The teleological argument (specifically the one I posted about recently from Robin Collins) relies on a premise that he calls the Prime Principle of Confirmation. I will be addressing an objection to this principle, but for the sake of dialectical clarity here is a formulation of the whole argument:

1. The fine-tuning data are not improbable under theism.
2. The fine-tuning data are very improbable under single-universe atheism.
3. If some evidence is not improbable under A but very improbable under B, then that evidence provides strong evidence for A.
4. So, the fine-tuning data provide strong evidence to favor the design hypothesis over single-universe atheism.

The Prime Principle of Confirmation is premise 3 in the above argument. It states that when we are considering two competing hypotheses, an observation counts as evidence for the hypothesis under which the observation is most likely. For example, let's say that I walk into my friend's room and there is a puddle of water on the floor and that my friend is nowhere to be seen. Next, I consider two possibilities (also for the sake of the example assume they are the only two possibilities). Either my friend spilled some water then stepped out to get a towel, or he spontaneously melted into a puddle of water. Since the former of these hypotheses seems more likely, the Principle states that the puddle counts as evidence that he spilled and left to get a towel.

This principle may seem intuitively obvious, but the Doomsday Argument provides a promising counterexample. It comes from Michael Huemer's Paradox Lost and goes like this:

If the human species continues to last for a good while longer, then the population will continue to increase and most of the humans who ever lived will live in a future advanced society. However, we find ourselves living in today's world, a primitive society by comparison. Since most of humanity will live in a more advanced society than ours under this hypothesis, it is surprising that we find ourselves living today. But, if the world is going to end in the near future, then it is not at all surprising that we find ourselves alive today. Therefore, since our being alive is more probable under the hypothesis that the world will end soon, it also counts as evidence that doomsday is close upon us.

Certainly, this is an absurd result that we reached by applying the Prime Principle of Confirmation, and thus a strong counterexample to premise 3 in the teleological argument at the top of this post. I do not think that it succeeds, however. The supporter of the teleological argument can sidestep this trouble by claiming that the Principle has causal constraints. Consider the puddle example from earlier. The observation I made, that there is a puddle on the floor, was an effect that I used as evidence for the most likely cause. This mode of reasoning works because of the causal link between the proposed cause and the effect that I observed. In the doomsday argument, the observation we make, that we are alive right now, has no causal link to the end of the world. And so the Principle doesn't apply. To make this distinction clear I suggest we alter the teleological argument like this:

1. The fine-tuning data are not improbable if they were caused by God.
2. The fine-tuning data are very improbable if caused by anything other than God.
3. If some evidence is not improbable under cause A but very improbable under cause B, then that evidence provides strong evidence for cause A.
4. So, the fine-tuning data provide strong evidence that they were caused by God.