Posts Tagged ‘unknowable’

The Liar Paradox can be resolved by the unknowable

October 17, 2017

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

  1. the assumptions were wrong or
  2. that the logic applied was not valid or
  3. 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 p is a truth that is never known then it is unknowable that p is 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” .


Known, unknown and unknowable

The unknowable is neither true nor false

Gödel’s Incompleteness Theorems



The unknowable is neither true nor false

September 22, 2017

Some things are unknowable (Known, unknown and unknowable).

In epistemology (study of knowledge and justified belief), unknowable is to be distinguished from not known.

Fitch’s Paradox of Knowability (Stanford Encyclopedia of Philosophy)

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 p is a truth that is never known then it is unknowable that p is a truth that is never known.

Some believe that everything in the universe can be known through the process of science. Everything that is not known today will be known eventually. I do not quite agree. Some things are, I think, unfathomable, unthinkable – which one can not know. My reasoning is quite simple. The capacity of our brains is limited. That seems undisputed. It is because of our cognitive limitations, we have found it necessary in our invented languages to invent the concept of infinity – for things that are beyond observable. If brain capacity was unlimited there would be no need for the concept of infinity or for words such as “incomprehensible”. Even in mathematics, which, after all, is another language for describing the universe, we also have the concept of infinity. Infinitely large and infinitely small. We can find finite limits for some convergent infinite series, but we can never get to infinity by making finite operations on finite numbers. We cannot fill an infinite volume with a finite bucket or measure an infinitely long line with a finite ruler.

To my layman’s understanding, Cantor’s Continuum Hypothesis (which still cannot be proved), and Gödel’s Incompleteness Theorems (which prove that it cannot be proved in an axiom based mathematics), only confirms for me that:

  1. even our most fundamental axioms in philosophy and mathematics are assumed “true” but cannot be proven to be so, and
  2. there are areas of Kant’s “noumena” which are not capable of being known

It is not just that we do not know what we do not know. We cannot know anything of what is unknowable. In the universe of the knowable, the unknowable lies in the black holes of the current universe and in the time before time began.

I find Kant’s descriptions persuasive

Noumenon, (plural Noumena), in the philosophy of Immanuel Kant, the thing-in-itself (das Ding an sich) as opposed to what Kant called the phenomenon—the thing as it appears to an observer. Though the noumenal holds the contents of the intelligible world, Kant claimed that man’s speculative reason can only know phenomena and can never penetrate to the noumenon.

All human logic and all human reasoning are based on the assumption that what is not true is false, and what is not false must be true. But why is it that there cannot be a state where something is neither true nor false? Why not both false and true at the same time? In fact, it is the logic inherent in our own language that prohibits this state. It is a limitation of language which we cannot avoid. But language is not discovered, it is invented. That “what is not true must be false” is just an assumption – a very basic and deep assumption but still just an assumption.

Following Fitch

in fact there are unknown truths, therefore there must be truths that couldn’t possibly be known.


“Truth” and “that which is not true is false” are assumptions and are definitions inbuilt in our languages. Truth and Falsity are not necessarily mutually exclusive quantum states. They may instead form part of a continuum which is unknowable.  Maybe a truth-time continuum?

(image Forbes)


Known, unknown and unknowable

July 22, 2017

Donald Rumsfeld was often the butt of cheap jokes after this quote. In reality, Rumsfeld was absolutely spot on and close to philosophic.

Starting from where Rumsfeld left off we come to the distinction between the knowable and the unknowable

These are things we don’t know that we don’t know. There are knowable unknowns. That is to say, there are things that we could know but we don’t know which we don’t know. But there are also unknowable unknowns. There are things we cannot know that we don’t know that we can never know. 

a la Rumsfeld

I am coming to the conclusion that the sum of all human cognition lacks some of the dimensions of the universe. It may be increasing with time, but human cognition is limited. The expanding universe may be infinite or it may be boundless. For human cognition to grasp the universe is then like trying to measure an infinite length with a ruler of finite length, or of trying to measure some unknown parameter with a ruler marked in inches. Those measurements will never reach a conclusion.


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