Most people who claim that argument from authority is fallacious would, perversely, argue for that claim by reference to the authority of common knowledge or of what were often taught. A fallacy is actually shown by demonstrating a conflict with a principle of logic or by an empirical counter-example. A case in which an authority proved to be wrong might be taken as the latter, but matters are not so simple.

When one normally makes a formal study of logic, that study is usually of assertoric logic, the logic in which every proposition is treated as if knowable to be true or knowable to be false, even if sometimes the study itself deliberately treats a propostion as false that is true or a proposition as true that is false. In the context of assertoric logic, an argument from authority is indeed fallacious.

But most of the propositions with which we deal are not known or knowable to be true or false; rather, we find that some propositions are relatively more plausible than others. Our everyday logic must be the logic of that ordering. Within that logic, showing that a proposition has one position in the ordering given some information does not show that it did not have a different position without that information. So we cannot show that arguments from authority are fallacious in the logic of plausibility simply by showing that what some particular authority claimed to be likely or even certainly true was later shown to be almost certainly false or simply false.

Arguments from authority, though often not recognized as such, are essential to our everyday reasoning. For example, most of us rely heavily upon the authority of others as to what they have experienced; we even rely heavily upon the authority of n-th-hand reports and distillations of reports of the experiences of others. And none of us has fully explored the theoretic structure of the scientific theories that the vast majority of us accept; instead, we rely upon the authority of those transmitting sketches, gists, or conclusions. Some of those authorities have failed us; some of those authorities will fail us in the future; those failures have not and will not make every such reliance upon authorities fallacious.

However, genuine fallacy would lie in over-reliance upon authorities — putting some authoritative claims higher in the plausibility ordering than any authoritative claims should be, or failing to account for factors that should lower the places in the plausibity ordering associated with authorities of various sorts, such as those with poor histories or with conflicts of interest.

By the way, I have occasionally been accused of arguing from authority when I've done no such thing, but instead have pointed to someone who was in some way important in development or useful in presentation of an argument that I wish to invoke.

Authoritarians and totalitarians of the political left and right, in arguing that they are not authoritarians or not totalitarians, often engage in a bait-and-switch, with little or no awareness that they are doing so. To argue that policy or programme X is justified is not somehow a contradiction of a claim that policy or programme X is authoritarian or even of a claim that policy or programme X is totalitarian At best, the authoritarians are arguing that authoritarianism is justified and the totalitarians are arguing that totalitarianism is justified but it seems that they are arguing that authoritarianism is not authoritarianism if it is justified or that totalitarianism is not totalitarianism if it is justified

[This entry may be superfluous, in that people who fall for any of the fallacies discussed are unlikely to read the entry, people who employ one of the fallacies are unlikely to reform if they do read the entry, and people who recognize that fallacies are involved may not see much use to analyzing them.]

I often encounter an argument, whose form is

P does A₁;
A₂ is better than A₁;
therefore it would be acceptable/desirable for P to do A₂.

It's easy to find P, A₁, and A₂ such that the intuïtion recoils from the conclusion that

it would be acceptable/desirable for P to do A₂.

and, in the face of such intuïtions, most people will acknowledge the non sequitur (acknowledged or otherwise) in the argument. We could even add a further premise, that

P ought to be persistently active (if not necessarily in their present manner).

and still find P, A₁, and A₂ such that the intuïtion recoils from that conclusion. (Consider that non-profit institutions do facilitate child abuse, and that child abuse is worse than many other things that are still themselves unacceptable.)

Yet one encounters this argument frequently with P as the state, A₂ is something that somebody wants done (such as space exploration) and A₁ is something disturbing that the state is doing or has done recently.

A variation on this can be found with form

P₁ approves when P₂ does A₁;
A₂ is better than A₁;
therefore P₃ should not object to P₂ doing A₂.

A non sequitur is evident in cases where P₃ is plainly no subset of P₁; but this argument is often presented in a manner so as to obscure a distinction, as when an everyone or a no one is used as-if loosely (which is to say inaccurately) in the first premise, but P₃ is some person or group of persons who aren't actually in the set labelled everyone or actually are in a non-empty set labelled no one.

However, this argument is fallacious even when P₃ simply is P₁. There may in fact be an incoherency in approving of A₁ while objecting to A₂, but that inconsistency could be resolved by changing one's position on A₁. For example, if forcing people to pay for birth control is better than forcing them to pay for war with Iraq, then perhaps someone who objects to the former should cease approving of the latter, rather than embracing the former.

(And resistance from P₁ to coherence wouldn't itself license A₂ when A₂ victimizes yet some additional party P₄. One doesn't force atheists to distribute copies of Al Qu'ran on the grounds that neoconservatives would object to such distribution even while supporting worse things.)

Sometimes one even sees an argument of the form

P₁ does not object when P₂ does A₁;
A₂ is better than A₁;
therefore P₃ should not object to P₂ doing A₂.

Variations of this even go so far as to replace objection with more active opposition.

P₁ does not actively oppose P₂ doing A₁;
A₂ is better than A₁;
therefore P₃ should not actively oppose P₂ doing A₂.

The appeal for those who present these arguments is that, if they were accepted, then almost no A₂ could be practicably challenged, as the objector could be dismissed for not having tackled each and every greater evil.

Of course, if this argument held, then it could virtually always be turned around against the claimant. For every P₂ and A₂, there is a P'₂, A'₁, and A'₂ such that A'₂ is the supposed ill addressed by A₂, P'₂ effects A'₂, and A'₁ is some greater ill effected by P'₂. In other words, even if A₂ were good, it would itself almost never address the greatest evil, so that there would always be something else that one would be required to do before ever getting to A₂.