For private arguments, even if logically impeccable, are not the be-all-and-end-all when it comes to central and defining matters of Christian doctrine, and may well turn out to be false. For one can have a logically valid but unsound argument. And one can have a private opinion, which, though internally consistent and beyond logical reproach, is inconsistent with other things, such as the teaching of an ecumenical council like Chalcedon, or -- more importantly -- the doctrine of Scripture.
Oliver Crisp, God Incarnate
Crisp makes a very basic, but often ignored or misapplied point concerning logic and individual viewpoints. Logic is tool of reasoning, a very useful and powerful tool of reasoning, and can be an effective arbiter of truth insofar as its use as a tool is correctly applied. But logic is not a source of or grounding for truth claims. One must bring one's assumptions concerning truth to the tool of logic; one cannot derive the truth of a given claim from the tool. This would be the same as using a hammer and nails to put together a chair without having the wood upon which to use the hammer and nails.
Additionally, one of the difficulties of man, and not just modern man, is submission to an external authority. Even in times when external authorities were obeyed out of fear or tradition, it was not necessarily the case that individuals believed in what the authority dictated. Also, even if they believed, it was not necessarily the case that they do so on the basis of that authority. In any case, the rub is that men have a desire to justify a perspective from within their own mind, according to the dictates of their own reason--and logic can be a tool to solidify such individual views from inconsistencies and error, such that one's claims are internally consistent with one's basic assumptions. But upon what authority such assumptions are based, or what claim to truth these assumptions have cannot be derived from logic, but must be taken from one's own "self-evidence" or upon the evidence of an external authority (such as Scripture and ecumenical councils and doctrinal confessions).