I'll give it a shot: an argument is universally compelling if no mind both a) has reasons to reject it, and b) has coherent beliefs. This is to say that a mind can only believe that the argument is false by believing a contradiction.
I think this may sound stronger than it actually is, for the same reasons that you can't convince an arbitrary mind who does not accept modus ponens that it is true.
More to the point, recall that one rationalist's modus tollens is another's modus ponens. This definition is defeated by any mind who possesses a strong prior that the given UCMA is false, and is willing to accept any and all consequences of that fact as true (even if doing so contradicts mathematical logic, Occam's Razor, Bayes, or anything else we take for granted). This prior is a reason to reject the argument (every decision to accept or reject a conclusion can be reduced to a choice of priors), and since it is willing to abandon all beliefs which contradict its rejection it will not hold any contradictory beliefs. It's worth noting that "contradiction" is a notion from formal logic which not all minds need to hold as true; this definition technically imposes a very strong restriction on the space of all minds which have to be persuaded. The law of non-contradiction (~(A ^ ~A) ) is a UCMA by definition under that requirement, even though I don't hold that belief with certainty.
The arbitrary choice of priors, even for rational minds, actually appears to defeat any UCMA definition that does not beg the question. Of course, it is also true that any coherent definition begs the question one way or another (by defining which minds have to be persuaded such that it either demands certain arguments be accepted by all, or such that it does not). Now that I think about it, that's the whole problem with the notion from the start. You have to define which minds have to be persuaded somewhere between a tape recorder shouting "2 + 2 = 5!" for eternity and including only your brain's algorithm. And where you draw that line determines exactly which arguments, if any, are UCMAs.
And if you don't have to persuade any minds, then I hesitate to permit you to call your argument "universally compelling" in any context where I can act to prevent it.
This prior is a reason to reject the argument (every decision to accept or reject a conclusion can be reduced to a choice of priors), and since it is willing to abandon all beliefs which contradict its rejection it will not hold any contradictory beliefs.
You're right that I am committed to denying this, though I would also point out that it does not follow a priori that it is always possible to resolve the state of having contradictory beliefs by rejecting either side of a contradiction arbitrarily. However, in order to deny the above, I must claim that... (read more)
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