What I did in my commentary at the Eastern APA is frame a debate about Fitch's paradox, and explain the significance of Salvatore Florio and Julien Murzi’s contribution to the intuitionistic reply. Along the way I tried to improve on their main argument.
Consider the following epistemic theories of truth, which are supposed to differ
precisely on the strength of the advertised relation between truth and knowledge.
Semantic Idealism (SI): p(p Kp)
Necessarily, all truths are in fact known (by some finite being at some time).
Strict Finitism (SF): p(p FKp)
Necessarily, all truths are feasibly knowable = necessarily, all truths are are knowable by beings who have precisely the cognitive capacities that we at some time happen to have.
Moderate Anti-realism/Weak Verificationism (WVER): p(p Kp)
Necessarily, all truths are knowable by us in principle (i.e., by beings whose capacities are at best finitely better than those we happen to have).
And consider the following brand of realism, which denies all three positions.
Realism (R): p(p & ~Kp)
There may be unknowable truths---i.e., truths that couldn’t be known given any finite extension of our cognitive capacities.
The three brands of anti-realism appear to be listed in the order of their logical strength, from strongest to weakest. (SI) entails (SF) entails (WVER), and the entailments are not meant to go the other way. Indeed, (WVER) gains its plausibility to the extent that it can distance itself from awkward forms of idealism and strict finitism.
The Church-Fitch paradox is a proof that threatens to show that moderate antirealism collapses into idealism. A classical formulation of Florio and Murzi’s paradox of idealization (presented at the APA) threatens to show that (WVER) collapses into (SF). The lesson of either is that so-called "moderate anti-realism" is an inherently unstable position. In the context of that epistemic theory of truth, the apparently modest idealization is equivalent to at least one of the immodest idealizations. Simply put, a so-called moderate anti-realist can’t distinguish between actual knowledge, feasible knowability, and knowability in principle.
I put the lessons this way for simplicity of exposition, although stating them as I
have presupposes excluded middle. The Florio-Murzi proof aims to draw related lessons without excluded middle, and thereby aims to show that an independent commitment to intuitionistic logic can't rescue the moderate anti-realist from the grips of Fitch-like paradoxes.
The key concept in the Florio-Murzi discussion is the concept of an ideal agent. They define it as any finite agent whose epistemic capacities are better than our own. Their proof requires that it be a priori that there are no ideal agents. Since humans may be cosmic hicks, as it were, F and M must not mean "human capacities" by "our capacities". After all, it is an a posteriori matter whether there are beings in the universe with epistemic capacities better than those of humans. So I take it that by 'ideal agent' they mean any finite agent whose capacities are better than any actual finite agent. This provides the desired strength to the first premise of the Florio-Murzi proof:
1. There are no ideal agents.
The second and most critical assumption in the Florio-Murzi proof is that
there is a truth q that isn’t feasibly knowable. Indeed, their assumption requires that it be necessary that anyone who knows q is an ideal agent. Call this assumption epistemic modesty. It is presumed that even the moderate intuitionistic anti-realist is epistemically modest in this sense.
2. (Epistemic Modesty) There is a feasibly unknowable truth; that is, a truth such that necessarily any being that knows it is ideal:
q(q & x(Kxq Ix))
Epistemic modesty is meant to be a more precise denial of Strict Finitism. F and M foreshadow the following kind of objection to (Epistemic Modesty). Can't we have a cognitive twin in a world with a more favorable set of epistemic resources or environment? In such a world subjects with our cognitive capacities are in a position to know q, even though we in the actual world are not. Beings internally like us, but in improved external circumstances, acquire knowledge more easily than we do. For instance, suppose that cognitive limitations prevent any actual being from determining the location of a particular distant star. Nevertheless, there will be possible worlds where our telescopes are better or the cosmic environment for whatever reason better preserves the brightness of stars over longer distances. In such worlds, beings with precisely our cognitive capacities come to know a truth that is feasibly unknowable in the actual world.
Such considerations threaten the plausibility of (Epistemic Modesty). I don’t believe that Florio and Murzi said enough in their paper to dispel the worry. But here is a quick fix. Redefine an ideal agent more generally as one who has a finitely improved epistemic state of information, where states of information include the subject’s cognitive capacities, resources and environment. And let q be a truth that can’t be known by beings in any actual epistemic state of information. Then (Epistemic Modesty) is more intuitive. When q is a truth that we're not in a position to know (owing to our cognitive, material and environmental limitations), then by definition knowing q necessitates being in a better epistemic state. That is, necessarily, if x knows q then x is an ideal agent in that she is epistemically better off---i.e., she has either increased cognitive capacities, better epistemic resources, or a more epistemically friendly environment.
