This past year I wrote a large, socio-political piece examining the illogic of certain bad-faith actors in popular culture. For me, it was just an excuse to get everything out of my head and on paper to stop thinking about it. What I did not expect, however, was the path it would take me on to seriously studying formal logic. As I wrote the piece and studied the logic around it I started finding newer and newer vocabulary to describe what I was examining and that vocabulary always seemed to be Latin, so I started asking questions and before I knew it I was knee deep in Aristotle's Syllogistics.
The door that studying that logic has opened for me hasn't made me any smarter, sadly, but it has made me far more capable. Before studying logic seriously I had a cursory understanding of how to talk about some fallacies and that was about it - I could identify a non-sequitur… sometimes. What I couldn't do was evaluate and identify every aspect of a statement, its logical form and coherence, and its soundness with a keen eye. That has since improved greatly and, while I'm still a novice, I'm now able to think about statements in terms of what they're actually doing to the argument with a large vocabulary to describe it. It's given me the power to understand what supports the arguments (and at roughly what strength it does so) and what has nothing to do with the argument at all.
What this has meant for me with regards to my examination of my own theology is a great relief from things that don't matter. Prior to understanding with formal language how arguments are supported I kind of held in my mind that archaeological or scriptural tensions would eventually resolve in favor of my beliefs. The error I was making, though, was that of not understanding how those tensions affected the truth claims of the church. I effectively had a grey cloud over my head that followed me around reminding me that some things must be held provisionally until they can be proven true in order for the church to be true.
The Clarity
Upon understanding how arguments, truth, and the laws of logic (like the law of excluded middle) come together, my eyes were opened. No longer did I worry about archeology or scriptural tensions or any other argument derived from an academic perspective, because I realized this truth like I never had before: all of them are non-sequitur to the claim of the church and ultimately if the claim of the church is true, then they resolve in the church's favor. Thus, there is no point in trying to prove the church right or wrong with archeology, scripture, or anything else tangible as the church doesn't claim to be true on the basis of our ability to discover that truth through those means. What this means is you must address the claim of the church on its terms and by no other means can you address it and still make a logically coherent argument.
(Un)Falsifiability
This is where falsifiability and the limits of science come into play. If science were capable of definitively proving something then not only could the truth claims be under fire by things like archeology and independent research, but they could also be vindicated by it. The problem, however, with invoking science and study in proving something is that it is philosophically incapable of proving anything to the point of certainty. Science can only act to disprove that which is disprovable, but never positively prove to a degree of complete certainty. Science can only get us what we think is close to the truth by narrowing the field of possible hypotheses, but it can never narrow that field down to only 1. That's not to disparage science - science is an amazing practice grounded in a very formidable philosophy for helping us understand our world - it's just that the very nature of what science can do is limited against positively proving anything and can do little to nothing for that which cannot be disproven (or in other words is unfalsifiable). Thus, at any point, regardless of how confident we may be in a finding and what that finding tells us, a new variable can sneak in there that we were never aware of and completely destroy or reshape our theories. Thus, there is no evidence, archeological or otherwise, that is so great that it can't be overcome if the truth claims of the church are accurate and thus there is no evidence that is load bearing on it outside of what can be experimented on regarding the claim itself - that you can receive revelation and know of its surety.
It's worth noting that a skeptic might look upon the unfalsifiability of the claims of the restoration as being to the theology's detriment. However, the claims of the restoration, Messiah, or His Gospel aren't unfalsifiable in the way that they can't be true or false - they're unfalsifiable in the way that they can't currently be measured. We don't have a God measuring tool and if one was gifted to us we likely wouldn't understand its measurements. That's not to say the gospel isn't somewhat falsifiable on its own terms, however - as the restored gospel comes with an invitation to read The Book of Mormon, ponder in your heart, and pray, that by those means, God willing, you will receive a witness.
The Claim With the Logic and Science Applied
So when you put together what logic and science are able to do against the claim you start to realize that their place may be useful for investigating or taking an interest in obtaining a testimony of Christ and His restored gospel, but they can do nothing, and by nothing I mean literally nothing, to prove it true or false and as such are completely auxiliary and require no attention other than that which you wish to give them by your own interest. Understanding how those function together has helped me remove issues from my periphery that would have sat there, presumably until death, and free myself from them. Instead, I can put those issues on a bookshelf somewhere and investigate them if they pique my interest. Unless and until I am interested and have the time, I'm afforded to not only comfortably, but reasonably and rationally forget about them. Trying to prove or disprove the claims of the church by those means is like trying to prove that Picasso's paintings weren't art by measuring the strokes - it is a categorical error that cannot be overcome.