Prepositional phrases and case in North American (heritage) Icelandic

Abstract The paper investigates the use of PPs, specifically prepositions and the case marking on their DP arguments, in moribund North American (heritage) Icelandic (NAmIce), using data from a map task experiment. Since prepositional phrases combine semantic properties with morpho-syntactic properties, PPs allow us to investigate the relative vulnerability of both domains at once. Our results show that while the prepositional inventory of NAmIce is not reduced as compared to Modern Icelandic, the choice of prepositions is subject to crosslinguistic influence from the dominant language English. For case, we find an increase in the use of nominative and accusative case at the expense of the dative; prepositions may take over case functions too. Our results are in line with previous research on case in heritage languages as well as studies on language change, while partially contradicting the assumption that loss is reversely related to acquisition.


Introduction
Heritage language speakers (HSs) are generally defined as speakers of a minority language in a majority language context. Our paper is concerned with a particular type of HSs, who may best be characterized as speakers of a 'moribund' language or variety (Putnam, Kupisch & Pascual y Cabo 2018). These individuals often represent the final or penultimate generation of speakers, typically the fourth or fifth generation since their ancestors' immigration, who have varying degrees of proficiency in the heritage language (HL) (Seliger & Vago 1991;Schmid & Köpke 2007, 2017. The US and Canada are home to many HLs of this kind, among them moribund North American Icelandic (NAmIce). Generally, the diasporic heritage varieties of Germanic languages have been well studied (e.g. Arnbjörnsdóttir 2006Arnbjörnsdóttir , 2015Boas 2009;Putnam 2011;Johannessen & Salmons 2015;Page & Putnam 2015;Lohndal & Westergaard 2016;Arnbjörnsdóttir, Thráinsson & Bragason 2018a;Arnbjörnsdóttir, Thráinsson & Nowenstein 2018b;Westergaard & Kupisch 2020). HSs of these varieties are typically elderly, live in rural areas and possess little if any rather than age of onset effects. Few studies so far have compared different domains based on the same data set to explore their relative vulnerability under reduced input. In this respect, PPs are a good testing ground, because they allow us to investigate interactions between syntax and morphology and those between syntax and semantics at the same time. With syntax being resilient, we do not expect case marking as such to disappear. However, the exponents of syntactic case marking, i.e. functional case morphemes, sit at the syntax-morphology interface, where they could be affected by crosslinguistic influence. Similarly, while we do not expect prepositions to lose their functions as assigners of semantic roles, we can expect a redistribution of roles due to external influences. Thus, case marking (syntax-morphology) and selection of thematic roles (syntax-semantics) are potentially vulnerable, possibly in different ways and to different degrees. This is what we are going to explore.
Previous studies have reported divergences between heritage grammars and monolingual grammars in nominal morphology more generally, but especially with case (Arnbjörnsdóttir 2006, Arnbjörnsdóttir & Thráinsson 2018, Björnsdóttir 2018 for NAmIce; see Benmamoun et al. 2013 for a general overview). Many of the observed patterns of case reduction happen when a case-marking language is in contact with English; see Polinsky (1995) for Russian; Montrul, Bhat & Bhatia (2012) for Hindi; Montrul, Bhat & Girju (2015) for a comparison of Spanish, Hindi and Romanian. For American Russian, Polinsky (1995Polinsky ( , 2006 has shown a general decline in the number of case forms, from six to two cases. The restructuring of morphological case has also been extensively studied in diasporic varieties of German, including varieties spoken in Eastern Europe, Brazil, Australia, South Africa, and across North America (see Yager et al. 2015 for an overview). In these varieties, changes in dative case marking are most typical. Often one can identify some sort of cause for restructuring. For example, the weakening of case systems may well be related to the fact that case assignment is dependent on multiple exponence, and exponence may vary between HSs and homeland speakers due to differences in input and intake. Along these lines, Yager et al. (2015) have proposed that changes in morphological case marking should not simply be viewed as a loss of inflectional morphology but rather in terms of the emergence of new semantic-morpho-syntactic mapping strategies. While at first sight, HSs seem to have lost the ability of using morphological case marking, a close analysis suggests that they are developing patterns of Differential Object Marking (DOM), following a systematic hierarchy where case tends to be marked on animate and definite arguments over inanimate and indefinite ones. This latter account implies that heritage bilingual grammars are complete grammatical systems that show structural innovations that should be understood in terms of reanalysis of structural systems (see also Putnam & Sánchez 2013) rather than acquisition failure or attrition. An interpretation along these lines has also been offered for NAmIce (Björnsdóttir 2018), and the present paper follows this line of thinking.

