I have read some works on Proto-Indo-European which mention different schools that advocate for different paradigms of reconstruction, such as the Leiden and the Erlangen schools. I'd like to know if there are any other such schools and a quick run-down of their proposed paradigms (as well as any scholarly works relevant to this question).
I know that this question has remained unanswered for over four years, but I have decided to revive it considering that it's currently the unanswered question with the most upvotes here on Linguistics Stack Exchange.
History of Proto-Indo-European Reconstruction
First of all, it is important to understand that reconstructions of Proto-Indo-European have changed over time, primarily due to the discovery of two language families: the Anatolian and Tocharian language families.
The discovery of the Anatolian languages led to the widespread acceptance of the laryngeal theory arguing that Proto-Indo-European possessed a series of consonants not found in any non-Anatolian languages. These consonants were initially believed to have had an articulation near the larynx but many modern-day linguists believe that they were velar or uvular.
As a result, the phoneme formerly reconstructed as a schwa is now reconstructed as one of three phonemes: *h₁, *h₂, and *h₃. For example, the Proto-Indo-European word for father is now believed to be *ph₂tḗr rather than the earlier *pəter.
The laryngeal theory has also affected the reconstruction of other vowel phonemes. The long vowels *ā, *ī, and *ū that were once believed to be found in Proto-Indo-European are now reconstructed as *eh₂, *iH, and *uH (the H signifies an unknown laryngeal). Furthermore, some phonemes that were formerly reconstructed as *ē are now reconstructed as *eh₁ and some phonemes formerly reconstructed as *ō are now reconstructed as *oH or *eh₃.
We now believe that words never began with a vowel in Proto-Indo-European because of the laryngeal theory. All words that were formerly reconstructed with an initial vowel now begin with one of the three phonemes described in the laryngeal theory (although which one of the three laryngeal phonemes exists there is often uncertain and is denoted with H in these cases).
The discovery of the Tocharian languages (a branch of centum languages spoken very far east from all other centum languages spoken at the time) dispelled the long-held belief that the centum-satem languages constituted a simple east-west division with centum languages in the west and satem languages in the east.
Modern Disputes on Proto-Indo-European Reconstruction
Today, there are still numerous disputed aspects of Proto-Indo-European language reconstruction, giving rise to the different schools of reconstruction mentioned by the OP. There are three major disputes: the glottalic theory, the pronunciation of the laryngeal consonants, and the reconstruction of pronouns.
The Glottalic Theory
The original glottalic theory argues that the consonants typically reconstructed as the unaspirated voiced stops were actually ejective consonants. Proponents of this theory also believe that allophonic aspiration was used for both the unvoiced and voiced consonants (which, according to them, correspond to the consonants typically reconstructed as breathy voiced).
An example of the differences between the traditional reconstruction and the original glottal theory is shown below:
|Traditional Reconstruction||Glottal Theory|
|/t/||/t ~ tʰ/|
|/dʱ/||/d ~ dʱ/|
However, there are numerous other theories on the nature of stops in Proto-Indo-European. Here is a chart showing the differences between them:
|/t/||/t ~ tʰ/||/t/||/t/||/t/||/t/|
|/d/||/tʼ/||/ˀt/||/ɗ/||/d̬ ~ d̰/||/ˀt/|
|/dʱ/||/d ~ dʱ/||/tʰ/||/d/||/dʱ/||/d̥/|
Proponents of non-traditional reconstructions for the Proto-Indo-European stops argue that it is unlikely that a language would create a three-way contrast between unvoiced, voiced, and breathy voiced stops. As of 2022, only one language is known to possess this system: the Kelabit language native to the Sarawak–North Kalimantan border of central Borneo. However, some people claim that the Kelabit's breathy voiced phonemes are not fully murmured in the same way that is reconstructed for Proto-Indo-European.
Furthermore, contrasts between consonants tend to be rather symmetrical across languages. The conventional reconstruction of Proto-Indo-European lacks /b/, yet possesses both /d/ and /g/. This is highly uncommon across languages, which typically use /g/ as the missing voiced consonant or lack /p/ rather than /b/ if a voicing contrast is used.
A third reason why some people support these theories lies in the unusual distribution of stops in the traditional system. No reconstructed root contains two voiced stops (ex. *deg) or both a voiceless stop and a voiced aspirate (ex. *dʰek or *tegʰ). However, plain voiced stops appear to be compatible with both other series (ex. *degʰ or *dek).
Both of these linguistic features are found in a few languages, so it is not implausible that they exist. However, many people are skeptical as a result of their rarity.
Although the primary argument for non-traditional stop reconstructions is their rarity among languages, several proponents have pointed towards grammatical features of descendant languages as further evidence.
The Sindhi language substitutes implosives for the aspirated voiced consonants found in other Indo-Iranian languages.
Some Danish dialects form glottal stop clusters after voiceless stops that correspond with the Proto-Germanic voiceless stops. Frederik Kortlandt claims that English word-final glottalization, preaspiration in Icelandic, German and Faroese, and gemination in Swedish and High German derive from Proto-Indo-European preglottalization.
In Latin and Balto-Slavic, vowels lengthen before a "voiced" consonant. This is the same mannerism that vowels have before Proto-Indo-European laryngeals believed to contain a glottal stop.
Geminalization is also found in the broken tone of Latvian and Žemaitian and in various Armenian dialects.
Opponents of the glottalic theory and alternative stop reconstructions point to the fact that all traditionally reconstructed features exist in at least one language, so it is not impossible for them to exist.
Another major argument for the traditional reconstruction lies in the fact that no modern Indo-European languages except for the Ossetian language possess ejectives. However, this developed because ejective consonants are found in nearby Northeast Caucasian and/or Kartvelian language families and is unrelated to any plausible ejective consonants found in Proto-Indo-European. It is quite likely that an unusual stop distribution in a language would evolve into a rather common distribution in its descendant languages, but the chances of relatively stable ejectives or glottal consonants evolving into a completely different system in all of its descendant languages is very unlikely.
The strongest piece of evidence against these unconventional reconstruction theories are the oldest set of Iranian loanwords borrowed into Armenian and early borrowings from Celtic languages into Proto-Germanic. These borrowings always occur voiced to voiceless, which is not possible under a glottalic theory where they would have been voiceless at the start.
It is, however, possible that an earlier stage of Proto-Indo-European used ejectives that later evolved into the traditionally reconstructed stops. The unstable system of the traditional stops would, as explained earlier, provide a reason as to why descendant languages have not preserved this system thoroughly.
Pronunciation of the Laryngeals
Although the existence of laryngeal consonants within Proto-Indo-European is now almost universally accepted, great debate on their pronunciations continues to exist. Determining their pronunciation with certainty is almost impossible because only one descendant language branch continues to possess these sounds.
Pronunciation of *h₁
Jens Elmegård Rasmussen suggested that *h₁ is pronounced /h/ with an /ə/ syllabic allophone. Evidence for this comes from the fact that /e/ (which it combines with in Ancient Greek) is close to /ə/. Furthermore, *h₁ does not create auxiliary vowels in Ancient Greek and the Tocharian languages between semivowels and consonants (unlike *h₂ and *h₃). The large number of aspirated consonants in Proto-Indo-European according to the traditional stop reconstruction would also suggest /h/ as a likely reconstruction of *h₁.
Winfred P. Lehmann suggested in 1993 that *h₁ actually consisted of two different phonemes (/ʔ/ and /h/) because Hittite reflexes are inconsistent.
Several people have proposed that /ʔ/ is the only value of *h₁. Alwin Kloekhorst argued in 2004 that Hieroglyphic Luwian sign 19 (𔐓) is pronounced /ʔa/ rather than the commonly reconstructed /a/, supporting the hypothesis that /ʔ/ is at least one value of *h₁.
Kloekhorst also suggested that many initial vowels in Hittite actually signified a glottal stop and that the language preserves *h₁ there. If this hypothesis is true, the third person singular Hittite copula e-eš-zi would actually be ʔe-eš-zi (compare Proto-Indo-European *h₁ésti). However, this has received criticism from many other linguists.
A final theory as to the pronunciation of *h₁ and the two other laryngeal consonants is that the laryngeals actually constituted dorsal fricatives and *h₁ was pronounced /ç/ or /çʁ/.
Pronunciation of *h₂
A common theory reconstructs *h₂ as a pharygeal fricative such as /ħ/ or /ʕ/. This is because many other languages (especially the Semitic languages) use these types of consonants to change other vowels into an /a/ allophone.
Vowel coloration is also found with uvular fricatives, so /χ/ and /x/ have been proposed as values of *h₂. /ɐ/ has also been suggested as an allophone of *h₂ rather than /a/.
Alwin Kloekhorst proposes that *h₂ signified /qː/ (he believes that the traditionally voiceless stops of Proto-Indo-European were universally geminalized). However, he believes that a fricative allophone of *h₂ could have existed in Proto-Indo-European.
Pronunciation of *h₃
Linguists generally believe *h₃ to have been a labalized consonant because it colors /e/ to /o/ (a rounded vowel). The linguist Jens Elmegård Rasmussen claims that *h₃ was pronounced /ɣʷ/, with a syllabic allophone of /ɵ/. However, Martin Kümmel suggests the value as /ʁ/.
Kloekhorst claims that *h₃ was pronounced /qʷː/ (the labialized counterpart to his value for *h₂).
Reconstruction of Pronouns
Reconstructing Proto-Indo-European pronouns is difficult due to the significant amount of variation in the pronouns found in its descendant languages. Reconstructing some pronouns are more widely agreed upon by linguists than others since there is less variation in their reflexes found in descendant languages.
For example, the Proto-Indo-European singular first person nominative pronoun has been variously reconstructed as *éǵh₂ (by Donald Ringe), *eǵóH (by Andrew Sihler), *h₁eǵH (by Alwin Kloekhorst), and *eǵh₂óm. Historically, it has also been reconstructed as *éǵ, *h₁eg (by Frederik Kortlandt), and *h₁eǵ(Ho, -Hom) (by Robert Beekes). The presence of a non-initial laryngeal is confirmed through the -h- in Sanskrit ahám. There is no reason to assume that h₁ appears in the pronoun, although it is certainly possible.
This answer will not explore the reconstruction of Proto-Indo-European pronouns in more detail, since this is not the main focus of the question and a detailed analysis of differences in Proto-Indo-European pronoun reconstruction would require an entire book of several hundred pages. However, pronouns are arguably the most disputed area of Proto-Indo-European reconstruction due to the significant variation in descendant pronouns. Furthermore, all Indo-European languages decline their pronouns irregularly.
Although the original question asked for the existence of several “schools” of Proto-Indo-European reconstruction, strict “schools” do not exist as reconstruction is highly debated and different linguists have very different opinions as to how a certain word is pronounced.