Basically, a logic statement is one you can reason with.
Suppose I read a beautiful piece of poetry and am taken out of myself in such a way that a desire is awoken which is not directed toward anything I know or apparently anything I have ever encountered in my earthly life. What prevents me from reasoning with this? What prevents me from reasoning about the nature of the human being, the sorts of experience open to us, the possibility of the desire being fulfilled, etc.?
Indeed both the atheist scientist and the Catholic contemplative (or scientist, for that matter) seek to "make sense" of such an experience of poetry. They both reason with it.
As for the bifurcation in my worldview.... perhaps. But rather than black/white, I suspect there may be more of a continuum. Our minds (and so our reasoning) are highly conditioned to life as animals on earth, per our ancient ancestry. Back then, what use was the eternal? None but a deadly distraction. Or it may have been the opposite, that the eternal was the un-noticed water in which we swam. Either way, it was not a suitable subject for reasoning, which does seem time-dependent.
I guess I would say that we are highly conditioned to life as animals, that the eternal is as much use today as it was then, and that it was as much a subject of reasoning then as it is now.
You say reasoning seems time-dependent and you also have a post on "ultimate proofs." You seem very focused on syllogistic, temporal reasoning. But pre-syllogistic reasoning--whether found in modern epistemology or ancient philosophers--has a very strong transcendent, contemplative quality. For example, what is "a triangle"? You could show a monkey a million different triangles and they would still have no idea what it means to be "a triangle." They would not understand the underlying principles that universally define triangularity,
apart from all material and concrete manifestations. Yet we do. We come to an act of understanding by which we understand what triangularity is, and it is far more than a matching and picking with a large memory bank (although, as Aristotle points out, experience with material manifestations is very helpful in making that "jump" to the abstract, universal concept).
Similarly with language, parents speak to infants all the time, and it is pure nonsense to them. They are just stimulus-response machines. Until one day, suddenly, mysteriously, they
understand. They see that meaning and language is abstract, meaningful apart from stimulus, response, desires, etc. And so they run around pointing at things, asking for their names. They want to know what it is called, what it means. Science itself, in its healthy form, is a maturation of this sense of wonder and desire for knowledge. The syllogisms, the logic, are just tools, they are just means to an end. The illuminations, realizations, understanding that such things help effect are the real goal, and they are really not "logical statements" at all. The act of understanding by which we appropriate what something is--its nature, the way it behaves, its characteristics, its mode of being--is not really a temporal or "logical" act at all. It is given in an eternal moment, not unlike that which is given to the child whose eyes are opened and who sees, at last and at once, that "mama" is not just a button you push when you want something, but is actually a subsistent being/object apart from itself with its own mysterious norms for movement and life, part of which include caring for the child itself.