Fitch's paradox of knowability is a testament against at least naive/moderate anti-realism as based on the principle that all truths are knowable. The derivation is split into two major sections, the first of which goes:

enter image description here

The second is:

enter image description here

If we take knowledge and understanding to be separate and on the same "level" (or as two distinct kinds of objects in one category, or whatever) (see the SEP/IEP articles on understanding), can we yet recapitulate Fitch's paradox of knowability for an understanding operator?

Here would be the first stretch of the deduction, if we merely replace K with u:

enter image description here

Now I assume that u is not (always) factive, since we can understand false propositions. Is that enough to block the second stretch of the argument template from being used to devise a "paradox of understandability," to the effect that even if not all truths are knowable, all propositions are understandable? The SEP article goes on to say that we can use non-factive operators like "it is rationally believed that" and still get a version of Fitch's paradox, though. I have also found San[20] to be particularly illuminating. San reasons that Fitch-like paradoxes can be generated not just by factivity or negative infallibility, but by what he calls "level-bridging principles." He seems to me to be identifying something to do with iterations of modal operators as the problem, here, which then I think I can see in my claim that u¬u = u.

Meaning-theoretic versions:

These are based on Conifold's suggestion. Let there be a meaningfulness operator m such that "mA" reads "it is meaningful that A." Let's rewrite the first stretch of the parallelism accordingly (I offer two parallels, actually, a presumably weaker and a presumably stronger version, although which is which I don't know, and I can envision that there might be yet other versions available, here, depending on alternate interleavings of u and m):

enter image description here

Offhand, the last line for (A) looks wrong, to me: how could we understand that a proposition is meaningful "from the outside" if the proposition's meaningfulness is not understood "from the inside"? So maybe there's where we could get a paradox, if we had plausible accompanying rules for m (again, not factivity, but perhaps something similar enough, maybe possible factivity?).

Replacing m with modality

Part of me (a large part!) doesn't want to have to introduce a totally new operator into the "game," though. Might we simply replace m with a condition of possibility, then? So here's a (C)-version:

enter image description here

I was thinking that instead of applying a u-counterpart of (B) to the (C)-counterpart of (4) (since we are waiving factivity), instead we could use negative infallibility to collapse the prefixture to just ¬uA ("if it is understood that it is not understood that A, then it is not understood that A). Then:

enter image description here

Then we give up on the absence of u-omniscience, or on the pertinent understandability principle. The presence of u-omniscience would mean that we understand all possibilities, that there are no possibilities that we don't understand. To be honest, this seems less objectionable than, "There are no truths that we don't already know," although it also seems intuitive to me that I, for example, don't understand what all the possibilities are, that there are (e.g. I don't fully understand that iterated possibility is redundant except relative to modal logics axiomatizing this redundancy, perhaps; or I don't understand how logical and metaphysical possibility fully differ; or etc. and so on and so forth).

  • 3
    It seems to me that a parallel paradox should not involve truth at all since it is irrelevant to understanding. The thesis should be not "if p is true then p is understandable", but rather "if p is meaningful then p is understandable". A parallel derivation would then produce "if p is meaningful then p is understood", which does sound paradoxical in Fitch's spirit.
    – Conifold
    Dec 13, 2023 at 8:25
  • I would love for you to join me here and we could work on redrafting this question to make it easier for potential answerers (like me) to respond to. Thank you. chat.stackexchange.com/rooms/150343/… Dec 15, 2023 at 17:27
  • @JuliusH. I will edit the post to try to make it crisper. I think that the San text that I found supplies the answer that I'm looking for, for all versions of the u-formulation that I came up with (and arbitrary others), though. Dec 15, 2023 at 17:55
  • 1
    @AgentSmith Proving that the expression you wrote is a contradiction is part of the proof of Fitch’s paradox. This forces one to discard one of the assumptions that leads to the contradiction. Dec 19, 2023 at 15:24
  • 1
    @AgentSmith reportedly it is still possible to derive a version of the paradox when waiving K-distribution: the SEP article reads at one point, "Though some have argued that knowing a conjunction does not entail knowing the conjuncts (Nozick 1981), Williamson (1993) and Jago (2010) have shown that versions of the paradox do not require this distributive assumption." Dec 20, 2023 at 3:31

2 Answers 2


Muchas gracias Kristian Berry.

