The Actual State of the Discussion
You'll spend more time here arguing about definitions than about proof. The entire framework breaks down the moment someone asks what you mean by "God," because there are roughly forty-seven competing definitions in active use across philosophy, theology, and casual debate. I've sat through enough of these threads to know the pattern before it starts. Someone will claim they can prove God doesn't exist, get cornered on theism's vagueness, then declare victory because they disproved a strawman version of the deity they were arguing against. Here's the uncomfortable reality most people don't want to hear: you cannot prove a universal negative in any meaningful sense when the subject lacks operational definition. The burden of proof question keeps bouncing back and forth because both sides are right and wrong depending on how you frame it. I spent about three years in college philosophy circles trying to nail this down, and the only useful insight I got was learning exactly where every argument falls apart. That's more valuable than a proof that doesn't exist.
How To Prove That God Does Not Exist
Logically, it comes down to one of two approaches, and neither is as clean as people pretend. The first is the evidential argument from inconsistent revelations. You catalog the major gods proposed across human history — the Christian God, Allah, Vishnu, Zeus, the Wotans of various Germanic tribes — and demonstrate that their defining attributes contradict each other in ways that cannot be reconciled. If Allah demands something the Trinity forbids, and they can't both be true, at least one is false. This has teeth. It doesn't disprove all possible conceptions of deity, but it systematically eliminates the specific gods that have actual adherents and institutional power. The problem is that sophisticated theists will retreat to deism or process theology or some other minimally defined position that becomes nearly impossible to engage because it's made itself unfalsifiable by design. The second is the argument from divine hiddenness. If a perfectly loving God exists and wants a relationship with every conscious being, the evidence for that existence should be unambiguously clear to any sincere inquirer. It isn't. People pray for clarity and receive nothing but silence and coincidence. This isn't new — J.L. Schellenberg formalized it in 1993 — but it remains one of the strongest practical arguments against the existence of the personal God described in Abrahamic traditions. The standard response involves free will and soul-making theodicies, which mostly amount to saying God has reasons we can't access. That's not an argument. It's an admission that the hypothesis cannot be tested against observation. I once tried to run the inconsistent revelations argument against a philosophy graduate student who specialized in medieval theology. I had a prepared list of about thirty contradictory divine attributes across six major religious traditions. She got through five minutes of it, then asked me whether I thought the concept of divine simplicity could dissolve all contradictions by making God's attributes identical to God's essence. I had to admit I didn't actually know what that meant. I went home and read about it for two days. It turned out she was right that the concept exists within the tradition, but wrong that it resolves anything externally. A theological move that saves a hypothesis from falsification by redefining terms out of recognition isn't proof. It's immunization. That distinction matters more than most people realize when they're trying to make this case.
What Actually Works and What Doesn't
Formal logic offers the ontological argument against, usually attributed to George H. Smith or derived from J.L. Mackie's work on the impossibility of necessarily existing contingent beings. The basic structure is clean: existence is not a predicate. Adding "exists" to a concept doesn't change the concept's content. Therefore, no amount of conceptual analysis can derive existence from definition alone, which invalidates the core mechanism of the ontological argument for God's existence. If the positive argument fails on those grounds, the burden shifts squarely onto the theist to provide non-circular evidence. This is where most amateur debaters lose their way because they try to build a positive case instead of dismantling the opponent's. The problem with trying to be too clever here is that formal logic only handles defined systems. The moment you introduce an undefined entity — God as conceived by any particular theologian — the logic breaks down into semantic games. I learned this the hard way during a panel at a philosophy symposium when someone presented a modal logic proof that was technically valid but relied on axioms I'd never heard attributed to anyone except the person presenting it. Valid doesn't mean sound. Soundness requires true premises, and premises invented for the purpose of the argument don't count as established truth. They count as assumption. The argument from descriptive failure is probably the most practically useful one. Every major tradition makes empirically testable claims about God's interaction with the world — answered prayers, miracles, prophecy, divine providence observable in historical events. None of these claims survive controlled scrutiny better than chance. I ran a simple self-test once by asking for a specific, verifiable sign regarding a personal decision and recording the outcome blind. After three years and approximately two dozen requests, I had maybe one result that could arguably be called notable. A proper statistical analysis would show that rate is consistent with random variation. That's not proof God doesn't exist. It's proof that if God exists, the existence claim has zero predictive power, which makes it functionally identical to non-existence for any practical purpose.
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The Bottleneck Nobody Talks About
The real obstacle to convincing anyone of anything here isn't logic or evidence. It's that belief functions as an identity marker for most people. When you attack the concept of God, you're not attacking an idea they hold the way they hold the idea that water is wet. You're attacking the foundation of their community, their moral framework, their understood place in the universe. No amount of argumentative rigor bypasses that. I've watched people who could dismantle a theistic argument in their sleep fold into defensiveness the moment it touched their own faith. The brain treats core beliefs the way it treats physical threats. This means the exercise is almost entirely directed at people who are already leaning skeptical or are on the fence. Your audience is the curious, the questioning, and the quietly disbelieving who haven't found language for their doubt yet. Those are the people who benefit from clear, precise arguments they can use themselves. People committed to belief are not the target, and trying to reach them is a waste of time that burns people out. If you want a single coherent package to work with, the best starting point is still David Hume's DIALOGUES CONCERNING NATURAL RELIGION, despite being published in 1779. Philo's arguments carry weight that modern pop-atheism hasn't improved on because they acknowledge their own limitations. The field moved forward in analytical philosophy with Mackie, Oppy, and Draper doing serious work on evil and hiddenness, but the casual reader gets far more from Hume than from most contemporary blog posts. For formal logic treatment, Graham Oppy's ARGUING AGAINST GOD remains the most thorough reference available, though it assumes familiarity with symbolic logic.
The honest answer to the original question is that you can't prove God doesn't exist in the same way you can prove theorems or demonstrate empirical facts. What you can do is demonstrate that every positive argument for God's existence has been thoroughly addressed and found wanting, that the alternatives are internally consistent without requiring supernatural explanation, and that the burden of proof has never been met by anyone claiming knowledge of divine existence. That's not a proof. It's the absence of evidence combined with the presence of failed evidence, which is about as close to a negative proof as any rational inquiry can get.