Can every truth be known?
A modest hope—that no truth is forever beyond discovery—has a startling logical consequence: every truth is already known.
0. The gap between possible and actual knowledge
Imagine a sealed box containing a red ball. Nobody has opened it. The ball’s color is unknown, but discovering it seems perfectly possible. “Every truth can be known” sounds compatible with a world full of unopened boxes.
Fitch’s paradox asks whether that compatibility survives when “every truth” includes truths about ignorance itself. It shows that a seemingly modest philosophical claim
Every truth could, in principle, be known.
logically implies a much stronger and implausible claim:
Every truth is already known.
Fitch’s paradox of knowability is probably one of the most surprising results in epistemic logic. Remarkably, it does not rely on self-reference, unlike the Liar paradox or Gödel’s incompleteness theorem.
The philosophical motivation
The idea stems from the difference between the two positions about the relationship between truth and knowledge.
Realism: Some facts may be true even if no one could ever discover them.
Anti-realism (verificationism): Truth is fundamentally connected to our capacity to verify it. If a statement is true, it must at least be possible for someone to know it. Everything true is knowable.
Let’s add some “math”
Let \(Kp\) mean that \(p\) is known and \(\Diamond Kp\) that it is possible for \(p\) to be known. The unrestricted knowability principle is the schema
\[p \longrightarrow \Diamond Kp. \tag{KP}\]This is the anti-realist’s knowability principle–seemingly much weaker than its counterpart of classical consequence, omniscience:
\[p \longrightarrow Kp.\]1. The proof in a few steps
Use two elementary principles: knowledge is factive, and knowing a conjunction entails knowing each conjunct.
\[Kq \longrightarrow q, \qquad K(q\land r)\longrightarrow (Kq\land Kr).\]Let’s consider a proposition \(p\) that is true but unknown. Think of something like “there is a buried Roman coin at a certain location, but nobody knows it exists”. Then another proposition is true:
\[q := p\land\neg Kp.\]In words:
“The coin is buried there, and nobody knows that it is buried there.”
Universal knowability says that this whole conjunction \(q\) can be known. But suppose it were known:
\[K(p\land\neg Kp).\]Conjunction elimination gives both \(Kp\) and \(K\neg Kp\). Factivity applied to the latter gives \(\neg Kp\). We obtain
\[Kp\land\neg Kp,\]a contradiction. Consequently,
\[\boxed{\neg\Diamond K(p\land\neg Kp)}.\]It is impossible to know that a particular truth is unknown.
Recall that KP applies to every true proposition. Therefore, it must also apply to \(q=p\land\neg Kp\) :
\[(p\land\neg Kp)\rightarrow \Diamond K(p\land\neg Kp).\]But we just proved that its consequent is impossible. Hence,
\[\boxed{\neg(p\land\neg Kp)}.\]In classical propositional logic, this is equivalent to
\[\boxed{p\rightarrow Kp}.\]Since $p$ was arbitrary:
\[\boxed{\forall p\,(p\rightarrow Kp)}.\]Every truth is known!
The crucial distinction: \(p\land\neg Kp\) can be true. What cannot be true is that this entire conjunction is known. Consistency of a proposition does not guarantee consistency of knowing it.
2. Tennant: restrict which truths must be knowable
Neil Tennant wants to preserve anti-realism without accepting that everything is known.
His proposed solution is ingenious: the knowability principle should apply only to propositions whose being known is logically consistent.
He calls these Cartesian propositions.
Neil Tennant proposes applying knowability only to Cartesian propositions: those for which assuming knowledge does not yield inconsistency. Schematically,
\[p\longrightarrow\Diamond Kp \quad\text{only when } Kp\nvdash\bot.\]This blocks the original substitution: \(p\land\neg Kp\) fails the eligibility test. The issue then becomes whether the restriction has an independent philosophical justification and how its consistency test treats necessary background truths.
3. Williamson: can ignorance be hidden inside an eligible truth?
In Tennant on Knowable Truth (2000), Timothy Williamson challenges the ad-hoc restriction with a modified construction.
Let \(n\) rigidly name an actual number of books, and let \(E(n)\) mean that this number is even. Since the number of books is fixed, in the relevant modal interpretation, either \(E(n)\) is necessarily true or \(\neg E(n)\) is necessarily true. But we may not know which.
Consider
\[r:=p\land(Kp\to E(n)).\]This is much less obviously problematic than Fitch’s original proposition \(q\).
If \(p\) is true but unknown, \(r\) is true because the conditional has a false antecedent. If \(r\) is known, conjunction elimination and factivity yield \(Kp\) and \(Kp\to E(n)\), hence \(E(n)\). Knowability of \(r\) would therefore imply possible evenness; rigid numerical parity makes that actual evenness. Repeating with oddness produces a contradiction—if both constructed propositions qualify as Cartesian.
That qualification is disputed. Tennant’s 2001 reply counts necessary truths in the consistency test: if \(n\) is necessarily odd, knowledge of the evenness construction is inconsistent, and conversely. On that reading, the restriction blocks one application.
4. Why this is different from a self-referential paradox
The liar sentence says, “This sentence is false.” It refers to itself, producing a feedback loop between its own content and its truth status. Fitch needs no sentence that names itself and no diagonal construction. Start with any ordinary \(p\)—a ball’s color, for example—and form a new statement about \(p\)’s epistemic status.
Fitch is closer to a Moore-style puzzle: “It is raining, but I do not know that it is raining” may describe reality correctly even though knowing its entire content is impossible. The obstruction is epistemic structure rather than semantic self-reference.
5. What the paradox leaves open
The result exposes how demanding the word “every” is. An assurance about discovering ordinary facts becomes a much stronger claim when it covers arbitrary compounds involving knowledge.
One can restrict knowability, revise its modal or temporal interpretation, or reconsider the logic. In intuitionistic logic the argument yields \(p\to\neg\neg Kp\), without generally licensing the final step to \(p\to Kp\).
Further reading
- Frederic B. Fitch (1963), “A Logical Analysis of Some Value Concepts,” The Journal of Symbolic Logic 28: 135–142.
- Neil Tennant (1997), The Taming of the True, Chapter 8.
- Timothy Williamson (2000), “Tennant on Knowable Truth,” Ratio 13: 99–114.
- Neil Tennant (2001), “Is Every Truth Knowable? Reply to Williamson,” Ratio 14: 263–280.
- Berit Brogaard and Joe Salerno, “Fitch’s Paradox of Knowability,” Stanford Encyclopedia of Philosophy (overview and further debate).