Prepositions and case in Icelandic
Modern Icelandic as spoken in Iceland (ModIce) has a rich prepositional inventory, consisting of simple and compound Ps, none of which inflect. They cover an array of semantic meanings, including spatial and temporal relations (see e.g. Thráinsson 2005:107-123). PPs have the following main syntactic functions: predicates in copular constructions, as illustrated in (1) below, objects of prepositional verbs, as in (2), or nouns, as in (3), and modifiers (syntactic adjuncts), as in (4); Ps in bold.
(1) PP as predicates in copular construction Kirkja-n er á horn-i-nu. church-NOM.DEF is on corner-DAT-DEF.DAT 'The church is on the corner.' (2) PP as object of prepositional verb Ég er að leita að Landakirkj-u. I am to search for Landakirkja-DAT 'I am looking for Landakirkja.' (3) PP as object of noun Lundasafn-ið er á horn-i-nu á Lambastraeti og : : : Lundasafn-DEF.NOM is on corner-DAT-DEF.DAT on Lambastraeti-DAT and 'The Lundasafn is on the corner of Lambastraeti and : : : ' (4) PP as syntactic adjunct (spatial modifier) Finnurðu Lundasafn-ið á kort-i-nu? find.you Lundasafn.ACC-DEF.ACC on map-DAT-DEF.DAT 'Do you find the Lundasafn on the map?' Ps in Icelandic typically select NPs, which they govern and assign case to. Icelandic has bound definite articles, which attach to the right edge of a noun that is already inflected for case. The bound definite article exhibits gender, number and case. For illustration, see the paradigm for garður 'garden MASC' (stem: garð-) in Table 1. 1 ModIce distinguishes four cases: nominative (NOM), accusative (ACC), dative (DAT) and genitive (GEN) (Einarsson 1973:32). Ps taking NP complements govern ACC, DAT and GEN case (for details see e.g. Einarsson 1973:106f., 109f., 113;Thráinsson 2005:113-120), shown in (5) Some Ps may govern more than one case, notably ACC and DAT, or ACC and GEN, often depending on the meaning in a given context (e.g. Thráinsson 2005:119). For example, the spatial prepositions á 'on, onto', í 'in, into', undir 'under', and yfir 'over, above' select ACC when expressing movement in a certain direction (directional P), but DAT when not expressing directional movement (stative P). This is illustrated for í 'in, into' in (8). Table A2 in the appendix provides an overview of Ps relevant for the present study, along with their ModIce meanings, the cases they govern and examples of their use.

Prepositions and case in the acquisition of Icelandic as a first language
It is generally assumed that properties that are acquired early are more resistant to processes such as attrition or crosslinguistic influence (CLI) from another language than late-acquired properties (e.g. Polinsky & Scontras 2020). This idea is compatible with the Regression Hypothesis (Jakobson 1941), which relates order of acquisition to order of language loss. For example, in a study of 45 first-generation Dutch emigrants in Anglophone Canada, Keijzer (2007) has revealed mirror symmetries between attriters and acquirers, thus providing support for the regression hypothesis. In the case of early bilingual acquisition, an additional factor is that the HL normally develops under less pressure from the dominant language until speakers start entering school. In a recent proposal, Tsimpli (2014) has considered timing differences in the monolingual development of features and structures, distinguishing between early and late acquired phenomena. This distinction reflects differences in the role of narrow syntax: Early acquired phenomena typically represent phenomena in narrow syntax that are associated with syntactic parameters and autonomous, i.e. independent of other linguistic or extra-linguistic domains. Late acquired phenomena, by contrast, represent properties that link syntax with syntax-external and language-external domains. These timing differences have implications for bilingual development.
Those phenomena that emerge early in monolinguals are sensitive to age of onset effects in bilinguals. Late phenomena, on the other hand, involve the contribution of semantics, pragmatics and lexical knowledge and as such they map more readily to the notion of interfaces. In bilingual acquisition, late phenomena are subject to input rather than age of onset effects. Thus, a first language acquisition perspective can help us understand the relative vulnerability of different linguistic properties in language contact situations.
In the monolingual acquisition of Icelandic, inflectional case marking is observed before the use of Ps, and production performance is generally better with bound functional elements (e.g. case suffixes) than with free ones (e.g. Ps) (Thordardóttir & Weismer 1998, Nicholas 2011. Monolingual children acquiring Icelandic use ACC and DAT inflectional case as early as with a mean length of utterance (MLU) of 1.1 (age 15 months), and GENwhile rarerfrom 15 months onwards (Thordardóttir & Weismer 1998). Thus, inflectional case marking in Icelandic is acquired before school entry, i.e. before the speakers of the present study were intensively exposed to English. The situation for Ps is slightly different. Some local Ps are present at age 3;11 (years;months) and their use increases with growing vocabulary size (Nicholas 2011). In Sigurjónsdóttir's (1986) study, nine out of 10 children at age 5;0-5;11 used some Ps describing local relations correctly (inn í 'in into', í 'in', ofan á 'up upon', á 'on', upp á 'up on', undir 'under', við hliðina (á)/hjá 'next to'). However, the use of the Ps fyrir ofan 'over', yfir 'over', ofan við, ofan 'above', which started at age 4;0-4;11, had not been acquired by all children of age group 5;0-5;11. This corresponds to the age when the speakers of the present study entered school and came into contact with English. Thus, based on what is known about the acquisition of case and Ps in Icelandic, we would expect both case categories and case forms to be robust because they are early acquired, while at least some prepositions are late acquired and potentially more affected by exposure to the dominant language English.
