The argument, "No proposition is certain; therefore, the proposition that no proposition is certain is itself uncertain," is not self-defeating unless the initial proposition were supposed to be prefaced by, "Certainly," as in:
- It is certain that no proposition is certain.
- Therefore, it is uncertain that it is certain that no proposition is certain.
But the unbeliever in certainty need not be thought of as implicitly making use of the concept of certainty as such; indeed, one might say that nothing is certain because one doesn't use the concept of beliefs-that-are-certain-or-uncertain in the first place.
Another "bad" retorsion argument is the Reformed epistemological argument against evolutionary cognitive science:
- If our belief-forming capacity resulted from survival-oriented evolution, then it resulted from a process that is not essentially directed at representing true beliefs so much as survival-conducive beliefs.
- If a belief-forming capacity is not essentially directed at representing truth, then it is not warranted that this capacity be trusted.
- Our belief-forming capacity is generically trustworthy.
- Therefore, our belief-forming capacity did not result from survival-oriented evolution.
The amount of misunderstanding of evolutionary cognitive science involved in this reasoning is enough to render (1) unsound from the get-go, though, so there are plenty of local reasons why this retorsion argument fails.
Generally, the problem seems to be of unrestricted quantification over propositions, and it's a sword that cuts both ways. For example, suppose one tried to quantify over all well-founded sets by quantifying over a set containing all and only well-founded elements. However, the result would be a set that wasn't an element of itself firstly, yet then by satisfying this description it would go on to be an element of itself secondly, which would convert it back into an element that wasn't an element of itself, and so on. The quantifier "all and only" creates this problem, since one could seemingly have a set X such that X is defined as having all well-founded sets as elements while having itself as its sole ill-founded element, say.
How is this related to retorsion arguments? The targets of these arguments are universal quantifications over types of propositions, e.g., "All meaningful sentences are either tautologies or are empirically verifiable." The supposition is that this universally quantified proposition is in conflict with itself. In theory, the verificationist axiom is neither tautological nor empirically verifiable; so it must belong to the class of meaningless sentences; therefore... Yet one wonders, from the outside, how it could be meaningless, if we can "act as though it were true" in order to go on to derive its self-defeat. If it were meaningless altogether, it doesn't seem as though we would have grounds for inferring anything specific from it, no more than "$%luo*09(+4f--jjl{" as an empty string of symbols licenses the inference of some other string. So perhaps the verificationist axiom is tautological, after all.
Or take, "The truth of every true proposition is relative to the standpoint of believers in any propositions." It will be objected that this whole assertion is itself relative to believers, too, then, and that there will turn out to be believers according to whose viewpoints there are absolutely true propositions. However, "absolute" can be defined as "relative to everything," so relativism doesn't have to be taken for denying the existence of absolutes. By contrast, however, "Nothing is absolute," does go to, "It is absolute that there are no absolutes," and this is an impeccable retorsion argument, after all.
At least, in-context it is; but note again the problem of the Russell set. It was in response to that problem that Russell developed his type theory. And now we have another way to block the usefulness of retorsion arguments: we can talk in terms of universally quantifying over nth-order propositions, while allowing that the universally quantified proposition itself belongs to an (n+1)th-order domain of discourse. "All first-order propositions are meaningful if and only if tautological or verifiable," then could be interpreted as a second-order proposition, with a second-order criterion of meaningfulness, etc.
The upshot is that absolutely unrestricted, or ORDth-order, quantification over propositions will often be hard to justify. The point of retorsion arguments is usually to defend theism-friendly philosophical ideas, with hidden inferences to the absolute grounding of supposed revelations. But specific axioms, like the axioms of choice and constructibility in set theory, seem to have identifiable initial exceptions (Reinhardt and ω1-Erdős cardinals, respectively), and one suspects that there is some category of axioms historically intended to range over all sets yet whose objects all end up admitting of counterexemplars. Thus though a given retorsion argument will be consonant with recognizing the dangers of unrestricted quantification in one sense, it will perchance implicitly fall prey to the very same danger in another.
Addendum. Consider the stereotypical version of a retorsive dialectic: debater A says, "Every rule has an exception," to which debater B replies, "Ah, but then the rule of every rule having an exception, has an exception!" But did A ever mean to say that it is a rule that every rule has exceptions? B will be thinking that universal quantifications are rules; Kant even shares this kind of assumption when he defines apriority in part in terms of universality. But is A thinking the same thing? A sophisticated A could say, "Every first-order rule has exceptions," whereas that "rule over other rules" is essentially second-order, however. Moreover, A might say, "Oh, no, there's no rule mandating that every rule has exceptions; they all just happen to have them. It's just the way things are. Of course it's 'possible' that there could be or at least has been an exceptionless rule, though if that rule is trivial, well..."
