Welcome back to "Doggerel," where I ramble on about words and phrases that are misused, abused, or just plain meaningless.
As far as meaningless goes, this strikes me as a particularly meaningless phrase. I may not have taken formal logic courses in philosophy, but I'm having a hard time imagining a universe that could have anything "beyond logic" that doesn't end up being eaten up in pure chaos, if such a thing can exist. Pardon me for being inarticulate on the matter.
I tend to doubt woos really grasp what they're saying. Sometimes they say this to mean things beyond human comprehension. They always seem so sure of where exactly those limits are, hence no need to push them by asking questions. I've seen some reference infinity and the non-termination of pi, as if I should be impressed. Yeah, it was amazing to think about when I was a kid, but it was within my comprehension even back then. Also, I may not grasp "higher" maths now, but I'm not about to presume any of my shortcomings are universal to my fellow beings.
What's worse is when I bump into woos who think we're all Randroids reading from one of Spock's bad scripts. "What's so logical about loving and protecting someone?" As if selfishness was any more "logical" a motivation than love. Goals are separate from accurately figuring out the best method for achieving them. Of course, there's plenty of rational explanations for why we have those emotional goals, usually involving evolution, and neurology's been doing steady work on how our brains produce them. The data are quite noisy, and the definitions vary, but they're within the grasp of logic and science.
Another use is when woos claim that something is "beyond science." I've covered this pretty well with Doggerel #50. The only ways I can imagine something being beyond science would be if it were completely without effect (and thus pretty much irrelevant), or completely random and unpredictable. Science studies effects and makes predictions based on the observation of those effects.