Given the assumption that early acquired properties are robust, the early acquisition of case in Icelandic appears to be somewhat at odds with earlier research on case marking in heritage speakers. Indeed, for NAmIce, Arnbjörnsdóttir (2006:100) observed that '[c]ase assignment by prepositions has undergone some attrition' and that the 'most consistent regularity is that proper names of people and places are always in the nominative case, regardless of the preposition that precedes them', 2 while in ModIce proper nouns do inflect. Differences in the case forms following prepositions between NAmIce and ModIce have also been observed by Björnsdóttir (2018), who studied 152 letters written by a female speaker of NAmIce, Jóna, a second-generation emigrant to Canada, between 1908 and 1980. As a child, Jóna had fully acquired Icelandic. According to Björnsdóttir (p.c.), she was highly proficient in Icelandic, despite receiving no formal education in Icelandic. In particular, Björnsdóttir observes that dative increasingly replaces genitive and that dative is used with meanings that it does not have in ModIce, also replacing the accusative. As an anonymous reviewer remarks, and we agree, the situation described here for NAmIce may be one of differences in the realization of exponency. In other words, the exponents of the individual cases are expanded to new semantic domains. If correct, then this suggests that we are observing extension of some forms rather than a full-scale reduction of the case system.
Regarding prepositional meaning in NAmIce, Arnbjörnsdóttir (2006:100) observes that 'English meaning' may be transferred onto the Icelandic prepositional form. For example, the ModIce preposition fyrir 'in front of, on account of, for' may take over functions of the English preposition 'for' which ModIce fyrir does not have, including ModIce í and um used in contexts relating to temporal duration (e.g. NAmIce: ég lenti á spítala fyrir tvaer naetur; ModIce: ég lenti á spítala í tvaer naetur 'I was in hospital for two nights'; NAmIce: þegar ég var búin með University þá fór ég til Evrópu fyrir tíma; ModIce: þegar ég var búin með University þá fór ég til Evrópu um tíma 'When I had finished university then I went to Europe for a while'; Arnbjörnsdóttir 2006:103). 3 Arnbjörnsdóttir (2006:100) further notes that the NAmIce preposition af 'off, of, from' functions like the English preposition 'of'. Finally, two or more ModIce Ps 'collapse : : : into one' in NAmIce (Arnbjörnsdóttir 2006:100). Arnbjörnsdóttir (2006:100-103) provides several examples of the use of Ps in NAmIce and the case they select, yet without identifying a systematic pattern or change in case assignment. For example, af 'off, of, from', which selects DAT in ModIce, is used with NOM and NOM/ACC in her examples. P til 'to', selecting GEN in ModIce, combines with DAT in her NAmIce data, and yfir 'over' occurs with DAT in NAmIce in cases where it would select ACC in ModIce. Björnsdóttir (2018), too, mentions innovative uses of Ps in the letters of her speaker.
In summary, while case is acquired early in Icelandic, i.e. ACC and DAT as early as aged 15 months, GEN about the same age, the acquisition of Ps is a protracted process, which is still ongoing after the age of five. Previous research on NAmIce has shown that the speakers may differ from baseline speakers in Iceland by means of their case-marking strategies (Arnbjörnsdóttir 2006, Björnsdóttir 2018, and that they also show transfer from English in their use of Ps.
Before moving on, let us stress once more that present-day speakers of ModIce are of course not the ideal baseline for the NAmIce speakers that migrated to North America several generations ago, because ModIce has also changed over the generations, and has done so under the impact of factors other than those that have potentially affected NAmIce. Nevertheless, in the absence of more appropriate baseline data and as long as we keep this caveat in mind, we believe that the comparison can provide valuable insights into the nature of language change.

Hypotheses
The present study investigates the use of Ps and their case assignment by speakers of NAmIce as compared to ModIce, and the implications for the relative vulnerability of semantic and morpho-syntactic properties of Ps in HSs. Based on the background provided above our hypotheses (Hs) are as follows: (9) H1: The prepositional inventory of NAmIce exhibits changes in use as compared to ModIce and there is influence from English. H2: Case marking by Ps is subject to restructuring. In particular, we observe changes with DAT marking.
