If all truths can be known, then all truths ARE known.
Many philosophers—especially verificationists and anti-realists—hold a seemingly reasonable belief:
For any truth, someone could, in principle, come to know it
This would be omniscience—clearly false
The knowability thesis seems modest: we're not claiming anyone knows everything, just that there are no unknowable truths. No truth is forever beyond discovery.
But in 1963, Frederic Fitch proved something shocking...
Fitch showed that if we accept the knowability thesis, we're forced to accept omniscience!
Suppose p is true but not known. In symbols: p ∧ ¬Kp
This seems harmless—surely there are truths nobody knows yet!
If p is true and not known, then the statement "p is an unknown truth" is true.
Let's call this compound truth q = "p ∧ ¬Kp"
If all truths are knowable, then it's possible to know q.
That is: ◊K(p ∧ ¬Kp) — It's possible to know "p is an unknown truth"
To know "p is an unknown truth," we would need to:
But if we know p, then p is no longer unknown!
Knowing "p is an unknown truth" would make it a known truth—contradiction!
So ¬◊K(p ∧ ¬Kp) — It's NOT possible to know "p is unknown"
If all truths are knowable, but "p is an unknown truth" can never be known...
Then there can be NO truths of the form "p is an unknown truth"!
Therefore: All truths must already be known. Omniscience follows!
Click on a truth to try to "know" it:
Accepting that all truths are merely knowable logically forces us to accept that all truths are actually known!
This paradox threatens several important philosophical positions:
The view that meaningful statements must be verifiable. Fitch's proof suggests verificationists must accept absurd omniscience.
The view that truth is tied to evidence or knowability. The paradox seems to refute this position directly.
Mathematical constructivism holds that mathematical truths must be constructible. Similar issues arise.
Tennant's restriction: The knowability principle shouldn't apply to self-refuting statements like "p is an unknown truth."
Question whether knowing a conjunction means knowing each conjunct. Perhaps K(p ∧ q) ≠ Kp ∧ Kq.
Perhaps "unknown" should be time-indexed: p is unknown at time t. Knowing it later doesn't create contradiction.
Some suggest using intuitionistic logic, which blocks certain inferences. But most of the proof still goes through!
Despite decades of work, there is no consensus on where (or if) the proof goes wrong.
Alonzo Church discovers the paradox while refereeing a paper—it remains unpublished
Frederic Fitch publishes the proof in "A Logical Analysis of Some Value Concepts"
W.D. Hart rediscovers the result and brings it to wider attention
Neil Tennant proposes his restriction strategy
Church's original referee report discovered, revealing the earlier history
The paradox remains a central problem in epistemic logic and philosophy of mind
Fitch's paradox forces us to ask:
Perhaps some truths are unknowable precisely because knowing them would change what is true. 🔮