The Eleventh Meditation, Paradoxos – The Philosophy of Paradoxes

Truth and Falsehood by Alfred Stevens

~ Our meeting today, my dear reader, is not one of coincidence, luck or blind chance. That we have met today means that we were meant to meet and our meeting could not have happened in any other way. It is inevitable that the past must be as it has been before the future can unfold. We may forget in the near future that this meeting has ever happened but we cannot change the fact that this encounter has already taken place.

~

Like all human inventions, language is an imperfect tool for human communication. In Logos – The Building Blocks of Philosophy, I have highlighted the importance of language as an instrument that allows human beings to communicate and work together. Subsequently, I have also stressed that although language is of unparallel importance to humanity, it contains certain imperfections to which the meaning of things can be distorted and ambiguous. In this short essay, I will provide some demonstrations of how our human language may be vague and self-contradictory.

Paradoxes are considered to be an anomaly in languages. Central to the concept of paradox is the idea of conflict. One interpretation of ‘paradox’ is a ‘statement conflicting with received opinion’. One of the most famous philosophical paradoxes is known as the liar paradox, which goes like this: Jill says, ‘I am now not speaking truly’. If Jill is not speaking truly when this is what she says she is up to, she is speaking truly. If she is speaking truly, then she must be doing what she says, that is in this case, not speaking truly. Consequently, what she says is true if, and only if, it is not true. This of course, is rather absurd.

There are many implications that the liar paradox reveals. First and foremost is the nature of truth. How can one determine the truth when a person states that he is not speaking truly? Aristotle was known to state, ‘to say what is that it is and of what is not that it is not, is true’. Yet the liar paradox shows us that the truth is often if not always elusive. For Tarski, ‘snow is white if and only if snow is white’ is the truth. Yet in reality, sentences are not always all so simple and a string of written words can be self-contradictory like the liar paradox.

The barber paradox is another paradox that shows that grammatically correct sentences can be self-contradictory. The barber in a certain village is a man who shaves all and only those men in the village who do not shave themselves. Is he a man who shaves himself? Here we see that if the barber is a man that does not shave himself, he shaves himself. On the other hand, if he does shave himself it would mean that he does not shave himself. Both ways, the sentence is totally illogical. Similar to the liar paradox, the barber paradox implies the truth may sometimes be incomprehensible even through the sentence is grammatically correct.

The next two paradoxes arise from vagueness. The bald man paradox goes like this: suppose a man has a full head of hair: if he loses one hair he will still have a full head of hair. But if he loses enough hairs he will become bald. Clearly, there is no particular number of hairs whose loss marks the transition to baldness. How can a series of changes, each of which makes no difference to his having a full head of hair, make a difference to his having a full head of hair?

Consider next the paradox of the heap. With a single grain of sand, you cannot make a heap. If you cannot make a heap with the grains you have, you cannot make a heap with just one more. So even with 10 million grains you cannot make a heap. Obviously we can make a heap from 10 million grains of sand but the sentence above implies that we cannot. Some people may argue that the definition of a ‘heap’ must be defined before this paradox can be solved (since having 5 grains of sand can technically be made into a ‘heap’ with one grain of sand on top of four other grains of sand).

However, a version of this paradox poses a problem. 1 is a small number, and any number bigger by 1 than a small number is also small; so all numbers are small. While stating that all numbers are small is rather ridiculous, one must note that there is no clear point to which a number can be said to be either big or small. The number 100 may be considered ‘big’ when we compare it to the number 1 but ‘small’ when we compare it to the number 10,000. With no clear distinction of a big or small number, vagueness poses a stumbling block in human communications.

The implications of paradoxes are clear. It is possible to come up with a sentence both self-contradictory and/or vague without being grammatically wrong. While the bald man paradox and the paradox of the heap shows us the importance of definitions to foster greater understanding, it also shows us that coming up with accurate definitions for certain things are extremely difficult. How does one define a ‘heap’ or ‘baldness’? Consequently, the barber paradox shows us that certain things that we can invoke or create through languages can be non-existent in reality. There cannot be a barber that shaves everyone in the village who does not shave himself unless the barber himself is exempted from this clause. This means that the sentence should go: the barber in a certain village is a man who shaves all and only those men in the village who do not shave themselves ‘except himself’.

The liar paradox on the other hand, is by far the most challenging of all these paradoxes. Here we must draw a line between what is true in reality and what is true in a sentence as perceived from an external observer. When Jill says that, ‘I am now not speaking truly’, the truth in reality is that the sentence is a meaningless statement that contains no property of truth in it because it is self-contradictory. For the external observer, the liar paradox is good gibberish as it contains a meaning that can be taken both ways, either true or false. One thing is for certain, human language is far from perfect.

~ Ee Suen Zheng