H1 is based on the fact that some Ps are acquired late in Icelandic, coinciding with the age at which our speakers had been exposed to English for the first time, i.e. upon school entry at the age of five or six years. As outlined above, late acquired phenomena are more affected by reduced input than early-acquired phenomena (Tsimpli 2014). Moreover, previous work by Arnbjörnsdóttir (2006) has already shown potential influence from English on the use of NAmIce Ps; see also Björnsdóttir (2018). Given Arnbjörnsdóttir's (2006) observations, we expect in particular that NAmIce fyrir and af adapt the meaning of English 'for' and 'of', respectively.
As to H2, following the rationale that early-acquired properties are generally less affected by reduced input (Tsimpli 2014), Icelandic case should arguably be robust. However, we know from previous work on other Germanic heritage languages that case marking, especially for the dative, can undergo changes with respect to homeland speakers (Yager et al. 2015). Moreover, previous work on NAmIce already observed differences in case marking between NAmIce and ModIce (Arnbjörnsdóttir 2006, Arnbjörnsdóttir & Thráinsson 2018, Björnsdóttir 2018. Since all NAmIce speakers in the present study were monolingual speakers of Icelandic until around age six when they started school and continued to use Icelandic after the onset of acquisition of English (see Section 3.1 below), we assume that they had already acquired case when intensive contact with English started. If nonetheless deviances between NAmIce and ModIce are observed, these are likely to result from reanalysis rather than incomplete acquisition or attrition. But how can we identify these conceptually different scenarios in the absence of longitudinal data? We conceive of an 'incompletely acquired' or 'attrited' grammar' as a grammar that lacks functionally relevant distinctions. The reanalyzed grammar, by contrast, features new ways of expressing functions that were also relevant in the 'old' grammar. While we expect the 'new' grammar to represent a (relatively) stable system, we should not be surprised to find instances of variation because (i) variation occurs naturally, including in monolingual speech, and (ii) restructuring is likely to be preceded by a period of variation.

Data and analysis
The data used here are taken from the Icelandic map task corpus collected in Iceland and in Manitoba, Canada, in 2013-2014(Dehé 2015.

Speakers
The present analysis is based on data from 12 speakers of NAmIce (six male, six female), as well as 22 speakers of ModIce for comparison.
The NAmIce speakers were recorded in three locations in Manitoba, Canada: Winnipeg (two speakers; speaker identifiers CW01, CW08), Gimli (on Lake Winnipeg; eight speakers; CG01 through CG08) and Lundar (Interlake area; two speakers; CL02, CL03). All 12 speakers were born and raised in Manitoba. They were sequential bilinguals, as they grew up with L1 Icelandic until age five or six years (school entry) when English started to become their dominant language. 4 All speakers continued to speak Icelandic at home until at least age 16. The speakers were between 64 and 89 years of age at the time of recording.
We further selected 12 ModIce speakers (ModIce1; five male, seven female) aged 64 to match the age range of the NAmIce group, and 10 ModIce speakers (ModIce2; five male, five female) aged 19-34 to address possible effects of language change and growing influence from English on ModIce (as documented in Kvaran 2004, Thórarinsdóttir 2011, Rehm & Uszkoreit 2012, Rögnvaldsson 2018, Sigurjónsdóttir & Rögnvaldsson 2018. All ModIce speakers were from the north of Iceland (H: Húsavík, Ó: Ólafsfjörður), a region which is the origin of many immigrants to North America (Arnbjörnsdóttir 2006; see also Dehé 2018).

Method
Map tasks elicit quasi-spontaneous speech in an interactive discourse; they have the advantage that the experimenter has some control over the linguistic materials used by the participants. Another advantage, concerning the topic of this paper, is that it elicits a relatively high number of PPs. The maps used in the present study differ from previous map task experiments in several ways, one of them being that the target words were written onto the maps. The target words were landmarks (10) and street names (11). The participants were given time to familiarize themselves with these written target words before the start of their experimental session. They were tested in pairs, i.e. the data are from map task conversations between participants. All map task dialogues were recorded using two Microtrack II (M-Audio) recorders and two Rode NT-5 condenser microphones. The dialogues were between 15 and 59 minutes long (NAmIce dialogues between 23 and 50 minutes). All map task dialogues were orthographically transcribed by native speakers of ModIce. If a transcriber judged a NAmIce utterance as deviant from a well-formed ModIce equivalent (e.g. for reasons of lexical choice, word order, or case), they added the closest well-formed ModIce version to the NAmIce transcription.

