John Witherspoon was an important Founding Father. He was President of Princeton University, then The College of New Jersey and taught James Madison, and many other Founders. Witherspoon was a Presbyterian minister (in fact the only signer of the Declaration who was a minister). Witherspoon indeed remained an orthodox Christian when he preached from the pulpit. Yet, when he taught principles of politics, he left his Calvinism at the door and instead turned to the Scottish Enlightenment.
Witherspoon imparted his political philosophy to his students at Princeton through his Lectures on Moral Philosophy. Those lectures lack orthodox Christian content and are naturalistic and rationalistic in their approach. As it were, likely Witherspoon the naturalist and rationalist, not Witherspoon the Calvinist influenced James Madison et al. As Mark Noll, Nathan Hatch, and George Marsden point out in "The Search For Christian America," though Witherspoon, in Scotland, defended orthodoxy against Enlightenment thinkers David Hume and Francis Hutcheson, in his Lectures he "turned instinctively to the books of his erstwhile theological opponents, Hume, Hutcheson, and other philosophers of the Scottish Enlightenment." pp. 88-89, see also Gregg Frazer, "The Political Theology of the American Founding," PhD Diss., p. 278.
In other words, like the theistic rationalists whom he influenced, Witherspoon attempted to synthesize Christianity with Enlightenment rationalism, but managed to do so while remaining an orthodox Trinitarian Christian. As Dr. Frazer points out, the difference between Witherspoon and the theistic rationalists was that Witherspoon confined his naturalism and rationalism to the realm of politics and morality, not theology. When such naturalism and rationalism was applied more generally to theological matters, it led Madison et al. to abandon orthodox Christianity for theistic rationalism. Ibid. p. 280.
As Witherspoon begins his Lectures:
MORAL Philosophy is that branch of Science which treats of the principles and laws of Duty or Morals. It is called Philosophy, because it is an inquiry into the nature and grounds of moral obligation by reason, as distinct from revelation.
Witherspoon then defends the use of reason in religion, but concludes that if applied properly reason and revelation will always agree:
Hence arises a question, is it lawful, and is it safe or useful, to separate moral philosophy from religion ? It will be said, it is either the same or different from revealed truth; if the same, unnecessary—if different, false and dangerous.
If the Scripture is true, the discoveries of reason cannot be contrary to it; and, therefore, it has nothing to fear from that quarter. And as we are certain it can do no evil, so there is a probability that it may do much good. There may be an illustration and confirmation of the inspired writings, from reason and observation, which will greatly add to their beauty and force.
The noble and eminent improvements in natural philosophy, which have been made since the end of the last century, have been far from hurting the interest of religion; on the contrary, they have greatly promoted it, Why should it not be the same with moral philosophy, which is indeed nothing else but the knowledge of human nature?
I do not know any thing that serves more for the support of religion than to see, from the different and opposite systems of philosophers, that there is nothing certain in their schemes, but what is coincident with the word of God.
Some there are, and perhaps more in the present than any former age, who deny the law of nature, and say, that all such sentiments as have been usually ascribed to the law of nature are from revelation and tradition. We must distinguish here between the light of nature and the law of nature: by the first is to be understood what we can or do discover by our own powers, without revelation or tradition : by the second, that which, when discovered, can be made appear to be agreeable to reason and nature.
So what we've seen is Witherspoon, in these Lectures, elevates the discoveries of reason to the same level as scripture -- infallible, states that reason and revelation will always agree, but proceeds to construct his political teachings on reason alone without citing verses and chapters of scripture at all.
As Noll, Hatch, and Marsden put it:
Witherspoon did not derive his politics from the Bible. He did not think the Christian God had a specific role to play in public life, where the rule of nature prevailed. And he did not worry about assuming an Enlightenment perspective on political matters. Noll, et al., 90-91
Witherspoon's rationalistic perspective certainly profoundly differed from Francis Schaffer's.