+1 - proponents of PSR often overreach/overstate its applicability (my opinion). We're taking an intuition developed (evolved) on earth, for mesoscopic objects and trying to make it some kind of law of the universe. As you say, we look for order where we expect it, not where the particular outcome is rare (you die example). This has implications for things such Cosmological Arguments for God, where PSR is leveraged in "sophisticated" theologies to force assent on some kind of "reason giver" like God.

The biggest weakness seems to me to be that quantum mechanics asserts many events occur without a (known) cause and many physicists believe there is no reason to believe there is one. For example, the idea of virtual particle formation requires no cause.

@JimmyJames Does it stand to reason that there would be a first virtual particle. If the vacuum is nothing then it would always be filled and there would never be a start.

@JimmyJames, that is partly what I had in mind when I noted that the argument doesn't work for random events.

@DavidGudeman I kind of figured that, but some people might argue that the way a die is thrown is the cause of what side it lands on. I have a friend who swears he can 'set' the dice in craps. A croupier once scolded him in my presence. I have a pet theory that 'true' quantum randomness may have more impact on things like this than we realize, though.

Re. "random events" - Heidegger narrows the PSR down to only applying to determinate things. Practically by definition: determinate means there is rational determination. His rationale is interpreting PSR as being about existence, i.e rationally determined phenonmena. There's got to be a rationalisation to say what the thing it. Radioactive decay is an interesting case. I can't help thinking there are just unknown causes. Obvs quantum theory uses probability.

@ChrisDegnen "I can't help thinking there are just unknown causes." You are in good company. The question is whether 'random' events are truly random or, if we knew everything about the state of the universe, we could precisely predict all events. The idea that true randomness exists does seem to resolve things such as infinite regress. It's also put out as a potential explanation for consciousness and other thorny philosophical problems.

From a previous post : [Finally] we hear the principle of reason in a different tonality. Instead of "Nothing is without reason," it now sounds like this: "Nothing is without reason."

@JimmyJames Consciousness sure feels pretty random.

@JimmyJames, "true randomness ... is also put out as a potential explanation for consciousness". I'm taking that in a direct sense, So free will vs determinism saga. As a conscious Dasein my existential experience and its Being, is indeterminate. Intuitively understandable but not explainable. So not a determinate thing. Obviously to you I am more determinate, but one can understand that other Daseins have experiential existence. The point being PSR applies to determinate, objective things. Consciousness is very indeterminate. (and maybe a bit random)

@ChrisDegnen So I take it you consider consciousness to be non-objective? I'm not great with the terminology but I think you are saying that consciousness is not part of objective reality. Something special, in other words. Supernatural perhaps?

@JimmyJames "is consciousness part of objective reality?" I can see some of my thoughts with thought, but the real wellspring is not visible, not objectively visible. And if I quell my thoughts then the objects fade out.

This answer is not a "proof". It just shows that PSR is a useful heuristic that works out more times than it doesn't. And no, that is not a "strong argument" for the PSR.

@Dcleve, if you were expecting a mathematical proof, then you were expecting too much. Philosophy doesn't work that way.

The OP asked if there were a proof. You have now agreed the answer is "no, and such a proof will never be provided". But that is not what your answer implies. The PSR as Leibnitz spelled it out required a logical proof to satisfy it, hence the PSR itself needed a logical proof to be justified in his terms. Modern PSRs are weakened vs. Leibnitz, but even weakenings of the PSR are refuted by the Munchausen Trilemma. A "principle of seeking partial justification" is all we can actually support, and this reality is a key step in the argument for pragmatic truth over logical truth.

@Dcleve, some people might think it's reasonable to respond to a question about whether there is a proof with an answer involving rational argumentation that is not a mathematical proof. Also, Leibniz wasn't the only philosopher to discuss the PSR and what he thought about it is not universal.

@DavidGudeman You are citing an argument for a "principle of seeking partial justification" for the PSR. But under logical truth, these are very different claims, and if you operate under logical truth and this is the best justification you can find for the PSR, the PSR is false. Switch frames to pragmatic truth, and this argument is a justification for treating the PSR as a frequently useful heuristic. It is only an argument FOR the PSR under a different truth standard than I think you are operating under, and you have not made that clear in your answer.

@Dcleve, If by "logical truth", you mean provable by deduction, then almost nothing can be justified under that standard--just a few laws of deduction. I suspect you didn't grasp the argument given your repeated claims that it's a "heuristic". The argument proves that believing n the PSR is rationally justified, at least in the cases that it covers.