NAmIce: Data treatment
All NAmIce transcriptions were manually searched for contexts in which Ps were produced as well as those in which they were omitted although ModIce would have required one (according to the Icelandic transcribers). The PPs had the syntactic functions given in (1)-(4) in Section 2 above. Overall, the 12 speakers produced 546 (environments for) Ps, most of them spatial (due to the nature of the map task). They were the Ps listed in (12) in the order of their frequency. (12) Ps occurring in the data set (N = 546) 5 299 á 'on, onto, to' (252 non-directional, 40 directional, 7 others); 59 til 'to, towards'; 37 af 'off, of'; 32 hjá 'at, with'; 29 í 'in, into' (23 non-directional, 6 directional); 16 fyrir 'on account of, for'; 10 að 'to, towards'; 8 frá 'from'; 11 á milli 'between' (including deviant forms); 5 yfir 'over' (4 directional); 4 upp 'up'; 3 með 'with'; 2 framhjá 'past'; 2 um 'about'; 2 út á 'out on'; 1 eftir 'along'; 1 við 'with'; 1 úr 'out (from)'; 1 á móti 'opposite'; and 23 contexts for prepositions but none realized For each of the 546 instances, the P as well as the morphological form (case and definiteness) of the following NP were coded, both as produced by the NAmIce speakers and as required in ModIce according to the transcribers (see Table 2 for examples). For ease of presentation, we will use the term 'target-like' when a form corresponds with what is expected according to ModIce norms and 'targetdeviant' when a form does not correspond with what is expected in ModIce. 6 The codings include lexical and morpho-syntactic features of prepositional use. The 546 occurrences were then classified according to the six categories in (13), which relate to the use of NAmIce Ps and case forms as compared to ModIce. They are illustrated in (14)-(18). (13) Categories a. Overall target-like: Both P and the morphological form of the following NP conform to ModIce. b. Only P target-like: The P is as required in ModIce but the morphological form of the following NP is deviant. c. Replacement of P: The NAmIce utterance contains a P that diverges from the one expected in ModIce. (The accompanying case form may also diverge from ModIce.) d. Omission of P: No P used, although P is required in ModIce.

NAmIce prepositions
For a first overview, the numbers of occurrences of the categories in (13) are presented in Figure 1, showing target-like and deviant use of Ps and case endings.
Categories (13a) and (13b) add up to 61% (N = 333) target-like, which means Ps are used in the same way as in ModIce in 61% of all cases. By contrast, only 22% of NPs were realized with a case form that corresponds with ModIce (N = 102 in category (13a); N = 26 in category (13c)). All speakers produce PPs and all speakers but one (CL02, who has fewest productions overall) produce PPs that are target-like in ModIce. Moreover, whether or not speakers produce PPs that correspond with ModIce forms is not a matter of place of recording 10 (see Table A3 in the appendix for distribution by speaker). The example in (19) shows that one and the same utterance may contain forms that are target-like in ModIce or target-deviant (here: target-like PP að Melabúðinni followed by Lindagata uninflected for case).
Finnurðu Lindagötu? (ModIce) Lindagata goes to Melabúð-DEF.DAT find.you Lindagata.ACC 'Lindagata goes towards the Melabúð. Do you find Lindagata?' When the P was target-like but case on the NP was not (N = 231), participants used mostly case forms that exist but deviate in use from those required in ModIce (N = 217). Non-existent forms (N = 8) or incomplete forms (N = 6) were the exception. In the case of P replacements (N = 157), the by far most frequent scenario was a case form that exists but deviates from that required in ModIce (N = 123). It was less common for case on the NP to be target-like (N = 26) or non-existent (N = 8).
The results according to syntactic functions are summarized in Figure 2 (see also Table A4 in the appendix). They show that PPs in the corpus function most often as predicates (N = 233, 42.7%), followed by complement to V (N = 167, 30.6%), complement to N (N = 73, 13.4%), and adjuncts (N = 56, 10.3%).
Examples are given in (20) (22), the predicted P is most often replaced by another P. Adjuncts, as in (23), have the highest percentage (37.5%, N = 21) of overall target-like forms.
Relating to our hypotheses, we zoom in on the results for categories 'only P target-like' (see (13b) above) and 'replacement of P' (see (13c)  specifically relates to changes in case-inflectional morphology (H2), while (13c) relates to the prepositional inventory (H1). We begin with category (13c), replacement of P (H1). Prepositions were replaced by another prepositional form 28.8% of the time (N = 157/546); see Figure 1 (and Table A3 in the appendix). The most frequent preposition in the corpus is nondirectional á. There are 308 environments for non-directional á in the data set, 240 (78%) of which were produced as expected in ModIce. Of the 68 non-target-like cases, 61 were replaced by another P, mostly by af (N = 31) and hjá (N = 24). All 31 replacements by af 'of' occurred in the context á horninu á 'on the corner of'; lit.: 'on the corner on' and 21 of 24 occurrences of hjá occurred in the same context. Examples are given in (24). Another interesting case of replacements is that of directional prepositions. Of the 546 environments for prepositions in the NAmIce corpus, 124 (23%) are environments for directional að, á, or í; see Figure 3. Of these, more than half (56%) were replaced by a preposition the transcriber judged as deviant from ModIce. Most of the time, að, á, and í were replaced by the Icelandic preposition til 'to, towards', which suggests influence from English (see discussion in Section 5 below). Examples are given in (25). (CL03) Við skulum fara í Lindakirkju.

