How strong of an argument for theism is the fine-tuning argument, and what is the current opinion of it?

Great question!
"How strong" is a difficult question to answer precisely, and I'm sure different philosophers will have very different takes on this. The answer will probably have to be comparative (that is, compare how good this argument is to others), and philosophers disagree about how good the other arguments are. For example, some philosophers still think that the old ontological argument is great! Others think it is decisively refuted. In addition, there is disagreement about the merits of the fine tuning argument itself. So, I don't think there's consensus about either side of the comparison that determines how strong the argument is.
That said, I think most of us can agree that it is among the most promising versions of an a posteriori (or empirical) argument for the existence of god, since it has some advantages over the more traditional argument from design. The argument for design appeals to the apparent complexity in nature and posits an intelligent designer on that basis. A major objection to that argument is that evolution by natural selection explains that complexity, so there's no need to posit an intelligent designer after all. That's a pretty good objection (assuming all the details work out, which they seem to). But the fine-tuning argument cannot be refuted by appeal to evolution by natural selection. So that's a huge advantage. So we can say this: the fine-tuning argument is relatively strong in the sense that it resists a major objection to the more traditional empirical argument for the existence of god, namely the design argument.

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