The fashionable theory of PIE laryngeals offers plausible explanations for many phenomena, but plausibility is not proof. Are any implications of the postulated laryngeals amenable to statistical test? This would be the standard of proof in hard disciplines.

What I have in mind is this: One could try to compare cognates from two branches of the IE family to find a statistically significant correlation between some contrasting features allegedly determined by an ancient laryngeal. I would call this the method of parallel watersheds.

I have tried to apply this strategy to distinctive reflexes of “heavy” CṚ:/CḶ:(C) roots. These are expected to give rise to ŪR and ĪR in Sanskrit, as in pūrņa, stīrņa, tīrņa, gīrņa, jīrņa, ūrņā, dīrgha, bhūrja; ΡΗ/ΡΩ and vowel-liquid-vowel (VLV) sequences in Greek, as in πληθος, βρωτος, γηραω, ληνος, στρωτος/ στορε-, δολιχος; and RĀ and (arguably) RACC sequences in Latin, as in plēnus, strātus, trātus, vorātus, lāna, longus, fraxinus. “Light” CṚ/CḶ(C) roots, by contrast, are expected to give rise to LV/VL sequences with a short vowel to one side only.

The key step in the correlation analysis is to pigeonhole paired cognates into heavy/heavy, heavy/light, light/heavy, and light/light boxes. Loosely speaking, matches should preponderate over mismatches, but the technical definition of correlation is a bit more complicated.

Applying a χ² test to limited data, I found correlations “significant at the 5% level”. This bit of jargon means that the odds against finding a spurious correlation due to sampling error, when none would actually be present in abundant data, are better than 20:1. This would be considered adequate in most disciplines, and allowances might be made for the limited nature of the fossil record, because they don’t make more classical languages while you sleep.

The degree of concordance is not as impressive. There are a fair number of heavy/light mismatches without obvious explanation, e.g., sūrkšati/στεργεται, pṛthus/πλατυς/plānus. (I tried not to cherry-pick the data, but I may have applied unconscious bias.)

I leave you two questions: Has a statistical strategy been tried, and what did it conclude? Are there any other implications of laryngeal theory that can be so tested?

I would think that some, maybe even most, supposed implications of laryngeals defy testing by this method. For example: The leading vowels on Greek words such as ακουω, ανηρ, αστηρ, ελευθερος, ελαχυς, εννεϝα, ερεβος, ερυθρος, ομιχω, οριγω, οφρυς are commonly attributed to laryngeal prefixes, but no other IE branch (Hittite aside) displays them. The number of possible exceptions is too small to support a statistical test, and ονομα, ομφαλος, ονυξ are easily explained in terms of syllabic nasals.

EDIT: The point of such tests is not to prove the existence of PIE laryngeals, but rather to assess the strength of evidence (in a mathematically rigorous fashion) for their supposed manifestations in daughter languages by ruling out the possibility of random transformations. The evidence for some phenomena is stronger than for others.

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    "Are there any other implications of laryngeal theory that can be so tested?" Well, Hittite shows direct attestations of *h₂, and also of *h₃ word-initially, if you count those. – Draconis Sep 6 '19 at 20:29
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    Hittite does attest a laryngeal of some sort in a very limited number of words. Everything else is hypothesis. – fdb Sep 6 '19 at 21:12
  • @Draconis, laryngeal theory was modelled around the discovery of those hittite words. that does not need testing – vectory Sep 8 '19 at 23:55
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    @vectory Perhaps surprisingly, laryngeal theory predates the discovery that Hittite was an Indo-European language! It was seen as a great success for the comparative method when Hittite was shown to be IE, and to have direct attestation of *h₂. – Draconis Sep 9 '19 at 0:00
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    @Draconis. Strictly speaking, that is not correct. de Saussure posited what he called “sonants coefficients”, not “laryngeals”. IE laryngeals were first proposed by Möller on the basis of a supposed genetic relationship between IE and Semitic. This is not generally accepted. – fdb Sep 12 '19 at 15:58

Your proposed statistical test would not actually provide evidence (much less proof) for the laryngeal theory. You might use statistical methods to compute the significance of certain correlations, for example is there a significant correlation between patterns in Sanskrit like pūrṇa and patterns in Latin like plēnus? At the level of individual letters, the test would not establish significance (single-word comparisons are not significant), but if you code the data for broader categories, you could ask "Does Sanskrit CV:rC correlate significantly with Latin CrV:C".

Such a statistical test does not support the laryngeal theory, but it is consistent with the theory. Well before the laryngeal theory was set forth, there was a hypothesis that there were long syllabic sonorants, just as the correlation bh-f-b vs b-b-p was firmly established. The statistical test is also consistent with the traditional reconstruction – it does not at all distinguish the theories. The phonetic interpretation of the comparative relations is what is at issue, and there is no statistical test that can resolve those controversies.

Proving statistical significance is unnecessarily complicated low-hanging fruit. In well-established historical reconstructions, the correlations should in principle be perfect – zero variation, once you've discerned all of the correct categories. Verner's Law was discovered because an earlier coarser-grained theory of the categories was wrong, and new categories simply needed to be added. When people slap together random n-tuples of words into a comparative series that has no consistent phonological correlation, we can tell that without calling on statistical methods.

There are some common ad hoc re-categorizations that historical linguists apply to data, in order to deal with known types of imperfect correlations. For example, the word "podiatrist" in English ought to have fVt, like "foot": but we know that it's a loan word, so it is categorized separately and doesn't count against Grimm's law as a valid law of Germanic historical phonology. It takes considerable specific knowledge of history, culture and contact languages to be able to make a valid claim about loan-word status, and this is not something that an automated or statistical procedure can do. Onomatopoeic words likewise mess up regular sound correlations, and we deal with them manually.

In principle, one could use statistical tests to decide that there isn't enough data to declare that a certain correlation is valid, for example if you have a hypothesized category that has exactly one member, that isn't statistically significant. But we already knew that. Where I think it might be useful to apply some informed statistical thinking is deciding how big N has to be, i.e. do I abandon all hope if there are only 4 examples of hypothesized proto-phoneme X – do I really need 8 examples?

As a separate intellectual exercise, you could take large-enough comparative series that are not pre-classified in terms of loan or onomatopoetic words, and see if statistical methods could find the loan words. I think this will fail, because e.g. in English we have so many loan words that Grimm's Law is overwhelmed by the "other" correlations (English p to Latin p).

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    Thanks for a thoughtful answer, but ... You have obviously bought into Saussure’s quaintly Platonist conceit of phonetic transformations sans exceptions. It runs contrary to my viewpoint (as a physicist) that theoretical deduction is not sufficient to establish truth. Data are riddled with anomalies, and efforts to explain away apparent exceptions by appealing to unascertainable factors are too often speculative. That’s why I want statistical tests. – Bert Barrois Sep 9 '19 at 13:05
  • I think your problem can be scrutinized with an even clearer example: how do you make the plural of a noun in English (and the allied question, how do you make the 3rd person verb form)? Can you propose a mechanical statistical procedure to establish the truth? If you can't do that, then clearly you can't test the laryngeal hypothesis statistically. – user6726 Sep 9 '19 at 16:14

The different effects of laryngeals are: vowel coloring (a e o), vowel lengthening, tones (in Balto-Slavic), word-initial vowel prothesis (in Greek and Armenian), aspiration of consonants (typically stH- in Old Indian), lengthening of rhotics (r: vs r in Old Indian). Anatolian languages explicitly keep traces of one laryngeal (namely *H2). We don't need statistical "tests". The case in favor of Laryngeals is closed.

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