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If we consider that there are phonological observations as to what is an English word and what probably isn't, one could come up with a dictionary of "all possible" English words, i.e. all words that somehow sound "English".

(This dictionary would probably contain "caterpillar", but it wouldn't contain "lkdsafj" -- what's important here is that it would also contain words that somehow seem to be proper English words, but have no real meaning at all.)

How many words would that dictionary contain (roughly) if we reasonably limit the maximum word length (i.e. no longer than average words in an actual dictionary)?

Now, we could take an actual, contemporary English dictionary and compare it to our imaginary dictionary -- say we take the 300,000 main entries of the OED as our actual English dictionary, which would be the 300,000 most frequent English words.

What would be the ratio of "words actually in use" (in the context of this question "in use" means: "defined in an actual dictionary") to "possible words" (words in our imaginary dictionary)?

What is the proper scientific terminology for the problem/question I'm describing here?

Note that I'm using English as an example -- this might have been discussed for other languages as well.

If there is any research about this topic, I'd also like to ask if the "actual" to "possible" ratio differs between languages or not, and if it does -- can those differences be explained somehow?

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If you want such a product, limiting entries to ones no longer than average word length in an actual dictionary would exclude quite a number of actual words, since long actual words are longer than average (that's how averages work). A better approach would be to limit string to something longer than the longest actual word, so that you won't exclude real words. You have to make policy decisions about what the words of English are, for instance is the chemical name of the protein titin a word with a couple hundred thousand letters; do you include pneumonoultramicroscopicsilicovolcanoconiosis which most people don't know, or antidisestablishmentarianism which people mostly know because it's the supposedly longest word of English?

You'd also need to decide what your unit of counting is. The two most obvious possibilities are phonemes, and syllables. It is widely believed (incorrectly, as it happens) that "possible word" is a function of "possible syllable" and "possible number of syllables" (there's more to it than that). So if you determine that the upper limit on syllable count is 15, then all you'd have to do is develop a syllable-generator that gives you all of the syllables of English, so that [pæt] and [præt] are generated, but *[tpæ] and *[rpæt] are not. Armed with all of the possible syllables plus a limit on word length, you have an approximation of all theoretically-possible words of English.

This syllable-generator would be a bit tricky, since you have to make a decision about what syllables are allowed. We know what [hɪl] is a possible syllable in English, since it is an actual syllable found in hill, Hilbert, Hillman and so on. It is not clear on those grounds whether [pæf] is a possible syllable, but we can also rely on native speaker intuitions that even if there aren't any words with that syllable, it's still a theoretically possible syllable – unless your theory of syllables is radically empiricist and if it doesn't exist somewhere in an actual word, it isn't possible. So you would need to devise some psychological test to validate your underlying theory of possible syllables.

Unfortunately, prospects for developing a sharp discriminating function for sorting into possible / impossible are dim. Nonexistent types like *[bma] seem to be generally rejected by all speakers, but there are also syllables like ?[zla] which make most speakers uncomfortable but are not as badd as *[bma]. Onset sequences like [zw] don't exist, except in Zwicky and zwieback, which are not usually household words. The sequence [CLV̆LC] where L=liquid and V̆=single lax vowel happens to be non-existent (i.e. *blort, *grolm, *plælk), and by perseverating on this gap, you can shift people's intuitions to the point that such sequences get judged as "not apparently allowed".

The famous *sCiV̆Ci generalization, that identical consonants separated by a short vowel and preceded by [s] are not possible, is a generalization that isn't limited to the syllable, so just as *[skɪk] is rejected as not a possible word, *[skɪkuwn] is also rejected, and that sequence spans two syllables. A final problem is that while *[fɪ] is not a possible syllable of English, using the usual test (asking speakers "Could there be a word [fɪ]?"), it shows up in a lot of words, like physics. Some people solve this by making the assumption about syllabification, that the word is the combination of the syllable [fɪz] and the syllable [ɪks], but the problem is that people generally do not accept that the second syllable of physics is "icks".

Another approach is to drop the syllable from the computation. Asking people for intuitions about possible phoneme sequences is even more problematic – you're not going to get good answers to questions like "can an English word contain the phoneme sequence [ændə]?".

The underlying issue is that "possible" is not a self-evident fact. "Possible" can only be determined relative to a theoretical model of the thing in question, and there is no clearly-correct theoretical model of "what a word of English is". Even the question of what is "phonotactically possible" is very unclear, and typically people use professionally-privileged introspection to determine this, where judgments are based on on-the-fly mental searches for "similar substrings".

An alternative approach that I would give more serious consideration to is developing probability tables based on actual words, which would tell you what the probability of a given sequence of English phonemes is which, by overlapping the computation, could lead you to conclude that arpantize is a theoretically possible word though not an actual one. If your machine has a relatively short longest sequence for building those tables, for example 6 phonemes, this could be somewhat informative. If you build a probability table based on actual words ranging from 1 to 30 phonemes in length found in some dictionary ranging from 1 to 30 phonemes, all this will tell you is what words are in the dictionary.

[ADDENDUM]

There's a version of the latter approach here for the article and here for the calculator.