NAmIce case marking
With respect to case marking (category (13b) above; H2), recall that overall, only 22% of relevant NPs were produced with a case form corresponding to the expected ModIce form. We focus on the subcategory where P as the case assigner is target-like, and the case form used in NAmIce exists in ModIce but a different form would be expected in ModIce (see (15)). The overall frequency of this subcategory was N = 217 (40%). A more detailed analysis is provided in Figure 4, which lists the expected cases from top to bottom and the actually realized cases from left to right. The most noticeable result concerns DAT case. The corpus contains 379 cases where the ModIce P would select DAT. Of these, almost half (46%) were realized differently in NAmIce. Most strikingly, DAT is frequently replaced by NOM case forms or by case forms that are ambiguous between NOM and ACC case due to case syncretism, e.g. in the feminine and neuter paradigms (see Figure 4, Appendix Table A5 and examples (26a, b)). There were comparatively few replacements (N = 18) of DAT Sg by NOM/ACC plural forms in NAmIce. These were 17 cases of hornin instead of horninu 'the corner DAT', and one case of kortin instead of kortinu 'the map DAT'; see (26c, d), for instance, which require ACC in ModIce but are realized with NOM in NAmIce.   Table A5 in the appendix). (CW08) and that is on corner-DEF.NOM/ACC.PL there Og það er á horn-i-nu þarna.
(ModIce) you come out and go onto Mímisgata.ACC 'You come out (of that road) and walk onto Mímisgata. ' Recall that case in Icelandic distinguishes between directional vs. non-directional (or: stative) uses of the same P (compare í kirkjuna (ACC) 'into the church' vs. í kirkjunni (DAT) 'in the church'). If both ACC and DAT are replaced by NOM, the morphological marking of the semantic difference disappears. We therefore take a closer look at the most frequent preposition in the corpus, á, which can be used as either a directional preposition selecting ACC case or a non-directional preposition selecting DAT case. As Table 3 shows, NOM or NOM/ACC replaces both DAT and ACC in more than half of the cases, obscuring the semantic difference between the two uses.
Regarding structural innovation, we observe six cases of af taking over the function of genitive case. One example is given in (27a). In ModIce, the preposition á milli 'between' selects genitive (see (27b) below). In the NAmIce data, GEN is reclining (see also Björnsdóttir 2018), and in this context it is replaced by af (27a), in analogy to English 'in the middle of'. (Note that the P á milli occurs in different forms in the NAmIce data, one of them being í miðjan.) Concerning potential influence from English, we add that of eight cases of 'search for' (ModIce leita að) in the NAmIce data set, five were realized as leita að and three as leita fyrir.
In summary, what we observe is an instance of extension of the P-exponents to different semantic domains. This is not a lexical issue per se but one in which the exponent is mapped onto other semantic domains.

ModIce
For comparison, transcriptions from 22 speakers of ModIce were searched for prepositions. First, we searched the transcriptions of all 22 speakers for the prepositions til, á and af, the Ps that featured most often as replacements (see (16) above) and analyzed their functions. Comments had been made by reviewers as well as audiences at various talks where this work was presented that in ModIce, til has come to be used in some of the contexts that NAmIce speakers use it in, although the meaning might be slightly different ('up to'). Since af and til were the prepositions that speakers used most often in a non-target-like way, we looked at all instances of these two to see whether ModIce speakers used them in innovative ways, too (including fixed phrases, e.g. til baka 'back', til vinstri 'left', and parts of phrasal verbs, e.g. vera til 'exist', segja til 'say to'). All but two speakers produced utterances with til, there were 116 instances overall, and all were target-like. None were used in the contexts found for til in NAmIce. That is, uses like those illustrated in (25) above were absent from the ModIce data set. Like in the NAmIce corpus, af was produced less frequently (N = 54). All but five ModIce speakers produced instances of af, and all were target-like, for example, there were no cases of af instead of/replacing GEN case.