The fault is with how The Knowability Thesis (p --> LKp) morphs from knowable to known.

There's an unknown truth = p & ~Kp = p is true & Unknown p

So, by the knowability thesis, (p & ~Kp) --> LK(p & ~Kp)

We can't apply the rule K(p & q) --> Kp & Kq to LK(p & ~Kp) because the rule is about K (the known) and the LK(p & ~Kp) is about LK (the knowable). As you can see, the best we can do is LKp & LK~Kp i.e. p is knowable & that we don't know p is itself knowable.

Fitch's argument boils down to ...

Fitch's Mini Proof

  1. p --> LKp (Knowability thesis)
  2. LKp --> Kp (the epistemic link)
  3. p --> Kp (1, 2 HS)
    where p is a proposition, LKp means p is knowable, Kp means p is known.

We can disprove LKp --> Kp. Aliens, existing/not, is knowable, but it is an unknown. Basically a knowable unknown

  • It’s a good start but needs development. I will help you with this. Dec 20, 2023 at 14:40
  • 1
    The switch from K to ◊K is mediated by a rule that says that if a sentence is provable, it is necessary, and then if necessary, then not possibly not. Since the contradiction in the "middle" of the derivation is impossible, we get a second contradiction assuming either (A) the knowability principle or (B) that there is some unknown truth. So either (A) or (B) has to go (or we accept one or both of the contradictions, but see Santos[17]). Dec 21, 2023 at 13:36
  • Thank you @JuliusH.
    – Hudjefa
    Dec 22, 2023 at 1:49
  • @KristianBerry, you said there are other Fitch-like arguments that are resistant to critiques on the knowability thesis and knowledge of conjunctions. Can you include them in your post, if possible.
    – Hudjefa
    Dec 22, 2023 at 1:56

This is not a final answer but an attempt at a draft.

There is the conceptual problem of to what extent the formal language we design provides commentary on the world. We do not know if what we call “knowing” is well-represented by a “modal operator” K in predicate logic. No systematic argumentation is given for why this should be, yet, that hypothesis could be seen as akin to a “lucky accident” in science where a phenomenon is observed before it is either expected or explained. Because we are trying to develop the premises and the conclusions at the same time, this counts as an example of abductive reasoning. (I think a good way to frame this problem more starkly is by posing yourself the thought-experiment/question, “What if there was a logical axiom for ‘knowing’ that I didn’t include?” Assume there is some such axiom. Can you prove or disprove that your axiomatization is equivalent to some assumed, ideal, “correct” one? This requires us to define some distinction between a correct and incorrect axiomitization.)

We want to prove by contradiction that “there exist propositions that are not understandable” (equivalently, “It is not true that all propositions are understandable”). We insert the concept of “understanding” into a formal logic (presumably) by abstracting away what is non-logical (i.e., treating any general conceptuality of it like a predicate in predicate logic), while specifying what logical properties it has. But the situation appears different from Fitch. Knowledge is taken to be factive under the axiom Kp -> p, but understanding may or may not be. Conifold defines “understandibility” in relation to “meaningfulness”, so now we would have two “conceptual” modal operators plus a standard alethic one (of necessary and possible). And recall that Fitch’s paradox is not a proof of realism, but a proof that antirealism and naive idealism entail one another (again, this seriously forces us to acknowledge that we may never have had a consistent definition of “knowing” - I think if someone wanted to defend naive idealism, their use of the word “know” would actually mean something different than if a person said, “Clearly we don’t yet know everything about the world,” since an idealist thinks that things that are not present in consciousness do not (yet) exist.) A “paradox of understandability” might hopefully say “If all notions are understandable, then all notions are understood.” What happens then? I think kind of like the above, we actually have to debate what we mean by “all notions are understood” before we can reject the idea and settle on the contrapositive. There might be some room for interpretation. They probably have more reasons why they aren’t that perfectly analogous to one another, perhaps drawing from their conceptual semantics (maybe in the way you can model them in external structures like possible worlds). So I guess the point here is opening the question if we really even want to try to adapt Fitch’s paradox to understandibility - at first glance, it’s interesting, but it might not actually fit “understandibility” very well after all. That is, we might pursue a totally different proof strategy / derivation, than the one lain out by Fitch. We don’t know if we should follow Fitch’s proof strategy until we know how logically similar or dissimilar “knowledge” and “understanding” are, and we don’t know that until we know what what a good logical model of “understanding” is. How do we know which model of “understanding” we should use? It may even appear that the question is backwards - deciding how to model a concept is not a preliminary, but would be so determining on the result that it is actually the most significant unknown; therefore, the central question (maybe).