  • 1
    I'm curious about the 'famous *sCiV̆Ci generalisation'—I hadn't heard of it, but surely there are a number such words... stot, stet, stat (x 2)? – Gaston Ümlaut Nov 27 '15 at 22:33
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    "The sequence [CLV̆LC] where L=liquid and V̆=single lax vowel happens to be non-existent" Pseudo-words like this definitely sound funny, but what about "blurb"? That's become used as a word with an actual meaning. Or are all rhotic vowels/vowels before /r/ considered tense for this purpose? In that case, the second liquid obviously would not be able to be /r/, so it seems like the actual relevant sequence would be [CLV̆lC]. Also, is this analysis based on excluding non-lemma forms like "trilled" (or considering the /d/ in these words to be extra-syllabical)? – sumelic Nov 28 '15 at 7:02
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    @GastonÜmlaut: well, we'd have to know how "lax vowels" are defined here. If /ɔː/ (THOUGHT), /ɔːr/ (NORTH), and /oʊ/ (GOAT) are considered tense, then "crawled," "trawled," "trolled," "crolled," and "implored" would all be excluded anyway. (Although this makes it hard to see what the listed word *blort is supposed to represent, since the natural way I would transcribe it is /blɔːrt/.) – sumelic Nov 28 '15 at 11:26
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    Words like "blurb" don't have a lax vowel plus liquid (or even a liquid in non-rhotic dialects). It is true though that this is a generalization about underived roots, so suffixes aren't counted, just as they don't count for the general coda limit. – user6726 Nov 28 '15 at 16:38
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    @sumelic, no, because there are still patterns with lax vowel before [r] which don't become schwa and then syllabic r, for example *[blɔrt], *[flarp]. The one "counterexample" I know of is [flarn], but it is a Mimbari word and I don't think we have to be responsible for non-human. borrowings. – user6726 Nov 29 '15 at 2:16
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I couldn’t find a cite quickly, but the phonotactic space in English is very sparsely filled. If you use a length interval of e.g. 3 to 6 phones, there are (many) millions of legal sequences, but only (as you say), some hundreds of thousands of used English words (possible 1 million, by some estimates). The ‘occupancy’ is greatest at the lowest end of the length scale, and decreases very quickly as length increases. I’m not prepared to say that the decrease is ‘exponential’, but the space of possible words gets huge as more combinations are possible.

There are a couple ways to approach this problem empirically, in the absence of citations. One, you could use/make a phonotactic word generator to enumerate the phonotactic space (then use a dictionary search to subtract the real words).

Two, you could use a measure like ‘lexical neighborhood density’ as a proxy for phonotactic occupancy. There exist dictionaries with neighborhood density measures calculated (or, you could calculate them for a new list, using e.g. CLEARPOND). The average neighborhood density across lengths might give a rough idea of the occupancy. This second option is a problem because average neighborhood density (as commonly implemented, with a phoneme edit distance of 1) drops to 0 very quickly as length increases. This happens precisely because the occupancy drops so fast, but it’s also annoying in terms of precision.

The question is actually more nuanced, because phonotactic constraints are gradient and probabilistic; there is the question of ‘more likely’ and ‘less likely’ beneath the binary ‘legal/illegal’ concept. Looking at this level, the real words tend to be much more ‘likely’ than sequences sampled at random from the phonotactic space, which you’d expect if real people prefer likelier phonotactics.

  • But people can learn and use borrowed words that don't conform to their own language's phonotactic constraints. For instance, Turkish has backness vowel harmony, so that words have only front vowels or only back vowels. Yet Turkish has borrowed many words from Arabic, French, and so on, which don't conform to this constraint. – Greg Lee Nov 27 '15 at 23:37
  • That’s a question for the phonotactic model, but I don’t think it strongly affects this answer? – Jeremy Needle Nov 27 '15 at 23:57
  • If you think that the question is about the relatonship between lexical items and phonotactic constraints, it seems obvious that the phonotactic model is crucial. How could it not strongly affect the answer? The SPE proposal, for instance, is that admissibility to the lexicon is defined in such a way that all lexical items are necessarily admissible, while what I said above implies that the lexicon can contain a large number of inadmissible forms (borrowed words). – Greg Lee Nov 28 '15 at 0:28
  • I think that, regardless of the precise model used to constrain ‘legal’/‘possible’ words, the general issue of occupancy is similar: real words fill the phonotactic space very sparsely (except possibly at very short lengths, though in practice issues of confusability probably present high occupancy even there). If a language has loanwords, then those words are phonotactically legal for that language. If the question is being asked about some subset of a language (‘native words only’, if that can even be coherently stated), then the model for that subset will still produce a similar result. – Jeremy Needle Nov 28 '15 at 0:30
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Zero percent.

Proof: The prefix "un-" is prefixed to an adjective to form another adjective. For instance, corresponding to the adjective "true", we have the adjective "untrue". But since "untrue" is also an adjective, we must also have "ununtrue". Similarly, "unununtrue", and so on, ad infinitum. Yet only one of these additional forms is actually used, viz. "untrue". (They all have meaning, however.)

One divided by infinity is zero.

  • The question says: "in the context of this question "in use" means: "defined in an actual dictionary." So presumably only "true" and "untrue" would count. – sumelic Nov 28 '15 at 7:12
  • @sumelic, yes, I'd count only those two as being in use. That's what I said in my answer. Please read again my sentence that begins "Yet ...". – Greg Lee Nov 28 '15 at 8:10
  • Got it. Well, the "ad infinitum" part does not work, because the question specifies that we "reasonably limit the maximum word length." – sumelic Nov 28 '15 at 8:12
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It depends on how you approach this. It would be relatively easy to generate English non-words using phonotactic rules and then calculate the proportion. As others have said, it is likely to be quite small even if we exclude arguments by recursion which generate unrealistic word forms no matter how supposedly well-formed.

However, something similar has been done in the Lexicon Projects for orthographic forms (for many languages these would be the same but for English they present completely different challenges). The American and British Lexicon projects. However, their aim was not to generate the largest possible number of non-words but rather the largest possible number of plausible forms that could be used in psycholinguistic testing requiring non-words.

You could possibly use their data to generate a more comprehensive list (e.g. by training a neural network on the patterns combined with some rules). However, I'm not sure if the pay off would be worth the effort - unless you can find some real application.

Also, the British Lexicon Project has done work on Dutch, German and French.

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