Many religions advocate the existence of some sort of afterlife. Usually, this promise of an eternal paradise is used to persuade followers to stick to the church's rules, to please its deities, or to serve its leaders. Even if doing so is awkward or morally questionable or even dangerous, it's hard to decide that it isn't worth it if you believe in an afterlife.
It took me a long time to abandon my agnosticism. Aside from the seemingly contradictory way the world worked (seriously, why does evil need to exist; why didn't the Christian God just think up a better solution than actively killing so many people in the Bible; evolution is almost certainly true, so why are creation stories so wrong), I couldn't fully convince myself of atheism. The following isn't anything close to a proof that souls don't exist; there's always a way to define souls or the afterlife to weasel out of the implications of the given argument. But given how I (and I think most people) imagine souls working, it's hard to go back to honestly expecting an afterlife to exist.
What's a soul?
The core assumption we'll make is that the soul is what thinks. I've got an internal monologue yapping for most of the day so that's the idea that comes to mind when I imagine "thinking". If it exists, my soul is what's behind that.
This is the point where most theists protest that it's your brain that does the thinking; your soul, they'll claim, is the essence of you. I've heard this in several different, wise-sounding phrasings, but I think this is a nonsensical claim. What else can even be you, if not the thoughts you think? Without those, you have no identity, you can't process information, you can't pray, you can't make decisions, you can't be happy, you can't rejoice in Heaven or suffer in Hell, you can't do anything!
Another common objection is that when you die, another thinking-complex takes over; or maybe while you're alive, your brain thinks, but when you die, your soul takes over and does the thinking. This also doesn't make much sense: if there's no continuation of self, why is the thing that wakes up after dying still you?
Ultimately, if there is to be a continuous experience of consciousness, if the same you that lives now will experience the afterlife, then an immortal soul must be the source of your thoughts.
Thought experiment
Suppose I hand you a box. It's pretty heavy, and when you talk to it, you're surprised to find out that it talks back! What's more, it seems to be able to reason, understand humor, and express emotions. In the middle of asking it how its day has been, you accidentally drop the box, nicking one of the corners. When the box answers, you're caught off guard with the previously-polite box's sudden use of profanity: "My fucking day's been really shitty actually, fuck you."
Damaging the box in other ways seems to cause other unexpected changes in its responses: the box might start st-st-stuttering after every other word, or fail to produce semantically correct sentences while getting the tone of the expected response completely right, or lose the ability to understand or make jokes, or immediately forget about the responses it's already given. In fact, when you're given other boxes, you find that particular kinds of damage cause the same sort of result in the way the box responds.
How would you expect the box works? At a basic level, do you think there's a microphone inside the box and someone is relaying answers from outside the box? Or do you think the structure that's doing the thinking is inside the box?
Conclusion
It seems very likely that the thing doing the thinking must be inside the box, because the types of changes to the box's responses are changes to the way it thinks. When we damage the box, we change the way the box's responses are put together; indeed, it seems even the thoughts the box can think are affected by the type of physical damage. Simply damaging an internal mic wouldn't cause those same sorts of changes.
The analogy here is that the box is the human body, and the damage that's being done is brain damage. If souls existed, we wouldn't expect physical damage to change the sort of thoughts we can think.
This argument isn't one I came up with myself; I came across it in this comment thread. What struck me was that the idea of brain damage is so commonly known, and yet so obviously antagonistic to the idea of souls. I hadn't ever been surprised that brain damage existed, even though its existence implied the non-existence of souls, which I truly believed in. Maybe that's because I sort of classified souls as non-physical and thereby unfalsifiable, or maybe I never thought about souls hard enough to really think about what I expected that belief to look like in the real world.
My grandmother recently passed. We've still got her picture up by the altar in our living room. She's had dementia for as long as I can remember. I didn't get to spend much time with her, because we live halfway around the world from her. Even when we did visit (at most once every 2 years), I didn't really look forward to spending time with her; she was always pleasant to be around, but it was too sad to see how reproducible her questions and responses became. Every day, during our family rosary, my Dad will pray for her soul. He genuinely loved his Mom, and she him from the stories he's told me. But whenever he says a prayer for her, I wonder what he expects to see in Heaven: will he see his Mom when she was younger and could think and reason without dementia hammering at her thoughts? If that's my real Grandma, who was the person I got to know?
What unfathomable evils brain damage and death are.