“It is the mark of an educated mind to expect that amount of exactness in each kind which the nature of the particular subject admits.” So says Aristotle at the beginning of his "Nichomachean Ethics", and he returns to this theme several times later in the work. “It is equally unreasonable to accept merely probable conclusions from a mathematician and to demand strict demonstration from an orator.”
I cannot think of a similarly concise expression of the problem in Scripture, but I believe it is acknowledged in the structure of Scripture. There are no ethical discussions there that could be described as philosophical demonstrations, but instead we find narratives, then laws, then judgments and prophetic exhortations, poetic reflections, and collections of proverbs and wise sayings. There is an emphasis in the scriptures upon kochma--Wisdom, or literally “skill”—that involves practice as much as, or possibly more than, simply knowledge.
Precise demonstrations, like geometric proofs, require unassailable first principles to build upon, and are only as certain as those first principles. Furthermore, each subsequent step in the chain of reasoning must be likewise certain and unassailable if we wish to have certainty in the conclusion. When such a demonstration is used to convince another person, the attempt fails at whatever point the hearer cannot or will not grant either the premise or the basis for reasoning. Even when the chain of reasoning is not used to convince others, but simply to increase one’s own understanding, our conclusions can only be as certain as our first principles and each step we used to get there.
It’s actually even worse than this. Each incorrect assumption, and each invalid or non-necessary step in the reasoning, will actually drive us off-course, so to speak, and the errors will multiply their effects down through the entire course of reasoning. If we are unable to know just exactly what each error is, we will have no idea how far off course we are. It may be that we periodically arrive at a location that we “recognize” and can do a course correction, but even in this case it is important to realize that the need for such correction demonstrates that at least one error was made in the reasoning up to that point.
Now for a quick and dirty demonstration of the problem. Suppose we are “75% certain” that our first principle is correctly understood by us. (This might mean that, among people whose opinion we respect, about 75% agree with us about this principle, and 25% think we are wrong.) For the first step in our reasoning, let’s assume we have 90% agreement. At this point, only 66% of people we respect will agree with our conclusion (90% of the original 75%). If we get 90% agreement on the validity of the second step, we’re down to 60% agreement, and after only three steps we’re down to 54%, a little better than a coin toss. If at any step we encounter the need for some certain knowledge that we do not have, we have no way of assessing how big the resultant error will be. We could be completely wrong.
In ethical reasoning, then, we must make every effort to be as certain as possible about our very first premise, and about each step of reasoning based upon that premise. Given that we are rarely 100% certain about any abstract idea, we must keep our chains of reasoning very short. To expand the navigational metaphor, we should never, if possible, go out of sight of the land, so that we will always have fixed and knowable landmarks by which to correct our course. Otherwise, we will find ourselves in a trackless sea, arguing with each other about how we got there and what to do next.