Second, we searched the transcriptions of the young ModIce speakers (ModIce2) for the Ps á, í and yfir, all of which select ACC case if used with a directional meaning, and DAT if used with a non-directional meaning, to test the possible change of case forms in ModIce due to influence from English, and accompanying loss of the case marking of the semantic distinction. The Ps á (N = 184), í (N = 155), and yfir (N = 40) were used with both DAT and ACC complements, and ACC complements were exclusively directional, while the use of DAT coincided with a non-directional meaning. 14 Yfir (N = 19) occurred exclusively with ACC and a directional meaning. (The proportion of syncretic forms was 35% from the total). We finally searched for Ps that select DAT case (að, frá, hjá, af) to see whether there were any indications that ACC would be extended to these functions. However, the Ps að (N = 95), frá (N = 44), hjá (N = 115) and af (N = 20) were used exclusively with DAT complements. In sum, in the ModIce map task data, we did not find any evidence for change in either the use of Ps or the use of case forms that would correspond to the observations for NAmIce.

Discussion
We now discuss the results reported on in the previous section against the hypotheses, repeated below, as well as the implications for the relative vulnerability of the semantics and morpho-syntax. H1: The prepositional inventory of NAmIce exhibits changes in use as compared to ModIce and there is influence from English. H2: Case marking by Ps is subject to restructuring. In particular, we observe changes with DAT marking.
Our results have shown that the use of Ps in NAmIce is overall relatively stable, as the majority of Ps (61%) were used like in ModIce. The prepositional inventory does not seem to be reduced either. Whenever transcribers corrected a preposition, they used a preposition that was used target-like by the same speakers elsewhere in the NAmIce corpus, just not in that particular instance. However, there are also differences in use as compared to ModIce, and the changes are of different kinds, both semantic and structural. Semantic changes are mostly manifested in replacements, where one P is replaced by another P.
One example of a structural change is the preposition af taking over the casemarking functions of GEN case. As we have seen in (24) and (27), an NP marked for genitive case is replaced by the preposition af plus an NP marked for nominative case. In (27), the complex P á milli governing genitive case loses its case-marking ability in these contexts and selects a PP instead. This may be interpreted as influence from English. The structure emerging in NAmIce resembles English, because Icelandic af is close to English of in both orthography and pronunciation. Thus, when the genitive is lost, speakers of NAmIce seem to choose the P that is closest to the English equivalent, and af takes over functions from English of, as first observed by Arnbjörnsdóttir (2006). NAmIce af as an equivalent of English of also appears in another context: non-directional á frequently gets replaced by af in the context 'on the corner of'; see (24). Another finding that may be due to influence from English is the occasional replacement of að by fyrir 'in front of, on account of, for' in leita að 'search for' (five out of eight cases). Also, NAmIce fyrir takes over functions of the English P 'for' which ModIce fyrir does not have. Finally, a NAmIce preposition that takes over functions from an English preposition is til 'to, towards', replacing various ModIce prepositions with directional meanings, specifically á, í and að. This suggests that til may, in the long run, take over directional meaning from other Ps, conflating the meanings of different prepositions into one (see also Arnbjörnsdóttir 2006), and possibly affecting the NAmIce prepositional inventory. NAmIce til is close to English to in meaning and phonological make-up. Note that while all instances of NAmIce til that entered the discussion here have been 'corrected' to either á, í or að by the native Icelandic speakers, til in ModIce may have a similar meaning. In fact, when the data were presented at the University of Iceland, Reykjavík, in November 2018, some members of the audience thought that in ModIce til was in fact possible in some of the contexts in which the transcribers had corrected NAmIce til to ModIce á, í or að. However, as reported above, we did not find any evidence for this in our own corpus, i.e. ModIce speakers did not use til in with directional meaning in the same way as NAmIce speakers did anywhere in the map task corpus. Nevertheless, there are many indications that directional Ps are in the process of conflating onto til in NAmIce. Moreover, in NAmIce this change from á, í, and að to til is accompanied by a change in the case system. Til in ModIce selects GEN, but in NAmIce it typically takes a NP as complement with a NOM case ending. 15 For Ps like á and í, the selection of DAT vs. ACC incorporates the semantic difference between non-directional P and directional P, respectively (e.g. í kirkjuna ACC 'into the church' vs. í kirkjunni DAT 'in the church'). If all directional á and í are replaced by til 'to, towards', the case distinction becomes unnecessary, as with English directional to. However, at least in our data, replacements of prepositions are less prominent than changes in morphological form or changes in syntactic case assignment. The latter result in utterances which contain the original preposition (instead of til), but for which the case distinction has been lost, so that the actual meaning (direction vs. state) will have to be inferred from the context. Note that one basic question is to what extent the changes are syntactic (restructuring of the case-marking system) and to what extent they are morphological (reflecting problems with the relevant morphological paradigms). As pointed out above, it can be difficult to distinguish the two.
