The problem with saying that the burden of proof is on the one making the assertion, is that both parties then have to decide on and agree about who is making "the assertion". To most Christians, the foundation is that "The Bible forbids polygamy". The assertion being made, then, in their eyes, is that "The Bible permits polygyny". From their perspective, our assertion is the absurd, new claim that requires proof.
I don't have much arguing experience, especially not on polygyny (I prefer to avoid conflict, unfortunately, though I'm hoping to man up on this over time), but these are my thoughts on arguing for polygyny. Note that they come from observation and contemplation, but not much from experience. In other words, weigh them accordingly.
The problem with arguing from our position is that we are arguing for nonexistence (the nonexistence of a prohibition against polygyny), which simply doesn't work. "There are no black swans"... until there are. "There are no verses which prohibit polygyny", until there are. Nonexistence can only be proved if the context is a finite set, or if it can be proved that it must not exist. While verses are finite, interpretations are not. And having someone accept that a prohibition against polygyny must not exist sounds challenging, to say the least. But these two paths are likely the only ones that exist.
There are other theoretical paths, but they don't really work. Claims that there is a requirement for polygyny or that there must be an option for polygyny are both arguments for the existence of something, and would be easier to prove, logically speaking. However, requirements for polygyny are circumstantial (and often could be waved away with "that was then, this is now" variants). I'm not sure I could come up with an argument that there must be an option for polygyny for the general man that would convince a monogamy-only proponent.
If we were able to catch this problem before it became common, however many centuries ago, it would have been easier, because the burden of proof would have been more clearly on the monogamy-only side, as they would be the ones making a new assertion of the existence of a prohibition. Alas, we no longer have that advantage, and are now in the opposition's home turf. Let that be a lesson to all of us to not let things slide and lose the advantage. That's a challenging lesson, especially for people like me who like to avoid conflict.
Anyway, it seems to me that the only arguments that can be made are to:
1. Refute the arguments given for existence of a prohibition. This would need to be an exhaustive refutation, and would only bring the argument for existence into "black swan" territory. "It must say it somewhere" is an easy fallback.
2. Demonstrate, as much as possible, that a prohibition must not exist. Show that God cannot sin, and yet does it Himself, and does not change (and work through the counterarguments there). Show that certain patriarchs did it and were called righteous (dealing with counterarguments here also). Etc.
When refuting the arguments for existence of a prohibition, our position then gets tasked with saying "no... it doesn't mean that", which can simply be countered with "but I believe it does". Which might be what's happening in this thread (albeit by proxy).
Each individual interpretation of a passage becomes a mini (logical) argument of its own. Who has the burden of proof when asserting a specific interpretation of a passage? There is the traditional interpretation, and the opposing, "absurd" interpretation which supports polygyny. There, too, we are at a disadvantage, playing on stolen ground.
Alternative interpretations might be dismissed out of hand ("why would you be right, instead of many educated scholars over hundreds or thousands of years?", "you're biased/motivated", etc.), or simply accepted as a possibility. But even if one alternative interpretation is accepted, the debate opponent could simply move on to a different verse, ad nauseam. And if every verse were to have an alternative interpretation which was accepted as a possibility, the person would need to accept that the alternatives for every verse interpretation are true. Then the person runs into a reality-shift problem: the reality that all of the supposed authorities on Biblical interpretation are not able to be fully trusted, because one person says so. One person who may be biased and be particularly motivated. What about the slippery slope? It's "safer" to stay on the side of the many, of the familiar. And that's still an option, because all of these verses only have alternative interpretations, which could be dismissed simply by the wave of a hand. The debater proposes a conflict: scholars, pastors, family, and friends vs the debater. Easiest resolution: the debater is wrong. Who cares about what people call fallacies, when a person's whole reality and world are being turned upside down?
We might have better luck getting as close as we can to the idea that a prohibition must not exist, that the Bible explicitly claims it to be permissible or righteous (examples: God and righteous patriarchs practiced it). That creates, or rather reveals, a contradiction, a conflict, that must be resolved. The conflict is discovered within their own mind, not proposed by an external source.
Then follow that up with explanations of how the verses don't conflict. Then perhaps (I'm likely being naively hopeful here) the other person might feel like you are helping to resolve a conflict, instead of creating a conflict. At the very least, you're not the one shouting "they're all wrong!", but are letting them ask the question themselves.
Anyway, those are my musings. Perhaps, when I talk about polygyny in the future with people, I'll try to keep this mind.
