Mathematical logic has axioms and/or assumptions as the foundation. I am not clear what you mean by SOUND ARGUMENT. If you as a math person require sense verifiable objects and awareness of truth value, you have made a nonsensical statement. Sense verification is needed for Mathematical truth but you want SOUND ARGUMENTS?
An issue is existential import and objective knowledge. How do we know objectively there are no unicorns? How does one disprove the proposition? All a sense verifiable person can do is require a single instance for a proposition to be true. We don't always have that luxury in reality.
Now I come to YOUR ASSUMPTIONS or AXIOMS in Mathematics. You are NOT saying x is a True proposition outright. You are being slick by placing it in a conditional form (if . . . Then . . . ). This way you can't be said to be WRONG or mistaken as it was just an assumption anyway. Philosophers often have a backbone. If I make a proposition I have to own it if I end up being wrong. If I say all swans are white and someone finds black swans in Australia my position is busted. Too many busted attempts lower my standings in a social circle whether you want to omit this fact folks. ( If I give you wrong directions two times most practical people will not ask me for directions a third time to see if I get it right.) Back to my point: saying x is a true proposition is CLEARLY not as psychological as If x is true then something else is true. You save yourself from embarrassment because you never committed to any REAL CLAIM in REALITY and said nothing basically. If I am wrong with a proposition, then judgement of me and the claim follows. When you blunder you cover it with that huge IF.
Help me understand what you mean by SOUND ARGUMENT when the fact is that many people fail to have a back bone to make a CLAIM about reality? For instance, my proposition IF there were unicorns in New York, they would all be white animals DOES NOT make a claim about reality. How is this format (IF . . .THEN . . .) you run to so reliable and SOUND? We know sometimes this works and other times it fails. If it fails even 1 percent it is not absolute or certain. Science has the same issue. So math taking the scientific approach may be a mistake. You both have the same issue. Senses can be wrong.
I am also a bit confused about the terms ASSUMPTION and Axiom, which where I am from, DOES NOT designate you have any awareness of the ACTUAL TRUTH VALUE. Science can demonstrate all dogs are animals. I do not need an IF Statement to cover myself like you would. I would just have a back bone and say out loud All dogs are animals & accept the backlash or correction when it arrives. You seem to want to avoid all of this action. Pretty slick to me. I find it a bit odd to say Mathematical logic is sound as you say because you can't even tell me upfront if the assumption is TRUE. I guess then it would not be an assumption huh? MAKE A CLAIM not a way out if you are wrong. For an argument to be SOUND the premises MUST BE TRUE -- not kind of iffy, kind of true, possibly true, I am not sure if it's TRUE, etc. Its ironic you want sound arguments but start with a claim you are not certain of. Make a claim first! Tell me the proposition you begin with IS TRUE absolutely. That is all I ask and then we can work on building a SOUND ARGUMENT.
Your understanding of the term proposition is not on par with old school philosophy. All propositions are expressions that have a TRUTH VALUE. Your problem as well as the scientific problem is without sense verification you take the liberty to say a proposition has a false TRUTH VALUE or better yet the UNKNOWN VALUE. Let me get this correct, because you don't know it or have sense verification the value is NOW unknown? The proposition Jimmy Hofa is buried at Shea Stadium under home plate is FALSE because there is no evidence right NOW? I capitalized NOW because time can change as well as technology. Suppose tomorrow Hofa's body was found. What was scientifically FALSE or UNKNOWN magically has another TRUTH VALUE. Is science and math magic? A more rational approach would be the proposition about Hofa being buried was OBJECTIVELY true or false from the beginning --regardless of time or technology. Because YOU WERE UNAWARE of the truth value does not dismiss the actual value nor does new information CREATE A truth value from nothing.
I am not sure what you mean by Sound argument when there is little accountability when there is an error and no necessary commitment to say something about reality. Axioms by definition often fail the sense verification requirements that WAS used about unicorn propositions. I guess we will let that slide and say its objective knowledge. Can other types of reasoning outside of math be useful yes. Is mathematics the most useful? No. The starting point is the problem-- as is the argument of GOD. I can prove almost anything if we always take the assumption as truth. We would have to question what is truth and define truth I reckon at some point. We can all agree bad assumptions will make poor reasoning about reality. What do we do when we assume the negative? Often times we lose our trail and the case goes cold. This further would put you in a place where you dont want to put yourself out in the wind for judgment, but what do I know. I know nothing: as I can't prove most things I claim to KNOW sufficiently and I can never be wrong if I never make a statement about reality; and knowledge is demonstrated by expressing some kind of claim.
Expressing something I am uncertain about does not seem a method that gives me 100 percent accuracy.