As to hypothesis H2, case marking is clearly subject to restructuring, in particular with DAT case. Only 22% of NPs in the data set were inflected for case like in ModIce. Of the deviant cases within subcategory (13b) (target-like use of P but non-target-like case form), 98% of all DAT forms selected by P were realized as either NOM or ACC (or a form which exhibits case syncretism between NOM and ACC). One striking consequence of this is the fusion of directional and non-directional meanings, which come to be expressed by ACC/NOM case. The results with respect to case are in line with previous research indicating that inflectional morphology shows great variability and instability (e.g. Benmamoun et al. 2013;Polinsky 2006Polinsky , 2008Montrul 2008;Rothman 2009). Montrul (2016:58) concluded that 'case marking is [one] candidate for erosion and non-native mastery in heritage language grammars' and earlier research on NAmIce maintains that '[c]ase assignment by preposition has undergone some attrition in [NAmIce] morphology' (Arnbjörnsdóttir 2006:100; see also Salmons 2012 for a historical perspective of Germanic). Besides replacements, there is also considerable variation in inflectional case marking, including 22% target-like forms. This finding confirms previous results showing that restructuring of case systems is preceded by a period of variation and alternation between case forms or argument structures (Barðdal & Kulikov 2008:470).
Our data are consistent with the observation that DAT case is vulnerable and might eventually fuse with ACC case marking. However, rather than interpreting this process in terms of loss of case-marking exponents, we prefer to see it as a readjustment in the mapping of case exponents. This is even more plausible if we think about the decomposition of P along the lines of Svenonius (2007), who proposed that the category of adpositions (prepositions) can be split into at least two parts: P, the Ground (expressing a spatial relation), and p, the Figure (an element that introduces theme or location). The underlying structure of prepositions in NAmIce may simply exhibit different underlying structural specification (i.e. exhibiting true representational differences), which had already been determined very early on in the speakers' acquisition of Icelandic in North America. Such cases of readjustment or restructuring are not unique to heritage languages, but they also happen in diachronic change. For example, the Latin cases ACC and DAT have disappeared in (most of) the Modern Romance languages, but the traces are still visible. In fact, they have turned into new structural distinctions, as indirect objects are marked by a preposition (e.g. French parler à qqn 'speak to someone'), while direct objects are not (appeler qqn 'call someone'). The difference to heritage language acquisition is that normally (i.e. in the absence of an extreme situation of language contact) such processes happen more slowly.
What are the implications of our results for the relative vulnerability of morphosyntax and semantics? We found that divergence between NAmIce and ModIce was more striking in inflectional case marking than in the use of Ps. Does this imply that in NAmIce, case morphology is more vulnerable to 'attrition' than the lexical aspects of prepositional use, in contrast to Tsimpli's (2014) proposal that earlier acquired properties are not subject to input effects? Not necessarily. Instead, the nature of change seems to differ depending on the module. For Ps, which are late acquired and have semantic relevance, there is no loss of forms but a change in functions, possibly due to contact with English. For case marking, at first sight, there seems to be a loss of forms, as has been observed in previous research (e.g. Montrul et al. 2012, Polinsky 2006, Yager et al. 2015. However, the fusion of DAT and ACC that we observe may only be a temporary stage in a longer process of restructuring, where eventually the original case functions will be taken over by other elements, such as Ps. Since case is early acquired and part of narrow syntax, the divergence we see in the steady-state is more likely due to the fact that the exponents of case may not be as accessible to these speakers as they are to monolingual speakers. If HSs perceive and process the exponents of case differently, they may end up with a different steady-state, not because their development is arrested but because their 'target' was perceived to be different to begin with.

Conclusion
In this paper we have investigated the use of prepositions and case marking in North American (heritage) Icelandic based on 12 speakers, aged 60-90 years, born and raised in Manitoba (Canada), where they grew up with Icelandic as their first language and English as their chronological L2 but dominant language. We were particularly interested in the comparison of choice of prepositions and case marking because the former is primarily semantically conditioned, while the latter is primarily conditioned by morpho-syntax. The two were elicited in a map task experiment, which results in the production of a substantial portion of prepositional phrases, so that data on case and prepositions could be elicited simultaneously. We found that the speakers' repertoire of prepositions was not reduced but the choice of prepositions was indeed subject to crosslinguistic influence from the dominant language English. In morpho-syntax, there seems to be an ongoing shift such that the use of dative case decreases, while use of nominative and accusative case takes over its functions. The latter is in keeping with earlier studies on the restructuring of case in heritage languages, indicating that dative case in Germanic has historically (and synchronically) been more vulnerable than other cases. 16 It is possible that this is related to the unique nature of dative in Germanic, being partially 'lexical' and partially 'structural'.