I was dining in a restaurant and overheard the following conversation between two men.
First man: You cannot call me bald.
Second man: Why not? You have just a few hairs left.
First man: Still not. Indeed, I can prove it by mathematical induction. You will agree that a man with a million hairs is not bald.
Second man: I Agree.
First man: A man with million hairs minus one is also not bald.
Second man: Agree.
First man: A man with million hairs minus two is also not bald.
Second man: (alarmed) Agree.
First man: Now continue to remove hairs one by one. You will come to the conclusion that if a man with million hairs is not bald, so is a man with a single hair.
The second man was dumbfounded. For a few moments he remained silent then retorted:
If your mathematics tells me that you are not bald then mathematics is nothing but balderdash.
I was enjoying the conversation and inwardly smiling. Possibly, my enjoyment had some outward expression because suddenly the first man turned to me and said; “Sir, why don’t you say some thing.”
Reluctantly, I said,
Sir, since you have asked, I will tell you about sorites paradox, because just now you have given a classic example of Sorites paradox. Sorites paradox is generally attributed to the Greek philosopher Eubulides of Miletus who lived in 4th century BCE. He is known for inventing many paradoxes. For example, the famous liar paradox is attributed to him. It is as follows:
A man says: “What I am saying now is a lie.” If the statement is true, then he is lying, even though the statement is true. If the statement is a lie, then he is not actually lying, even though the statement is a lie. Thus, if the speaker is lying, he tells the truth, and vice versa.
Now sorites paradox is also called heap paradox. In Greek, the word “sorites” means heap. Originally, the paradox was posed as follows:
(1) 1 grain of sand does not make a heap.
(2) If 1 grain of sand does not make a heap, then 1+1=2 grains of sand do not.
(3) If 2 grains of sand do not make a heap, then 2+1=3 grains of sand do not.
—–
(10000) If 9999 grains of sand do not make a heap, then 9999+1=10000 grains of sand do not.
10000 grains of sand do not make a heap.
The number 10000 here is for demonstration purpose. You can increase it to any arbitrary number.
An alternate statement of the paradox can be,
If 10000 grains of sand is a heap, then so is 1 grain of sand.
Or more generally you can say,
“for any number n, if n grains of sand is a heap, then n-1 grains of sand is a heap. ”
Certainly, the arguments for the sorites paradox seems to be valid, yet the conclusion seems to be false. The paradox attracted little interest until late 19th century. Marxist philosophers took it is a failure “customary” logic and triumph of the dialectic. In more recent years the paradox is understood as more of a semantic problem than a mathematical problem. Core of the issue is vagueness of predicate. For example, in the sentence: “ 100000 grains of sand is a heap,” the predicate ‘heap’ is vague. It does not have any clear definition. Indeed, vagueness is a defect of ordinary languages. At one time, mathematician and logician Bertrand Russell and Gottlob Ferge dreamt of an ideal or perfect language, devoid of any such defect. Sorties paradox will dissolve in an ideal language. However, their dream was not realised.
Modern response to sorites paradox generally fall into four categories:
1. Nihilism or rejection of initial premise:
This is the simplest and most drastic solution, reject the initial premise. The paradox then collapses. However, the solution is highly unsatisfactory. For example, in heap paradox, it denies the existence of the word heap.
2. Epistemic view or rejection of one or more of the other premises:
It is assumed that the fault lies in one of the premises other than the initial one. For example, in the sorites argument,
(1) 1 grain of sand does not make a heap.
(2) If 1 grain of sand does not make a heap, then 1+1=2 grains of sand do not.
(3) If 2 grains of sand do not make a heap, then 2+1=3 grains of sand do not.
—–
(100000) If 99999 grains of sand do not make a heap, then 99999+1=100000 grains of sand do not.
100000 grains of sand do not make a heap.
Say, we reject one of the premises other than the first. Which one it should be? Say it is number 5000 premise;
(5000) If 4999 grains of sand do not make a heap, then 4999+1=5000 grains of sand do not.
If we reject this premise, sorites problem disappear. However, it is also unsatisfactory, on what ground we say 5000 grains of sand make a heap but 4999 grains of sand do not? Indeed, the epistemic view introduces a sharp boundary between a “heap” and “not heap.”
3. Truth value gap theory:
It tries to avoid the epistemic view of sharp boundary. Intuitively, grains of sand is divided into two groups, one group is a heap, the other is not. Say, n grains of sand is a heap and m grains of sand is not. However, there are grains of sand in between m and n. What about them? In truth value gap theory, the word ‘heap’ does not apply to them. It is undefined for them. Thus for p (between n and m) grains of sand , if you say it is a heap, you are not saying the truth, but you are also not saying false. For p (between n and m) grains of sand, the word “heap” is undefined.
Truth gap value theory raises certain interesting issues. For example, it is common to say,
Either it is raining or it is not raining.
The sentence appears to be logically true. But in the truth value gap theory, in certain cases, the sentence may be false. There may be borderline cases when, “is raining” is a vague predicate.
4. Degrees of Truth approach:
In truth gap value, there are groups to which the predicate ‘heap’ does not apply. To avoid the problem, in degrees of truth approach, sentences are assigned with a degree of truth. A sentence is simply not true or false, but true to a certain degree. For example the consider the statement,
An adult man of height 7’ is tall.
It is true to a high degree. Say we assign it a degree of truth 0.99.
Then the statement, “An adult man of height 6’11’’ is tall.” is true to a lesser degree, say 0.98 and so on.
Cooking Cosmos 