This article will be a bit more philosophical than my usual content. However I think that sometimes we can really benefit from thinking about things at a higher level, even if the abstraction initially seems to take us away from the real world, we might find that it does give us some practical insight.
The famous Gödel's First Incompleteness Theorem states:
Any consistent system of axioms whose theorems are recursively enumerable contains statements that can neither be proved nor disproved.
In other words, a model of arithmetic can’t always resolve whether a statement is true or false. Indeed in Gödel's construction these sentences are actually true. This doesn’t just apply to artificial statements, as, famously, both the continuum hypothesis and its negation were shown to be consistent with set theory, thus proving that set theory cannot resolve it one way or the other.
This seems fairly unsatisfying because we’d really like to know whether something is true or false. And if we aren’t so inclined to be interested in mathematical logic, we might ignore this argument completely as being something artificially constructed and not very relevant to everyday life, for it assumes certain limitations on the models used, that we might not apply to our own human minds.
Fitch’s Paradox of Knowability can be seen as an extension of Gödel's argument to philosophy (specifically, epistemology, the philosophy of knowledge). It doesn’t require any complex mathematical machinery and reaches a similar conclusion. The paradox rests on two assumptions:
Everything true can be known.
Some truths are not (yet) known.
So now using assumption (2) let’s take P to be some truth that is not known (of course I can’t give a specific example, since that would require me to know it to be true). Now consider Q = “P is an unknown truth”, and by our premise, Q is true. By assumption (1) above, it should be possible to know that Q is true. But that can’t be, since if we know that Q is true, then we also know that P is true. This means that P cannot possibly be an unknown truth, which leads us to a contradiction.
We are left to conclude that one of the two assumptions is wrong. It seems completely absurd to reject (2), since that would mean we already know everything that is true. So we must reject (1), i.e. that everything true can be known. This is Fitch’s paradox. Despite the name, it’s not really a paradox at all, it’s just a clever logical argument that shows that (1) cannot be true.
So we must conclude that not everything true can be known. Yet it seems not-uncommon to act in life as if that were not the case. Many of us seem to believe that we could know potentially anything, had we but data enough and time. We run experiments to collect ever-and-ever increasing amounts of data, and apply complex algorithms, statistics, and machine learning to glean from it the truths we desire.
And of course this does often work and provides us with valuable insights. But what I take away from Fitch’s argument is that we should be a little bit more humble in our endeavor. We should consider that some things we might really want to know, may be part of that unknowable swath of truth that we’ll never have access to. In a world of uncertainty, we should strive to become comfortable in making decisions with incomplete knowledge.
Hi Nicholas,
I found your post on Fitch's paradox of knowability during a literature sweep for some research I'm working on. I'm intrigued by a few points in your post. I'd like to pick your brain a bit--would you be up for an email exchange or brief Zoom convo? Feel free to message me on here--although I'm new to Substack, I *believe* I've turned messaging on.
Best,
Ben