A Paradox of Modal Logic

Fitch's Paradox:
The Paradox of Knowability

If all truths are knowable, then all truths are actually known. A modest claim about human knowledge leads to omniscience.

Consider this seemingly modest philosophical claim:

"All truths are knowable."

For any true proposition, it is at least possible for someone to know it.
This is the anti-realist position: truth is connected to our ability to verify it.

This sounds reasonable. We are not claiming that all truths are actually known, just that they could be known in principle. What could be wrong with that?

In 1963, Frederic Fitch published a proof that this modest claim leads to a shocking conclusion: all truths are actually known.

Knowability implies Omniscience

If every truth can be known, then every truth is known.
Something has gone very wrong.

PART I

The Anti-Realist Thesis

The philosophical debate starts with a question: What is truth?

Realism

Truth is independent of our ability to know it. A proposition can be true even if no one can ever verify it. Truth "transcends" our epistemic capacities.

Anti-Realism

Truth is connected to verifiability. A proposition is true only if it is at least possible to know it. Truth cannot completely outstrip our ability to recognize it.

Anti-realism captures an intuitive idea: what sense does it make to say something is "true" if no one could ever, even in principle, verify it?

The Knowability Principle expresses this idea formally:

For all propositions p:

p → ◊Kp

"If p is true, then it is possible that p is known"

Here, K is the knowledge operator and ◊ is the possibility operator. Let's understand these before diving into the proof.

Modal Logic Operators

Knowledge Operator

Kp means "Someone knows p"

Key Properties:

-Kp -> p (Factivity: knowledge implies truth)

-K(p & q) -> Kp & Kq (Distribution)

-Kp -> KKp (Positive introspection)

-~Kp -> K~Kp (Negative introspection)

Kp is true at w0 if p is true in all worlds epistemically accessible from w0

In Fitch's Proof:

The key formula is <>K(p & ~Kp) - "It is possible that someone knows both p and that p is unknown." The proof shows this leads to contradiction.

PART II

Fitch's Proof

The proof is surprisingly short. It uses only basic principles of logic and knowledge. Walk through each step and watch how the contradiction emerges.

Step 1 of 9PREMISE

The Knowability Principle

All truths are knowable

p -> <>Kp

For any proposition p, if p is true, then it is possible that someone knows p. This is the anti-realist thesis.

Jump to:

The Result

If the Knowability Principle is true, there cannot be any unknown truths.
But clearly there are unknown truths! So the principle must be false.
Not all truths are knowable.

The key insight is that "p is true and p is unknown" is itself a proposition. If all truths are knowable, this conjunction must be knowable. But knowing it creates a contradiction.

PART III

Unknown Truths Are Everywhere

The paradox hinges on the existence of unknown truths. Do such things exist? Absolutely. The world is full of facts that no one knows.

Unknown Truths Exist Everywhere

Example 1 of 3Unknown but computable

"The 10^100th digit of pi"

This digit has a definite value, but no one has computed it. It is a truth, but unknown.

p = "The 10^100th digit of pi"

~Kp = No one knows this truth

p & ~Kp = This is an unknown truth

The Problem for Anti-Realism:

These unknown truths seem obviously to exist. But Fitch's proof says: if all truths are knowable, there cannot be any unknown truths!

The existence of unknown truths seems undeniable. There are billions of facts about the universe that no one has discovered yet. Each of these is an "unknown truth"—a proposition of the form p & ~Kp.

But Fitch's proof says: if all truths are knowable, then no such propositions can exist. This is the paradox.

PART IV

What Position Do You Hold?

Different philosophers have reacted differently to Fitch's paradox. Some abandon anti-realism, others try to modify the knowability principle, and some even accept the paradoxical conclusion.

What position do you find most defensible?

Test Your Anti-Realist Position

Select the anti-realist position closest to your view:

PART V

Attempts to Escape the Paradox

Philosophers have proposed various ways to avoid Fitch's paradox while preserving some form of anti-realism. Each approach has its advantages and problems.

Attempts to Avoid the Paradox

Tennant's Restriction

controversial

Core Idea:

Only "Cartesian" truths are knowable - truths whose knowledge doesn't lead to absurdity

Formal Expression:

If <>Kp doesn't lead to contradiction, then p -> <>Kp

Main Problem:

This seems circular - we define knowable truths as those that don't cause the paradox.

Restrictions try to exclude problematic truths like "p & ~Kp" from the knowability principle

Advantage

Preserves some form of the knowability thesis while avoiding the paradox

Disadvantage

Often seems ad hoc or changes the thesis so much it loses its original appeal

The debate continues. Some philosophers believe a satisfactory solution exists; others see Fitch's paradox as a decisive refutation of anti-realism.

PART VI

Explore Modal Logic

Modal logic is the formal framework underlying Fitch's proof. Build formulas and see how the operators interact.

Build a Modal Formula

Current Formula:

Key Formulas in Fitch's Proof:

p -> <>Kp : Knowability Principle

p & ~Kp : An unknown truth

<>K(p & ~Kp) : The fatal step

PART VII

The Philosophical Landscape

Since Fitch published his proof in 1963, philosophers have debated its significance. Here are some notable responses.

How Philosophers Respond

What does Fitch Really Show?

Perhaps the paradox reveals that our intuitive notion of "knowability" is confused. The concept itself may need clarification before we can properly evaluate anti-realism.

The Self-Referential Problem

The troublesome proposition p & ~Kp is self-referential in a sense - it refers to its own epistemic status. Perhaps we should restrict knowability to non-self-referential truths.

Temporal Considerations

Knowledge changes over time. Perhaps "knowable" should mean "could have been known at some point" rather than "can be known simultaneously with being unknown."

The Anti-Realist Dilemma

Anti-realists must either accept the paradox, restrict their principle in potentially ad hoc ways, or abandon the thesis entirely. None of these options is fully satisfying.

The Takeaway

Fitch's paradox reveals a deep tension between two intuitions:

1. Truth connects to verification

We want truth to be accessible

2. Unknown truths exist

Not everything is known

The proof shows these cannot both be maintained in their simplest forms. Something must give.

Whether Fitch's paradox refutes anti-realism or merely shows we need a more sophisticated version remains one of the central debates in philosophy of logic and epistemology.

Explore More Paradoxes

Fitch's paradox is one of many surprising results in logic and philosophy. Explore more thought experiments that challenge our intuitions.

Thomson's Lamp Newcomb's Paradox All Explainers

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Reference: Fitch (1963), "A Logical Analysis of Some Value Concepts"