Gödel introduced his notion of what has come to be called extrinsic justification in the following terms:
Furthermore, however, even disregarding the intrinsic necessity of some new axiom, and even in case it had no intrinsic necessity at all, a decision about its truth is possible also in another way, namely, inductively by studying its “success”, that is, its fruitfulness in consequences and in particular in “verifiable” consequences, i.e., consequences demonstrable without the new axiom, whose proofs by means of the new axiom, however, are considerably simpler and easier to discover, and make it possible to condense into one proof many different proofs.
This is a second-hand quote from Barton, et. al., "On Forms of Justification in Set Theory". The authors go on to quote Maddy:
Ultimately we aim for consistent theories, for effective ways of organizing and extending our mathematical thinking, for useful heuristics for generating productive new hypotheses, and so on...
They later quote Tiles' objection to such talk:
It would be circular indeed to justify the logical foundations by appeal to their logical consequences, i.e., by appeal to the propositions for which they are going to provide the foundations.
Now, according to one theory of erotetic logic, two general kinds of cases of erotetic inference are:
- Inferring a question from another question.
- Inferring a question from an assertion.
I've seen at least one example of
- Inferring an assertion from a question.
... but not a theory of (3). The example: there is a book whose title is What is the Name of This Book?. But it is easy to construct other examples: take a list of sentences, the third of which is a question, and have this question be, "What is the first question on this list?" The answer is, "That question is itself the first question on the list."
If erotetic logic is reasonable, then I wonder: can we construe extrinsic justification in these terms? The template of extrinsic justification might (as per the above quotes) be condensed to: justification by (logical) consequences. Tiles' objection seems entirely apropos if we think of these consequences as assertions. However, what if we characterized the consequences in question, as questions? Furthermore, allow instances of (3) as well. Then we might say things like: assertion A evokes question B, and question B entails assertion C.
For example, take the axiom of infinity. A finitist is liable to say that such an axiom does not admit of intrinsic justification, modulo an understanding of "intrinsic" as "intuitive." That is, they will say that we have no such "infinite intuitions." However, it seems clear enough that, given the axiom of infinity, we can go on to ask questions that would be unintelligible otherwise. Examples include, "What is the proof-theoretic ordinal of ZFC?" or, "What is the cofinality of ℵω?" If these questions can be inferred in a valid way from the axiom of infinity (plus replacement, say), does this fact (of valid erotetic inference) extrinsically justify the axiom(s) at issue?†
One possible problem with this picture: presupposition failure. For example, let the existence of a divine nature be an axiom of some system. Sans this axiom, questions such as, "Did the divine nature become Incarnate?" or, "Does the divine nature forgive sins?" are rather empty. It doesn't necessarily seem that merely enabling us to ask novel questions justifies this axiom of deity.
Rejoinder: here we can bring in the original Gödelian reference to "verifiable consequences." There is a reliable deductive method of proof-theoretic ordinal analysis and of possible cofinalization, such that questions of ordinal analysis and possible cofinality are deductively answerable. By contrast, questions about the Incarnation of a divine nature, or the propensity of deity to forgive sins, do not have answers that can be deduced from the bare axiom of deity.
On the other hand, modulo the theological concept, maybe we're on the wrong level of generality: maybe the axiom of deity rather evokes questions such as, "Is the divine nature simple or complex?" And theologians might say that they have deductions of answers to such a question at hand (e.g. they might refer us to the classical argument for divine simplicity). So maybe the axiom of deity is extrinsically justifiable, too, on erotetic grounds, after all. I myself am a (trans)theist, so I'm not motivated to outright reject such a proposal. At any rate, this is a mathematics, not a theology, site. As far as mathematical justification goes, what role does (or can) erotetic inference play in justifying assertions?
†In the course of my studies, I read an article which, if I remember it correctly, literally said something about V = L being wrong because restrictive, with "restrictive" represented in terms of something like "limits the questions we can ask." I'm not entirely sure but I think what I'm remembering is Steel's(?), "Thus adding V = L settles no questions not already settled by ZFC. It just prevents us from asking as many questions!"