Department of Linguistics
INFINITE NUMBER OF SENTENCES
Robert Mannell (1999)
The claim, often repeated by linguists, that we can potentially produce an infinite number of sentences in a language, is not strictly true. For such a claim to be true there must either be the possibility of sentences of infinite length or there must be an infinite number of words in the language. Of course, neither of these requirements are possible for any human language (or for any imaginable communication system made up of discrete word-like units of meaning). We will examine the "infinite sentences" claim for English. If we assume:-
- that English has about 500,000 words (there are about 450,000 in the 20 volume Oxford English Dictionary, but this excludes many colloquial forms - although it does include many obsolete forms),
- that English sentences can be up to 100 words in length (a fairly reasonable working assumption)
- that any individual word can occur 0 to 100 times in a single sentence (an unrealistic assumption)
- that words can be combined in any order (a false assumption)
then we can determine that there could be as many as about 10570 possible sentences (very much greater than estimates of the number of atoms in the observable universe). Grammatical rules would greatly reduce this number of sentences, as would the requirement that all sentences be meaningful, but the resulting number of possibilities would still be extremely large (more than could ever be spoken in the entire history of human languages let alone during the much shorter life span of an individual language). So for all practical, non-mathematical, purposes we can say that the English language, or any other living language (1), is an open system. It's actually quite easy to come up with a unique, never before produced, sentence. To do so, for example, combine an unlikely (or impossible, or meaningless) event with a particular named person on a particular date. For example: "On 31st October 1999, whilst writing a lecture on animal communication, Robert had a colourless green idea." (2) Once this sentence has been written or spoken, subsequent productions of this same sentence are not unique, but unique sentences may potentially be generated from it by making slight changes to it (eg. change "green" to "red").
If we consider spoken language, then we would come to a similar conclusion if we examine only the word content of spoken sentences. We might additionally consider the manipulation of vocal resonance (frequencies of spectral peaks), vocal pitch (fundamental frequency), vocal loudness (intensity), rate of utterance and the placement and timing of pauses that occur as a consequence of the combined effects of prosody, vocal emotion, and size, age and gender differences. There are potentially an infinite number of infinitesimally different productions of any sentence (infinitesimal differences of frequency, intensity and timing). It is well known, however, that the human brain is only able to discriminate discrete (step-wise) changes in each of these dimensions. Across the possible human vocal range of these acoustic dimensions there is only a finite number of discriminable (just noticeable) steps. Additionally, it is also well established that meaningful changes in each of these dimensions tend to involve significantly larger changes than those changes that are just noticeable perceptually. This means that all of the meaningful vocal nuances of all of the possible sentences in English would be a large, but finite, number.
I've recently (2011) been alterted to the following web page by Ken Wais that I found quite interesting:-
Also have a look at the following page that was suggested to me by John Fry. Especially look at the section entitled "4.4 Infinite vs unbounded". In that section it is argued that the length of English sentences (or sentences in any other human language) is not potentially infinite, but is unbounded in that we cannot define an upper limit to sentence length. Unbounded does not directly prohibit the possibility of an infinite set of sentences, but it does suggest that sentence length is always finite. In order for there to be an infinite number of sentences in a language there must either be an infinite number of words in the language (clearly not true) or there must be the possibility of infinite length sentences. The product of two finite numbers is always a finite number.
1. A "living language" is a fully operational language being passed on to a new generation. Some people also require that it be the first language of at least some speakers. This second requirement ensures that it isn't merely being transmitted as a fragmentary partial language (isolated words and phrases).
2. When Noam Chomsky came up with his famous sentence "Colorless green ideas sleep furiously" it was also, as Chomsky suggested, extremely likely to have been a unique, never before produced, sentence. (Chomsky, Noam (1957). Syntactic Structures. The Hague/Paris: Mouton)