Any logical argument by definition follows a sequence of statements that are presented in a specified order. Therefore, the logical argument must proceed in time.
Actually, a logical argument, by definition, is a collection of statements (called premises) and another statement (called the conclusion). What you are stating here is a method for determining whether an argument is valid (that is, whether the conclusion must necessarily follow from the premises).
But the "real" way of determining whether an argument is valid is to construct its truth table. Now it is true that it takes time for a person to do this, but any argument consists of only a finite number of simple statements and so can (in principle) be done in a finite amount of time, and the results are, unlike the linear proof you are talking about, unambiguous.
The reason that you don't see truth tables very often is that in real arguments, consisting of a very large number of premises (most of which are not even explicitly stated in the proposition), and so it is actually impossible to do a truth table in a reasonable amount of time.
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The real reason that logic cannot prove that God exists or that God does not exist is that logic cannot prove anything about the real world. All logic does is determine whether or not a conclusion does necessarily follow the given premises. In the linear method of proof that you mentioned (the only one available in practice), one can (one hopes) show that a conclusion does follow the premises, and, in some cases, one might show that the conclusion does not necessarily follow from the premises.
But that says nothing about the real world since there is no guarantee that the premises are accurate statements about the real world. As one example, given Newton's laws of motion and the law of gravity, scientists of the 19th century proved that the orbit of Uranus will follow a particular path. In actuality, Uranus did not follow this path. Clearly this meant that one or more of the premises were wrong: it could be that the law of gravity was incorrect, or that the laws of motion were not correct. Well, what actually happened was that the
unspoken premise that there were only seven planets was changed to explicity state that there is an eighth planet...and Neptune was subsequently discovered. In fact, the history of science is really the history of how our premises about the real world have changed. Some of these changes were so profound, using Kuhn's ideas of paradigm shifts, that our very definitions and conceptions ended up changing.
So the problem with the proofs for or against the existence of God is that, just like in the sciences, we cannot be certain that our premises, the assumptions that we make about the nature of God or reality, are correct. In reality, God either exists or he does not, and all we can do is try to come to the most reasonable conclusion whether he does or does not based on what we know or what we think we know about the real world around us. Logic is merely a tool to help us draw the correct conclusions
based on what we know or what we think we know, but it will never be definitive since at all times our knowledge about the real world is limited and usually inaccurate to some degree.
Actually, if their god makes better pancakes, I'm totally switching sides. --
Charley the Australopithecine