The Knowability Paradox: Can All Truths Be Known?

The Knowability Paradox is a philosophical puzzle that challenges our understanding of knowledge, truth, and the limits of human understanding. It raises a deep and troubling question:

If every truth is knowable, does that mean every truth is already known?

This paradox suggests that if we accept the reasonable idea that all truths can, in principle, be discovered, then we must also accept the seemingly absurd conclusion that all truths are already known—which is obviously false.

The paradox has profound implications for epistemology (the study of knowledge), logic, and the philosophy of science. It forces us to reconsider our assumptions about what can be known and whether some truths are forever unknowable.

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1. The Basic Idea of the Knowability Paradox

The paradox was first introduced by philosophers Frederic Fitch in 1963 and is sometimes called Fitch’s Paradox of Knowability. It is based on a simple but striking argument:

1. Suppose all truths are knowable (this is called the Knowability Principle).

2. If a truth is knowable, that means it must be possible for someone, somewhere, at some time to know it.

3. But this leads to a strange consequence: If it is possible to know a truth, then it must be actually known.

4. This would mean that there are no unknown truths, which contradicts reality—since we clearly do not know everything.

Thus, we arrive at a paradox: If all truths are knowable, then all truths are known—but that’s clearly false.

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2. Breaking Down the Paradox in Logical Steps

The paradox can be expressed in formal logic, but we can understand it with a simple example.

Step 1: A Simple Unknown Truth

Let’s assume that there is an unknown truth—let’s call it P.

For example, suppose P is:

“There is a species of deep-sea fish that no human has ever discovered.”

Right now, P is true, but no one knows it because the fish is still undiscovered.

Step 2: The Knowability Assumption

We assume that all truths are knowable. This means:

"If P is true, then P can, in principle, be known."

This seems reasonable—we could send deep-sea explorers to find the fish, and someday, someone might know P.

Step 3: The Problem Arises

If P is knowable, then there must be a possible future state where someone actually knows P.

• But right now, P is unknown.

• If P is knowable but never actually known, then it was never truly knowable in the first place!

• If P is knowable, then it must be known—which contradicts our assumption that P is currently unknown.

Thus, we are forced to conclude that there can be no unknown truths—which is clearly false, because there are many truths we do not know.

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3. What This Paradox Means for Knowledge and Truth

This paradox challenges the way we think about potential knowledge versus actual knowledge. It raises several important questions:

• Are there truths that are forever unknowable?

• Does knowledge require someone to actually know something, or is it enough that something could be known?

• Is there a limit to human knowledge, even in principle?

Some truths seem impossible to know, such as:

• The exact number of grains of sand on Earth at this moment.

• The thoughts of a person who lived and died thousands of years ago, if they left no record.

• Whether there is intelligent life on a planet 10 billion light-years away, if we can never observe it.

If we accept that some truths are unknowable, then the Knowability Principle (that all truths are knowable) must be false. But if we accept that all truths are knowable, then we run into the paradox that all truths are already known.

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4. Possible Ways to Resolve the Knowability Paradox

Philosophers have proposed several ways to escape this paradox.

A. Rejecting the Knowability Principle

One way to resolve the paradox is to deny the idea that all truths are knowable.

• Some truths might be forever hidden from human knowledge.

• The universe might contain mysteries that no one will ever uncover.

This is a realistic but unsettling answer—it means we must accept that some things will never be known, no matter how advanced our science becomes.

B. Redefining "Knowability"

Another approach is to change what we mean by "knowable".

• Instead of saying "all truths are knowable", we could say "all truths are potentially knowable by someone, but not necessarily at the same time or by the same person."

• This avoids the paradox because it does not require every truth to be known all at once.

C. Using Modal Logic: "Possibly Known" vs. "Necessarily Known"

Some philosophers argue that the paradox arises from a confusion between possibility and necessity:

• The original paradox assumes that if something can be known, it must be known.

• But in logic, just because something is possible does not mean it actually happens.

For example:

• It is possible for you to become a billionaire, but that does not mean you will become a billionaire.

• Similarly, just because a truth could be known does not mean it must be known.

Using this reasoning, some philosophers reject the paradox as a misinterpretation of logic.

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5. The Knowability Paradox in Science and Mathematics

The paradox has interesting implications in science and mathematics, where we often deal with the limits of knowledge.

A. Unprovable Theorems in Mathematics

In mathematics, Gödel’s Incompleteness Theorems show that some truths about numbers can never be proven, even if they are true. This suggests that not all mathematical truths are knowable.

B. The Limits of Science

Scientists often assume that given enough time, all scientific questions can be answered. But the Knowability Paradox suggests this might not be true.

• Are there scientific truths that are forever out of our reach?

• If we cannot observe something (like the interior of a black hole), is it still meaningful to call it "knowable"?

C. Quantum Mechanics and the Uncertainty Principle

In quantum physics, Heisenberg’s Uncertainty Principle states that we cannot simultaneously know both the exact position and momentum of a particle.

• This suggests that some truths about the universe are fundamentally unknowable.

• If these truths exist but can never be known, then the Knowability Principle must be false.

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6. Final Thoughts: What Can We Learn from the Knowability Paradox?

The Knowability Paradox forces us to think deeply about:

1. The nature of truth and knowledge—Is knowledge something that exists independent of human discovery?

2. The limits of human understanding—Are there truths that we will never know, even in principle?

3. The philosophy of science and mathematics—Does scientific progress have an ultimate limit?

This paradox, like many in philosophy, does not have an easy answer. But it reminds us of the fragility of knowledge and the mysteries that still await discovery.

Perhaps some truths will always remain beyond our grasp, hidden in the vast and unknowable depths of the universe. Or perhaps, one day, we will unlock every secret—and the paradox will no longer be a paradox.