A paradox appears when reasonable assumptions together with apparently valid logic lead to a seeming contradiction. When that happens, then applying the same rules of logic lead to the further conclusion that either

- the assumptions were wrong or
- that the logic applied was not valid or
- that the seeming contradiction was not a contradiction.

Where a paradox lies in wrongly identifying a contradiction, it is just an error. If the paradox is due to an error of applying the rules of logic it is also just a mistake. However sometimes the paradox shows that the logic itself is flawed or inconsistent. That can lead to a fundamental revision of the assumptions or rules of the logic itself. Paradoxes have contributed to whole realm of the philosophy of knowledge (and of the unknowable). Many paradoxes can only be resolved if perfectly reasonable assumptions can be shown to be wrong. Many scientific advances can be traced back to the confrontation of the starting assumptions.

Confronting paradoxes has led to many advances in knowledge. Niels Bohr once wrote, *“How wonderful that we have met with a paradox. Now we have some hope of making progress.” ***A paradox is just an invitation to think again. **

Classical Logic is digital. It allows no grey zone between true and false. What is not true is false and what is not false must be true. This requirement of logic is built into the fabric of language itself. Classical Logic does not allow a statement to be, *true and false* simultaneously, or *neither false nor true.* Yet this logic gives rise to the **Liar Paradox** in its many formulations. The paradox dates back to antiquity and in its simplest forms are the statements

*“I am lying”*, or

*“This statement is false”*

There are many proposed ways out of this paradox but they all require some change to the rules of Classical Logic. Some require a term *“meaningless”* being introduced which requires that a *proposition* may be *neither true nor false*. Others merely evade the paradox by claiming that the statement is not a *proposition*. Bertrand Russel took the radical step of excluding all self reference from the playing field.

Experts in the field of philosophical logic have never agreed on the way out of the trouble despite 2,300 years of attention. Here is the trouble—a sketch of the paradox, the argument that reveals the contradiction:

Let L be the Classical Liar Sentence. If L is true, then L is false. But we can also establish the converse, as follows. Assume L is false. Because the Liar Sentence is just the sentence “L is false,” the Liar Sentence is therefore true, so L is true. We have now shown that L is true if, and only if, it is false. Since L must be one or the other, it is both.That contradictory result apparently throws us into the lion’s den of semantic incoherence. The incoherence is due to the fact that, according to the rules of classical logic, anything follows from a contradiction, even “1 + 1 = 3.”

It seems to me that the paradox vanishes if we allow that what is not true is not necessarily false, and what is not false is not necessarily true. The fault lies in our self-determined assumption of the absence of a continuum between *true* and *false*. We can as well create a grey zone – some undefined state *between* **and ***outside* the realms of *true* and *false – *which would be an unknown state. Perhaps even an unknowable state. It is a strong indication – even if not a proof – that the unknowable exists and that logic needs to be fuzzy rather than digital.

The Liar Paradox is connected to the **Fitch Paradox of Knowability**

The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn’t possibly be known.

More specifically, if

pis a truth that is never known then it is unknowable thatpis a truth that is never known. The proof has been used to argue against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge.

Language is invented. Both words and the rules of grammar are invented – not discovered. We are forced to define words such as “infinite” and “endless” and “timeless” attempting to describe concepts which we cannot encompass with our finite brains and our limited physical senses. It is not possible to measure an endless line with a finite ruler.

The unknowable exists and it is therefore that we need the word “unknowable” .

Related:

The unknowable is neither true nor false

Gödel’s Incompleteness Theorems