Last Updated 08 May 2020

Number of possible sentences

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Noam Chomsky has infamously stated, "There is no such thing as the probability of a sentence. " For that, he is roundly mocked by computational linguists, which brings us to the final argument I want to discuss in this essay is as follows: "probability of a sentence is zero, so philosophy is dead! " Let's not take into account the fact that word dead in context of philosophy is quite ambiguous. As the former President of the USA, Bill Clinton (August 17, 1998) said: "That depends on what the meaning of 'is' is".

Therefore, in order to disambiguate the argument, I am going to explicitly say what I mean by word dead, is that there does not exist any philosophy at all. Important assumption is that sentences are uniformly distributed. As in the case of a fair coin or a dice, probabilities are 1/2 and 1/6 respectively, where 2 and 6 are the number of possible outcomes. As a result, probability of a sentence is one divided by the number of outcomes (number of possible sentences). We can easily show that number of sentences in any natural language is infinite.

Natural numbers are a subset of the set of all possible words, while the later is a subset of the set of all possible sentences. Archimedes in 250 BC proved that there is no upper bound for natural numbers; in other words, number of natural numbers is infinite. Therefore, number of possible sentences is infinite. Then probability of a sentence is mathematical limit of one divided by the number of sentences in a natural language. Since number of sentences in a natural language tends to infinity, hence the limit is zero. So we proved that probability of a sentence is zero.

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If probability of a sentence is zero then we cannot make any sentence. If we cannot make any sentence then we cannot speak. If we cannot speak then philosophy is dead. By Hypothetical syllogism we claim that: "If probability of a sentence is zero then philosophy is dead". Applying Modus ponens we get that philosophy is dead, since as it is shown above that probability of a sentence is zero. To summarise, we almost formally proved that philosophy is dead! What went wrong in the argument, where we concluded that philosophy is dead?

Even though all the steps were rigorously explained and formal methods such as Modus ponens and Hypothetical syllogism were used, the crucial thing is assumption we made! We made a fairly obvious assumption that all the sentences are uniformly distributed; in other words we said that probability of a sentence is one divided by the number of possible sentences. Since number of possible sentences in any natural language is infinite, as proved above, hence we obtained that the probability of a sentence is zero.

For instance, let's consider the following anecdote: A man and a blond were asked: "What is the probability that you will see now a real dinosaur walking towards you? ". The man said, that the probability is going to be one in a billion, whereas blond said that it is fifty-to-fifty, either I am going to see it or not. So this example showed that assuming something is quite dangerous thing to do, since we obtained a valid argument, but it is not sound. So this situation is also known as false or misleading presuppositions.

In conclusion, I argued both for and against the main argument of this article, mentioned in the introduction passage. Using common sense in reasoning about conditional probabilities is an example of false dilemma, whereas making almost obvious assumptions is an example of unjustified premises. These two arguments were carefully analysed and showed that proper way of using formal methods and rigorous proof will validate some issues, where common sense might mislead. However, the main argument is unsound, because it is not valid, which was shown using the fact that averages hide a lot of information.

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