Well, in that case, you should be made aware of the fact that the "old-school" logic of Aristotle contained two syllogistic errors of set containment that were not detected until the 20th century.
Occum's Razor is not a rule of logic, it is a highly fallable guideline which, if followed exclusively, would quickly lead science into the crapper. Few of the painstaking, generations-spanning breakthroughs in science followed from Occum's Razor. From Mendeleev's peas to Woese's re-organization of the tree of life at it's root is as twisty and convoluted a story as one could imagine. Continents glued to the planet is an orders of magnetude simpler explanation for their existence than the theory of plate tektonics.
Whatever you may think of under-determination, it is a varifiable presence in science. Nothing in the world was more "determined" than Newton's laws of gravity, but, as it turned out, Newton's laws were not the final word on the subject, and the present betting is that Einstein's aren't either. In like manner, no amount of verifying inductive evidence in favor the natural causes for the Big Bang, or life, in any manner eliminates the possibility of a supernatural Prime Mover--science only deals in physical evidence and only finds proximate physical causes--and it makes no claim as to the exclusive closure of those causes--is it's own history makes all too plain. You put a weight on logic it cannot bear when you claim it is dispositive against the existence of God. Like Ayn Rand, you have no evidence, and you have no proof, you just have a great deal of chutzpah, and the capacity to throw around the vocabulary of formal proof, but it means butkis. If Ayn Rand, or any of her erstwhile accolytes actually had a proof of the non-existence of God, they'd trot it out, instead of wordy razzle-dazzle peppered with logical techno-ese, delivered with a patronizing air.
I certainly welcome you not to believe in God, if you like, but trying to give the impression that science or logic makes it a decided issue everyone else is just too benighted to see is stretching the available evidence a great deal farther than it can reach.
Underdetermination is an embarassingly historically evident condition in science, and philosophers cannot wish it out of existence merely because they don't like it. As objectivists have been wont to point out for half a century--the social (or philosophical) consequences of an idea about the natural world are no kind of evidence in favor of, or against it.
The recognition of the problem of under-determination does not in the the least "deny that one theory can be superior or more accurate than another".
It does not put an end to, or even damage, critical thinking to recognize that our data and our reasoning is potentially fallible, and finite in scope, and to take that into account--because it happens to be true. And perceptions that happen to be true are more likely to yield useful results than perceptions that happen to be false.
It is in no small measure due to this problem of underdetermination that we have taken, at the end of the twentieth century, so enthusiastically to Karl Popper's notion of falsifiability as the central correct criteria to apply in our critical analyses of scientific evidence and theories. Proof is beyond our resources, but falsifiability makes useful sense as a major criteria for judging scientific theses. Or, putting it another way--falsifiability lets us evaluate theses usefully, in the face of under-determination. We, in fact, apply a very refined set of criteria in critically judging scientific evidence and theories, none of which is the least deterred by the painfully obvious fact of underdetermination. Simply because it is not a proof, and is fallable, does not mean it is irrational, or unproductive by any means. This is an extremely parocial, and unlikely role you've assigned to logic, to suggest we are bereft of wits unless we cast every attempt at thinking into a formal attempt at Aristotalian, deductive proof, and abandon the field if there is no proof to be found. This is a fairy tale, not a reasonable theory about how scientific, or, for that matter, any critical reasoning can or should operate.
Odd that we should think we can make progress this ragtag way, despite the conspicuous lack of proof of anything, eh? Or do you think physics has backslid greatly from the underdetermination of the law of gravity that has been flagrantly extant for the last 300 years?