When people argue, both sides always claim to use facts and logic...

I wish I could formulate this in proper logic notation, but not if they're using different axioms :p Regarding facts, pretty much the FIRST illustration they do in any sort of debate/science thing is the one where a sphere is unveiled between two people who will argue whether it's white or black, until it's turned over to reveal the other half is white and the other is black.

Not necessarily, but in most cases you're right.

Because facts and logic would inevitably point to a certain set conclusion.

There's always that thang called interpretation 🤷🏼‍♀️

Someone is claiming erroneously :)