With these adjustments the Florio-Murzi proof is much stronger. It has the following structure. Assumptions I through III entail a contradiction:
I. (No Ideal Agents) There are no ideal agents.
II. (Epistemic Modesty): There is a feasibly unknowable truth.
q(q & x(Kxq Ix))
III. (WVER): All truths are knowable in principle.
An illuminating version of the proof goes like this:
|0||0. p(p Kp)|| [WVER]|
|1||1. ~xIx|| [No Ideal Agents]|
|2||2. q|| [A for CP]|
|3||3. x(Kxq Ix)|| [A for Reductio]|
|3||4. x(Kxq Ix)|| [from 3]|
|3||5. Kaq Ia|| [from 4]|
|6||6. Ka(q & ~xIx)|| [A for reductio]|
|6||7. Kaq & Ka~xIx|| [6 by K-distributivity]|
|6||8. Kaq|| [from 7]|
|3,6||9. Ia|| [from 5 and 8]|
|3,6||10. xIx|| [from 9]|
|6||11. q & ~xIx|| [6 by K-factivity]|
|6||12. ~xIx|| [from 11]|
|3,6||13. Contradiction|| [from 10 and 12]|
|3||14. ~Ka(q & ~xIx)|| [6-13 by reductio]|
|3||15. ~Ka(q & ~xIx)|| [from 3, 4-14, since only necessities follow from necessities]|
|3||16. ~Ka(q & ~xIx)|| [15 by def. of ]|
|0||17. (q & ~xIx) Ka(q & ~xIx)|| [from 0]|
|0,3||18. ~(q & ~xIx)|| [from 16, 17]|
|1,2||19. q & ~xIx|| [from 1, 2]|
|0,1,2,3||20. Contradiction|| [18, 19]|
|0,1,2||21. ~x(Kxq Ix)|| [3-20 by Reductio]|
|0,1||22. q ~x(Kxq Ix)|| [2-21 by CP]|
|0,1||23. q(q ~x(Kxq Ix))|| [22 by -Intro]|
|24||24. q(q & x(Kxq Ix))|| [Epistemic Modesty]|
|0,1,24||25. Contradiction||[from 23 and 24]|
As F and M point out, the proof is intuitionistically valid. However, I don’t think it gains any ground, over and above Fitch's paradox, against the intuitionistic strategy. The intuitionist is happy to deny the epistemic modesty principle---i.e., that there are feasibly unknowable truths:
i. ~q(q & x(Kxq Ix))
After all, existence for the intuitionist is constructive existence, and we can’t construct an example of a feasibly unknowable truth.
The typical charge against this sort of maneuver is epistemic hubris. But the typical reply is for the intuitionist to regain her modesty by replacing the original modest assumption with a claim that is classically but not intuitionistically equivalent. For instance, she might deny that every truth is feasibly knowable:
ii. ~q(q x(Kxq & ~Ix))
Classically i. and ii. contradict. But not intuitionistically. There are more intuitionistic distinctions than there are classical distinctions, and the intuitionist usually takes advantage of this fact. Contrary to what I thought at the APA, the move won't work in this case, since ii contradicts line 23 of the proof. Line 23 rests only upon WVER and No Ideal Agents.
The intuitionist at this point might chose some other more complicated classically (but not intuitionistically) equivalent formula with which to express her modesty. Surely there is at least one that doesn't intuitionistically contradict line 23. The move should be followed with an explanation of why this, rather than the original formula, best expresses her epistemic modesty.
There is, however, a recommendation to preempt this intuitionistic maneuver. It is based on a suggestion raised by F and M in their paper. First and foremost, don’t assume the existence of a feasibly unknowable truth. Instead begin with a proposition that would be feasibility knowable, regardless of its truth value. For instance, let q be the sentence ”There is life on x”, where x is some planet that is epistemically inaccessible in the relevant sense. Our cognitive capacities, or overall epistemic state of information, is inadequate for the determination of whether or not q. So we should be modest about q and about ~q. The commitment is to a proposition q such that: necessarily if an agent knows q then she is ideal, and necessarily if an agent knows ~q then she is ideal.
iii. x(Kxq Ix)
iv. x(Kx~q Ix)
Notice that we don’t presuppose the constructive existence of a feasibly unknowable truth. But we still get a contradiction. Line 23, which rests just on WVER and NO IDEAL AGENTS, shows us that the assumption q together with iii jointly entail a contradiction. Hence, ~q. And by analogous reasoning ~q and iv jointly entail a contradiction.
If the intuitionist already has some other way of expressing her modesty she can give up at least one of iii and iv. She is still not committed to Strict Finitism, but she will be committed to some principle classically, but not intuitionistically, equivalent to Strict Finitism. There is logical space for her to do so. But she will have to tell some complicated story to regain an epistemically modest footing.
Contrary to what I thought at the APA, I don't believe that the Florio-Murzi paradox of idealization raises new difficulties for moderate intuitionistic anti-realism---that is, difficulties over and above those already raised by the Church-Fitch paradox. However, I think it highlights the significance of knowability paradoxes more generally. Such paradoxes show us that if we treat truth as an epistemic notion, then we blur modal epistemic distinctions that are needed to make such theories plausible.