I'm sure some of the ministers on here can give more depth and clarity to the discussion, but Calvin's commentary is helpful here. The passage needs to be situated in the chapter's contrast between the wisdom of the world and the hidden wisdom of God. It is no reason to doubt the Gospel that the wise of the world reject it--even those who appear to be experts in the Torah--rather, it is enough that the Spirit testifies to the truth of the Gospel. And who can sit in judgment over that which the Spirit testifies to? The Spirit is the ultimate and sufficient witness to Christ (think of Jesus' teachings on the Spirit as a witness in John's Gospel). The spiritual man (insofar as he is characterized by the Spirit) is not subject to "any human wisdom or reason", as Calvin puts it. This of course does not mean he is thus independent from the oversight and care of the visible church, as that care is itself spiritual. Calvin's caveat is useful:
"But here it may be asked, who is the spiritual man, and where we may find one that is endowed with so much light as to be prepared to judge of all things, feeling as we do, that we are at all times encompassed with much ignorance, and are in danger of erring: nay more, even those who come nearest to perfection from time to time fall and bruise themselves. The answer is easy: Paul does not extend this faculty to everything, so as to represent all that are renewed by the Spirit of God as exempt from every kind of error, but simply designs to teach, that the wisdom of the flesh is of no avail for judging of the doctrine of piety, and that this right of judgment and authority belong exclusively to the Spirit of God. In so far, therefore, as any one is regenerated, and according to the measure of grace conferred upon him, does he judge with accuracy and certainty, and no farther. "
