Basically Dave, a syllogism, or conditional or any argument, must obey the ponen and tollens rules, the only "two ways" of logic. So if a syllogism is of the correct form it is valid, but to make an argument sound, the premises also have to be true.
This is always the point of dispute. A lot of people can create a valid argument (correct form), by learning how to construct a syllogism, and obeying the two ways/rules of logic, but the premises themselves can be speculative, which doesn't offer proof.
This is why the term, "proof" is so misunderstood by people. 100% proof is something that is demonstrably factual, IMHO. If you can dispute the thing in question, it isn't proven. For example, would anyone dispute the existence of grass or trees? (if we rule out crackpots or the mentally challenged).
Naturally we as Christians would dearly LOVE to say, "we have proof of God" but when you think about it, such a statement would contradict the New Testament. For, "we live by faith not by sight", which means we don't have the proof, because most things that can be seen, are proven. We know the moon is there, and Everest, and humans and grass, just by sight alone, we don't need an experiment or an argument to tell us these things exist.
In the same way I think we can interpret that scripture this way, I believe; "for we live by faith not by proof". Not that I want to change scripture, but I think it's pretty fair to interpret it that way as it is making a delineation between the most obvious things to be proven, by sight, and things we don't see, therefore having no proof of.
"Now faith is the substance of things hoped for, the evidence of things not seen."
How can faith be valid if we see God or God is proven? If God is on every TV station in the world, and you are standing next to Him, and the whole planet hears Him say to you, "believe I exist by faith and you will be saved", how is that possible? For do we need faith to believe the moon and the grass exist? Yet we are told that, "the only way to please God" is by faith. It's a great mystery of life, and a great mystery of God. He gets pleasure only if we believe He is there when we can't fully know or prove He is. There's something mysterious there, don't you agree? Something special about God's nature I can't put my finger on. Perhaps you have the answer.
I hope you understand my point; as soon as God proves Himself, the show is over and the curtain goes across the stage.