Base atheism (aka soft atheism) is *not* the assertion there are no gods, but the rejection of godly claims which have so far failed to meet their burden of proof, which all have so far failed. Hard atheism is where you start seeing the claim there are no gods.
I'm of the position that soft atheism is actually agnosticism.
And you'd be wrong. A/gnosticism speaks to *knowledge* specifically. A/theism deals with belief. Using them this way makes for better distinctions. If you keep to your current system, how would you label a person who believes there are zero gods but recognizes they don't know for sure? Using *my*..
system, I'd simply call them an agnostic hard atheist.