must accept the italicized principle.  It seems to me that I am rationally justified in believing that Peter is an Other Mind on the basis of my social interaction with him despite my not having solved the problem of Other Minds.  It seems to me that I am rationally justified, on the basis of memory, that he ate at my table on Thursday night despite my not having solved all the problems thrown up by memory.  And so on.