"Atheism is a lack of belief. Only the person who claims God exists bears the burden of proof."
Burden of proof is a concept from legal philosophy that says anyone who makes a claim must justify it with evidence. "Lack of belief" is merely a description of the atheist’s mental state and doesn’t require a defense since it’s not a claim. However, the moment atheists makes an objective claim about reality, they do bear a burden of proof and must provide reasons and evidence to justify their position. Here are a few common atheist claims that require evidence:
"God doesn’t exist"
"Belief in God is irrational"
"The material universe is all that exists"
"The universe came from nothing"
"The universe isn’t designed"
"Life can arise from non-living matter without a designer"
Since "lack of belief" often has irrational implications, it’s important to examine the reasons behind our non-beliefs. Skeptics who lack a belief in a round earth or the moon landings may not have a burden of proof since they haven’t made a claim, but their skepticism has irrational Implications that can’t simply be assumed (a flat earth and mass NASA conspiracies.). The same is true with atheism. Rejecting the possibility of a designer requires the circular assumption that the physical universe caused it’s own existence, that life and consciousness randomly arise from lifeless matter, and that morality is merely a social construct to aid survival. Atheists must provide good reasons and evidence for these underlying assumptions in order for their lack of belief to be rational.
"Only the person who makes a positive claim bears the burden of proof."
This arbitrary rule has no grounding in logic or legal philosophy. Negative claims still require evidence because they’re objective statements about reality. The person who claims George Washington never existed or that the holocaust never happened isn’t free from the burden of proof simply because they’ve made claims of negation. Like all negative claims, their statements have unspoken assumptions that can’t simply be assumed true until proven otherwise. This is an example of a logical fallacy known as shifting the burden of proof.
Another problem with the rule is that all positive claims can be phrased in a negative way. A theist could say "undesigned universes don’t exist", which would put the burden of proof on the atheist who makes the positive claim that we live in an undesigned universe. The arbitrary "positive claim rule" reduces rational debate to a game of who is better at using language to avoid defending their position.
"Extraordinary claims require extraordinary evidence."
Some atheists think this is a law of logic, but it’s actually just a slogan invented by Carl Sagan. It sounds reasonable because of it’s symmetry, but it leads to illogical and unscientific practices. As geophysicist David Deming has pointed out, "science doesn’t contemplate two types of evidence". Introducing a second category of "extraordinary evidence" turns truth seeking into a subjective, ambiguous practice that can’t be measured by science or logic. There are no objective, scientific tests you run to figure out if it’s more extraordinary to claim that the universe is the result design or that it appeared out of nothing for no apparent reason.
In science, claims only require enough evidence to demonstrate their truth value. Claiming that I won the lottery is a lot more extraordinary than claiming I bought a loaf of bread. However, both can be proven with the same exact evidence: a printed receipt. It’s possible Sagan was trying to say we should investigate extraordinary claims more carefully, but the way the quote is worded has lead many atheists to use it as justification for a double standard of evidence.
If atheists want to apply Sagan’s rule, then the sword cuts both ways. We shouldn’t let Richard Dawkins off the hook when he makes the extraordinary claim that that matter, energy, and the laws of physics popped into existence out of nothing for no apparent reason. When people ask him for evidence, he always responds by saying "scientists are working on it" and is content assuming a logical impossibility until then. It’s fair to atheists don’t have extraordinary evidence when it comes to claims about origins.