It appears that the linked paper from San helps us take a very large step forward. I believe that San discovered that the semantics of “knowing” do not actually explain Fitch’s paradox, but that this phenomenon is fully “formal” (as in, it appears in abstract logical form, and does not require a semantics). If we focus on purely formal proofs of how various axioms lead to various derivations, we can embrace a potentially more pluralistic and insightful viewpoint where the point is never to definitively prove something about the world, but to prove how particular characterizations of the world would be conditional on particular characterizations of properties of the world. As a first step, I would like to lay out how the formal proof of “modal collapse” by San works, before exploring novel directions in which it could be modified, or applied.

San’s proof of their General Collapse Theorem is simple. Here is an abridged version.

San proves that for any normal modal logic with an n-level bridging principle and in which the Fitch principle holds, one of the modal operators experiences n-degree modal collapse. A normal modal logic is a logic including the axioms K (modal an implies b implies modal an implies modal b) and T (that every p is modal p). Normal modal logics exhibit duality, modal p implies p entails negative p implies negative conjugate modal p, for example, necessarily p implies p entails not p implies not possible p.(?)

Duality allows any n-level bridging principle to be rewritten with more modal operators on the left-hand side. The fact that possibility operators can be written in terms of necessity operators allows any sequence of modal operators to be made homogeneous, or, expressed in terms of one modal symbol. It is proved by induction that the Fitch principle allows the cancellation of a necessary-modal and a possible-modal from the left and right sides of an ‘implies’-expression, respectively. These are used to show that possible (conjugate) modal entails necessary modal. These together show that for any n-level bridging principle, possibility becomes identical to necessity.

The next step is to examine what kinds of interesting level-bridging principles there are, and what alternatives to Fitch’s principle would or would not have related effects.

To be continued…

  • 1
    One way to approach the adequate-definition problem is to suspend it, i.e. to generalize over possible operators and simply ask what happens when various generic operators are specified in various ways. I was pursuing this with a notion of "paraepistemic logic" at one point, although it was hard to corral the results (at one point I had something like 120, IIRC, operators in play; I found a reason to reduce those to just 4, which left me with knowledge, understanding, and two others of a similar "flavor," but the proof operator wasn't among them!). Dec 19, 2023 at 17:28
  • I would like to explore further in that direction so let me know if you want to discuss the details of that approach. Dec 20, 2023 at 14:55
  • Julius -- My tentative view of the world takes your first paragraph, and plausibly extends it into a general principle that brings the outcome of this project into doubt. There are infinite logics, and the correspondence of one of them with our world is -- undemonstrated. We can be fairly confident that the logic of our world is not classical logic, there are just too many paradoxes that classical logic suffers from,. And between too much of our world not operating under either/or, and both our world in general plus most items in it do not satisfy A=A,
    – Dcleve
    Dec 22, 2023 at 14:21
  • and the Munchausen trilemma preventing the full justifying of any of our beliefs, and the failure of Verisimilitude for our knowledge, and our not yet finding a coherent logic that accounts for free will as free despite our knowing we have it -- I have strong suspicions that no specific logic can solve all of these issues. Our desire to find a One True Logic does not mean there IS one. I now treat logics as useful engineering approximations, and accept that they all have limitations/failing cases.
    – Dcleve
    Dec 22, 2023 at 14:27
  • The original purpose of philosophy was to learn how to live well. To achieve wisdom in how to live life. I see the recent focus of philosophical thought on logic -- to try to solve the problems of how to discover the increasingly elusive "truth" by abstracting our reasoning ever more distantly from our world and pragmatic life issues, to be a fruitless dead end, that destroys the value of philosophy. Selves, relationships, creativity, values, and valuing are what matter in this world, and the abstraction of logicians can tell us nothing about these too-vague concepts.
    – Dcleve
    Dec 22, 2023 at 14:33

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .