A. The PU comparative *-mpV has sometimes been related to PIE *-m̥Ho- (or *-m̥mo-, depending on whose reconstruction is followed). However, its use within Uralic is not only for comparatives. Jussi Ylikoski (https://www.academia.edu/40666645):
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The paper describes a previously little-known grammatical category in South Saami. Termed here as “relation forms”, the phenomenon in question is etymologically related to the comparative and superlative marking of adjectives, but synchronically quite distinct from it. The suffix -be/-åbpoe can be attached not only to adjectives (e.g., nuerebe ‘younger’, båarasåbpoe ‘older’), but also to nouns and kinship terms in particular (e.g., tjidtjebe ‘(the) mother’, vuanavåbpoe ‘(the) mother-in-law’), and the superlative marker -mes/-ommes can be used similarly, albeit to a lesser extent. The paper discusses the position of such forms in South Saami morphology, syntax and sentential semantics, especially in relation to markers of definiteness and possession. From a diachronic perspective, South Saami sheds new light on the origin of the Saami-Finnic (and Hungarian) comparative marker *-mpV, and from a typological point of view, it is proposed that the closest analogues to the Saami phenomenon can be found in Tungusic, which also adds to our understanding of the development of *-mpV comparatives – possibly from a more original contrastive function of the suffix.
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To provide a concise and straightforward introduction to the subject matter of this study, it is instructive to reproduce the three examples of the phenomenon as presented in the best-known grammatical description of South Saami by Bergsland (1982: 107; 1994: 110), who briefly states that when speaking of two persons or things, it is possible to refer to their distinct identities by using a morpheme that is formally identical to the comarative degree marker -be/-åbpoe (cf. urre-be ‘new-er’ and båaras-åbpoe ‘old-er’) [Daktar-åbpoe darjoeji guktie tjidtje-be jeehti ‘The daughter[1-of-2] did as her mother[1-of-2] said’]
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As mentioned at the very beginning of the present study, and in almost all studies on the subject, the Saami-Finnic comparative marker *-mpV (e.g., South Saami -be/-åbpoe and Finnish -mpi : -mpA-) seems to correspond to the Hungarian -bb, but the relations of the superlatives (South Saami -mes/-ommes, Finnish -in : -impA-, Hungarian leg-A-bb) are much less straightforward. Moreover, one of the oft-repeated truths is that there is little evidence of a pan-Uralic or pan-Finno-Ugric comparative in *-mpV; instead, the Saami-Finnic and Hungarian comparatives are usually regarded as results of the convergent development of an element that was originally a derivational suffix with other functions. Opinions – and interests of Eurocentric Finno-Ugricists – differ as to whether Tundra Nenets moderative adjectives (səwa ‘good’ →) səwa-mpoy° [good-moder] ‘rather good’ (cf. Finnish some-mpi [nice-cmpv] ‘nicer’) and (ŋarka ‘big’) → ŋarka-mpoy° [big-moder] ‘rather big’ ought to be considered cognates of the Finno-Ugric comparatives (see Bergsland 1946: 204; Hajdú 1976: 146; Korhonen 1981: 247; Janhunen 2018: 51–53; Aikio, forthcoming).
To mention but a few of the most comprehensive descriptions of Saami, Finnic and Uralic historical morphology, scholars like Hakulinen (1979: 116), Korhonen (1981: 247) and Janhunen (1982: 29) refer to originally contrastive and especially spatial functions of *-mpV, as evidenced by Saami pronominal adverbs such as South Saami daebpene ‘here’, debpene ‘there’, duebpene ‘there (further away)’ and dubpene ‘there (far way)’, as well as East Mari adjectives and adverbs like umbal ‘distant’, umbalne ‘at a distance; far away’ and ümbal ‘top; surface’, ümbalne ‘above, on the top’. The moderative functions of the Nenets suffix are often ignored in this context.
Some authors such as Itkonen (1966: 270), Häkkinen (1985: 91–92; 2002: 86–87) and Abondolo (1998: 18) give more attention to the contrastive or oppositive (and not necessarily spatial) functions of the suffix. The most commonly cited examples include the above-mentioned North Saami interrogative pronoun goabbá ‘which (of two)’ (~ South Saami gåabpa/gåabpetje id.) and its transparent Finnic cognates such as Finnish kumpi id. and Estonian kumb id. (< Pre-Proto-Saami, (Pre-)Proto-Finnic *ku-mpa [q-*mpV]). Another common example is North Saami nubbi ‘other; second’ (~ South Saami mubpie id.), going back to Pre-Proto-Saami *muu-mpa [other-*mpV] and paralleled by (Erzya) Mordvin ombo ‘other’ and omboće ‘second; following’, apparently based on another, pronominal stem. Example (35) also contains South Saami jeatjabidie [other.pl.ill] (nominative jeatjebe), in which the comparative cum relation form suffix -be is attached to the pronominal-adjectival stem jeatja- ‘other’, resulting in the pronominal meaning ‘other; else’ (cf. Bergsland 1946: 182).
It may also be added that in addition to the well-known interrogative pronoun kumpi ‘which (of two)’ and the partly reduplicative indefinite pronoun jompikumpi ‘either one (of two)’ in present-day Finnish (~ Estonian emb-kumb), earlier Finnish also had a corresponding demonstrative pronoun sempi ‘that; it (of two)’ and a relative pronoun jompi ‘that (of the two)’...
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As for the origin of PU *-mpV 'more', I don't know if anyone has mentioned Finno-Permic *umpe 'shut, closed; whole, entire; rather (large)'. For its ety., (https://en.wiktionary.org/wiki/Reconstruction:Proto-Finnic/umpi):
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Cognate with Proto-Sami *ompë, probably borrowed from Proto-Germanic *umbi... From Proto-Indo-European *h₂m̥bʰi (“by, around”)... Cognate with Old Latin am (“around”), Latin amb- (“around, about”), ambi- (“both”), Ancient Greek ἀμφί (amphí, “around, about”), Sanskrit (abhí, “against, about”).
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A loan from Gmc would not fit the 'closed (off) > closed by snow, heavy snowfall' found even outside of Finnic. If also matching the IE range 'both > very, more (S. abhi+), one of 2' in PU *-mpV 'more, one of 2 (relatives/contrasting things)' then it surely could not be a loan to ALL the Uralic branches with it. Their *umpe might just as well be *umpi (if any contrast of e \ i existed in non-initial syllables), and since PIE *H2(a)nt-bhi and dual *(H2ant)-bhoH1 'both' often have the same range of meaning, the vowel of *-mpV is not assured even if this relation is true.
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B. This could tie into why supposed PU *-mpV > F. -(e)-mpi ‘more, -er’, Hn. -bb contrasts with other Hn. outcomes of *mp.
*kumpë ‘rounded & swollen thing’ > F. kumpu ‘hummock / hillock / mound / high rounded wave’, X. xump ‘wave’, Hn. hab ‘foam / froth’
*tompë > Hn. domb ‘hill / mound / hump’, *towmb > Mi. tō̆mp ‘hill / island’, Es. tomp ‘clod’ (or *tump-?, V's not with uncontroversial outcomes; several with good reason for certainty in other branches seem to have 2 outcomes in Hn.)
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Though *mp > mb, bb, b is completely irregular, the usual response of linguists that any irregularity must be "fixed" has not led to any new theory. However, each of these words would correspond to an IE one with *mb, *mbh, *mp (if the source of *umpi, *-mpV is accepted).
*mb > mb, PIE *tumbo- > G. túmbos ‘mound / cairn’, MI tomm, I. tom ‘hillock’; PU *tombë > Hn. domb ‘hill / mound / hump’, *towmb > Mi. tō̆mp ‘hill / island’, Es. tomp ‘clod’
*mbh > bb, PIE *H2nt-bhi, PU *-mbhV > F. -(e)-mpi ‘more, -er’, Hn. -bb
*mp > b, PIE *kump- > Lt. kùmpt 'become hunchbacked, crooked', Po. kępa 'river island covered with bush, shrubbery' (compare range of *tomp-), R. kupiná ‘shrubbery, mound, knoll'; PU *kumpë > F. kumpu ‘hummock / hillock / mound / high rounded wave’, X. xump ‘wave’, Hn. hab ‘foam / froth’
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This simple scheme might need to be modified since PIE almost certainly had *mpH, etc., and the outcome of *mbr is not known in PU. Accent in IE also had many effects, so looking for some here would be the most cautious approach. Since other IE branches also contrast changes to *mp vs. *mb(h) that don't have to do with voice (Celtic had a number of these), I don't think my ideas for PU wouldn't fit.
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The forms with *ë are intended to extend the contrast between PU *a & *ë in initials with *a1 & *a2 in finals (https://www.academia.edu/8196109).
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C. I think there's more of an IE relation here than the paper could say. If Latin māter-tera 'maternal aunt' came from 'on the mother's side (of 2) > mother's relative', then there would be several IE parallels. In Skt. aśvatará-s 'mule' or 'better horse' the suffix -tara- is both 'more, better' and 'relative (of a horse); with a horse on one (of 2) sides of the family'. The same in Armenian *jio- 'horse', *jio-tero-s > ǰor 'mule' (Hrach Martirosyan). I also wrote (https://www.academia.edu/172639045):
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IERR 144 ‘cross’
The need for PIE *terH2- 'cross, go beyond, overcome, conquer' to form *+t(e)r(H2)o- 'beyond, other, one of two; more, -er' is seen in variants with H-met. (143), like *H1wi-terH2o- > *witH2aro- > Slavic *vьtorъ 'other, 2nd', *wiH2tro- -> L. vītricus 'stepfather' ( < '2nd (father)'; likely used with *pater- at one time, alone when *witr- fell out of use as an independent word). Since compounds often lost *H, some IE branches might have *+tro-.
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IERR 145 ‘thirst’
Though PIE *ters- ‘thirst’ is the standard rec., Baltic requires *terH2s- \ *traH2s- (*tra:-sk- > Li. troškulys 'thirsty', trókšti 'to be thirsty; thirst, hope, desire'). Since some cognates also mean 'desire', I think this is older, with the relation *terH2- 'cross, reach (for), seek, desire', *terH2-s- ‘desire, thirst’. Again, H-met. (143, 144) allows variants *H2ters-, *tersH2-, etc., which would hide the *H2 in some or all IE branches. This might also explain L. trem-; several IE languages seem to show n-infixed verbs, but with -m- not -n- when next to *H2 (*staH2- 'stand (upright)', *stamH2-).
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Taking the ev. for PU *-mpV into account, it seems likely that all basic IE family members were named with *-terH2- > *-H2ter- (*ma-H2ter- 'mother', etc.).
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D. In this vein, comparing Finnish ku-mpi ‘which (of two)’ with IE *kWu-tero- or *kWo-tero- not only shows the same suffix that can be applied to contrast, comparison, & family, but also the initial *ku- vs. *kWu- (or *kWo-, both found in the original paradigms). It would lead most undecided readers to a reconsideration of the common origin of PU *k- & PIE *kW- 'wh-' words. Veps kudamb 'which, what' might also point to PIE *kWi-d 'what' > *kWut (by sound change or analogy with *kWu- in other derivatives).