I see that some flat earth proponents do positively claim the earth is flat, therefore taking on the burden of proof, and support their claim with "look out of your window" or "look down". This is the evidence upon which they rely in support of their positive assertion. I see the logic, but it is an extremely basic argument. Perhaps it is not the intention of the society or of the flat earth movement to be taken seriously. I anticipate a counter-argument along the lines of "why would we want to be taken seriously by people who refuse to question inconsistencies between what they see and what they are told?", and that may be why this debate will never get any further than these message boards.
If this forum were a courtroom, and the flat earthers had brought their claim that the earth is a disc, what finding would the judge make? He or she would be bound to consider evidence from both sides, but ultimately the burden of proof would be upon the claimant. Let's assume that the burden of proof is "on the balance of probabilities", such as in civil courts in England and Wales.