Knowledge of God doesn't disappear among men after the fall. We still have His image stamped on us, to know of His power and being (Rom 1:20). At the same time unbelievers suppress and distort that knowledge. Goodwin calls it a false knowledge and he (or Beeke/Jones) appeal to 1 John 2:3-4. I think this is off base. Knowing God in the context of that verse is more of a profession of faith in God, not an awareness of His existence and attributes that leaves one without excuse. As the old joke goes, the atheist claims God isn't there, and hates Him. The atheist can't suppress the truth that God exists, even if he rejects the Christ who is the sole means of knowing God.
This is the old saw between emphasizing the antithesis, the difference between believer and unbeliever, and emphasizing the common truth implanted in all men and in nature about who God is. Theologians and philosophers will debate this endlessly and systematize all thought to fit their preference. I tend to think that the apologetic circumstance should direct what we emphasize. If an unbelieving friend wrestles with an impending sense of doom, or finds a philosophical argument proving God's existence compelling, seize that "natural" knowledge of God within him, instead of denying it is there and trying to supply a more certain epistemological foundation! But if a friend is in the throes of postmodern uncertainty, it may help to point out that all our knowledge is beating the air without some anchor point outside us and our reason. Both aspects are true and both have their advantages in different conversations.
John Owen wrote most and best about this of any Puritan.
Scripture was inspired by God, and the Spirit testifies to our hearts that it is so.
God also revealed himself through audible voice, dreams, visions. These have ceased since the New Testament time (Heb 1:1).
It's interesting that older theologians tended toward a mechanical view of inspiration - they wrote down the words God suggested to their minds. With the advent of lower criticism, noting literary differences between various human authors, we've become more attuned to the personalities and style of those authors. The authors do a good job showing that this sensitivity was not lacking even in Owen, though, before we got so smart. This is sarcasm aimed at self-congratulating Christian academics enamored with liberal textual criticism of the Bible. We've learned some things, to be sure, but we weren't benighted imbeciles before the nineteenth century, either.