Dear Pen and Sy,
I have benefited from your exchange. I’ll try to add some input.
Sy: I have wanted to say something about your proposal. But I am still very unclear on how you understand the philosophical landscape, in particular, on how you understand the distinction between intrinsic and extrinsic justification.
One of the distinctive aspects of your view — a selling point, from a philosophical point of view — is that it involves pursuit of “intrinsic justifications”, as opposed to “extrinsic justifications”. But I am not sure how you are using these terms. From your exchange with Pen it seems that your usage is quite different from the original, namely, that of Gödel.
For Gödel a statements S is intrinsically justified relative to a concept C (like the concept of set) if it “follows from” (or it “unfolds”, or is “part of”) that concept. The precise concept intended is far from clear but it seems clear that whatever it is intrinsic justifications are supposed to be very secure, not easily open to revision, and qualify as analytic. In contrast, on your usage it appears that intrinsic justifications need not be secure, are easily open to revision, and (so) are (probably) not analytic.
For Gödel a statement S is “extrinsically justified” relative to a concept C (like the concept of set) if it is justified (on the basis of reasons grounded in that concept) in terms of its consequences (especially its “verifiable” consequences), just as in physics. Again this is far from precise but it seems clear that extrinsic justifications are not as secure as intrinsic justifications but instead offer “probable”, defeasible evidence. In contrast, on your usage it appears that you do not understand “extrinsic justification” as an epistemic notion, but rather you understand it as a practical notion, one having to do with meeting the aims of a pre-established practice.
So, you appear to use “intrinsic justification” for an epistemic notion that is not as secure as the traditional notion but rather merely gives epistemic weight that falls short of being conclusive. Moreover, at points, when talking about intrinsic justifications you talk of testing them in terms of their consequences. So I think that by “intrinsically justified” you mean either “intrinsically plausible” or “extrinsically justified”.
I think you need to be more precise about how you use these terms and how your usage relates to the standard usage. This is especially important if the main philosophical selling point of your proposal is that it is re-invigorating “intrinsic justifications” in the sense of Gödel. (Good places to start in getting clear on these notions are the papers of Tony Martin and Charles Parsons.)
In what I say next I will use “intrinsic justification” in the standard sense, both for the sake of definiteness and because it is on this understanding that your view is distinctive from a philosophical point of view.
Let me begin with a qualification. I am generally wary of appeals to
“intrinsic justification”, for the same reason I am generally wary of
appeals to “self-evidence”, the reason being that in each case the
notion is too absolute — it pretends to be a final certificate, an
epistemic high-ground, a final court of appeal. But in fact there is
little agreement on what is intrinsically justified (and on what is
self-evident). For this reason, in the end, discussions that employ
these notions tend to degenerate into foot-stamping. It is much
better, I think, to employ notions where there is widespread
intersubjective agreement, such as the relativized versions of these notions, notions like “A is more intrinsically plausible than B” and “A is more (intrinsically) evident than B”. This is one reason I find
extrinsic justifications to be more helpful. They are piecemeal and
modest and open to revision under systematic investigation. (I think
you agree, since I think that ultimately by “intrinsic justification”
you mean what is normally meant by “extrinsic justification”).
But let me set that qualification aside and proceed, employing the notion of “intrinsic justification” in the standard sense, for the
reasons given above.
There is an initial puzzle that arises with your view.
- You claim that IMH is intrinsically justified.
- You claim that inaccessible cardinals — and much more — are intrinsically justified
- FACT: IMH is implies there are no inaccessibles.
The natural reaction at this point would be to think that there is
something fundamentally problematic about this approach.
But perhaps there is a subtlety. Perhaps in (1) and (2) intrinsic
justifications are relative to different conceptions.
When you claim that IMH is intrinsically justified what exactly are you saying and what is the case for the claim? Are you saying IMH (a) intrinsically justified relative to our concept of set (which, on the face of it, concerns V) or (b) the concept of being a countable transitive model of ZFC, or (c) the concept of being a countable transitive model of ZFC that meets certain other constraints? Let’s go through these options one by one.
(a) IMH is intrinsically justified relative to the concept of set. I don’t see the basis for this claim. To the extent that I have a grasp on the notion of being intrinsically justified relative to the concept of set I can go along with the claims that Extensionality and Foundation are so justified and even the claims that Infinity and Replacement and Inaccessibles are so justified (thus following Gödel and others) but I lose grip when it comes to IMH. Moreover, IMH implies that there are no inaccessibles. Does that not undermine the claim that IMH is intrinsically justified on the basis of the concept of set? Assuming it does (and that this is not what you claim) let’s move on.
(b) IMH is intrinsically justified relative to the concept of being a
countable transitive model of ZFC. I have a good grasp on the notion of being a countable transitive model of ZFC. And I think it is interesting to study this space. But when I reflection this space –when I try to unfold the content implicit in this idea — I can reach nothing like IMH.
(c) IMH is intrinsically justified relative to the concept of being a countable transitive model of ZFC that meets certain other constraints. I can certainly see going along with this. But, of course, it depends on what the other constraints are. We have two options: (i) We can be precise about what we mean. For example, we can build into the notion that we are talking about the concept of being a countable transitive model of ZFC that satisfies X, where X is
something precise. We might then deduce IMH from X. In this case we know what we are talking about — that is, we know the subject matter — but we merely “get out as much as we put in”. Not so interesting. (ii) We can be vague about what we mean: For example, we can say that we are talking about countable transitive models of ZFC that are “maximal” (with respect to something). But in that case we have little idea of what we are talking about (our subject matter) and it seems that “anything goes”.
You seem to want to resolve the conflict in (a) — between the claim
that inaccessibles are intrinsically justified and the claim that IMH
is intrinsically justified — by resorting to both intrinsic justifications on the basis of our concept of set (which gives inaccessibles) and intrinsic justifications on the basis of the hyperuniverse (understood as either (i) or (ii) under (c)) and which gives IMH) and you seem to want to leverage the interplay between these two in such a way that it gives us information about our concept of set (which concerns V). But what can you say about the relationship between these two forms of intrinsic justification? Is there some kind of “meta” (or “cross-domain”) form of intrinsic justification that is supposed to give us confidence about why intrinsic justifications on the basis of the hyperuniverse should be accurate indicators of truth (or intrinsic justifications on the basis of) our concept of set?
One final comment: Here is an “intuition pump” regarding the claim that IMH is intrinsically justified.
- If there is a Woodin cardinal with an inaccessible above then IMH
- If IMH holds then measurable cardinals are consistent.
So, if IMH is intrinsically justified (in the standard sense) then we can lean on it to ground our confidence in the consistency of measurable cardinals. For my part, the epistemic grounding runs the other way: IMH provides me with no confidence in the consistency of measurable cardinals (or of anything). Instead, the consistency of IMH is something in need of grounding. Fact (1) above provides me with evidence that IMH is consistent. Fact (2) does not provide me with evidence that measurable cardinals are consistent. I think most would agree. If I am correct about this then it raises further problems for the claim that IMH is intrinsically justified (in the standard sense).
I have further comments and questions about your notion of “sharp-generated reflection” and how you use it to modify IMH to . But those questions seem premature at this point, given that I am not on board with the basics. Let me just say this: The fact that you are readily modify (intrinsically justified) IMH to in light of the fact that IMH is incompatible with (intrinsically justified) inaccessibles indicates that your notion of intrinsic justification is quite revisable and, I think, best regarded as “intrinsic plausibility” or “extrinsic justification” or something else