In this post I will explain how anthropic paradoxes are connected with self-reference.
Sleeping Beauty Problem
The contention is how to treat the fact that "I am awake now/today". To briefly summarize the debate at the cost of oversimplification: SIA suggests treating today as a random sample of all days. While SSA suggests treating today as a random sample of all awakening days in the experiment. This debate can be considered as the dispute over the correct way of interpreting/defining today.
It should be noted the direct use of the word "today" is not necessary. For example, in Technicolor Beauty due to Titelbaum, the experimenter would randomly pick a day and paint the room blue and paint the room red the other day. Say I wake up and sees the room is blue. How should I consider the fact that "The blue day has an awakening?". Here the direct reference is averted yet the problem remains the same. i.e. blue day is being asked only because that's what I'm seeing now.
The doomsday argument essentially says, considering myself as a random sample from all humans (SSA), then I am more likely to have my current birth rank if the total number of humans is low. This should shift my forecast of the species' future towards "doom soon". The SIA counter-argument suggests I shall regard myself to be a random sample from all observers that could potentially exist. Then I actually do exist (as a human) is evidence that favors a higher number of humans. It exactly cancels out the pessimistic shift due to my birth rank.
The debate is about the correct way to define myself. Again, a direct reference to "myself" is not necessary. Often the argument is presented with wordings such as "us", "the current generation", or even "people in 2021". Yet the argument remains the same because those wordings are chosen because that's what you and I (the people discussing this problem) are.
Fine-tuning is the idea that the existence of life in this universe seems rather surprising. E.g. even if a few free parameters in the standard model have slightly different values, then life as we know it would not be possible. Fine-tuning has been used to promote two types of claims. One is the teleological argument that the universe is indeed fine-tuned by some unknown actor to specifically support life. The other is the multiverse claim that there exist a (near) infinite number of universes with different parameter values. Both theories make the existence of life (nearly) guaranteed which warrants a Bayesian shift favoring them given life exists. Safe to say they are controversial.
Self-reference is pivotal in these arguments too, albeit an indirect one. "Life" is focused on in these arguments because that is what we are. Similar to "the blue day" or "people in 2021" in the above examples, the arguments ponder on our own existence and its implications about the universe. If there exists some kind of objective outsider who studies the universe, then there is no inherent reason for it to pay special attention to life.
"This statement is false"
To recap, self-reference is a common element among anthropic paradoxes. And debates are over how to correctly interpret words like "I", "now" in probability calculations. This interpretation of "I" and "now" may appear necessary (though I disagree with its necessity, yet it seems I'm alone on this opinion). Because the paradoxes involve thinking about the possibilities of "I am not awake now" or "I do not exist". Statements as such are troublesome since they negate the validity of themselves, similar to the liar's paradox "This statement is false". Changing"I" and "now" to something objectively defined avoid this problem.
Given this connection to the classical problem of self-reference, anthropic paradoxes maybe even more complicated than they are already considered to be. But looking on the bright side, they could give some insights about our understanding of logic and reasoning once solved.