3.1. Collecting stimuli

If my hypothesis about the existence of four different construction types governing the processing of two-element complex words were correct, I would expect to find that speakers assess respective words’ complexity differently. Specifically, if one assumes that subunits of complex words are more easily recognisable when they are variables within particular constructions, then one needs to take into account that HH words have no open slots, both LH and HL words have one, and LL words have two. Therefore, given that LL words are more complex than HH words, it naturally follows that LH and HL words should be placed somewhere in between in this hierarchy.

The experiments described below were conducted on English and Russian data. For each language, I selected 40 stimuli: eight prefixes of different linguistic productivity (this will be described further below) and five construction types (HH, LH, HL, and LL plus one pseudo-affixed word) with each prefix. Words were matched for the number of morphemes, and every effort was made to match them for junctural phonotactics, stress patterns, syllable counts, and the frequency of the derived form as well. However, in some cases, not all restrictions could be applied simultaneously. It is possible that the polysemy/homonymy of prefixes could have a non-negligible effect on the results. Nevertheless, because in Russian, LH and HL constructions are semantically differentiated (see below), controlling for their meanings seemed infeasible, so I did not do this for English data either.

Words of both languages were assigned to construction types based on the values of their transitional probabilities’ log ratios: (i) LH: log ratio < −1 (with the exception of out-, for which −0.73 was the lowest value in my data); (ii) HH: −1 < log ratio < 0; (iii) LL: 0 < log ratio < 1; and (iv) HL: log ratio > 1. The frequencies used to calculate transitional probabilities were obtained from two internet corpora provided by Sketch Engine (Jakubíček et al. Reference Jakubíček, Kilgarriff, Kovář, Rychlý, Suchomel, Hardie and Love2013): English Web 2018 corpus (enTenTen18, more than 21 billion words) and Russian Web 2017 corpus (ruTenTen17, more than nine billion words). English stimuli can be found in Appendix 1 and Russian stimuli in Appendix 2.

One can make some important observations concerning two types of derivation to base frequency ratios by looking at the tables with frequencies and transitional probabilities in the appendices. Not controlling for morphological family size and taking into account only occurrences of the base as a free element lead to a highly unstable and unbounded measure, which ranged from 0.003 to 499.4 in my English data and from 0.008 to 55.7 in my Russian data. This is not to mention the dubiousness of the assumption that modern language users are aware of the historical links between some bases and their derivations, for example, that between tact and contact.

One can also estimate the base frequency of a particular derivation in a different way, calculating first the cumulative root frequency (Cole, Beauvillain & Segui Reference Cole, Beauvillain and Segui1989), that is, the overall frequency of all lemmas in which this base occurs, either in a free or bound form, and then subtracting the frequency of the derivation itself (De Jong, Schreuder & Baayen Reference De Jong, Schreuder and Harald Baayen2000). This method has some desirable properties. First, it allows stabilising the derivation to base frequency ratio. For example, the variance of values in my data was reduced by a factor of 11,075 for English and a factor of 435 for Russian. Moreover, by treating each base as a representative of the whole morphological family, one does not gloss over the fact that speakers, for example, might be able to parse the element -cede- out of precede not only because it exists as a free form but also (and more importantly) because they encounter it in multiple words with related meanings (accede, concede, recede, secede, etc.).

On the other hand, the actual analysability measures of the complex words in my sample, when calculated as derivation to base family frequency ratios, revealed for both languages the multimodal distribution I expected to find (Figure 3, left panel). The form of this distribution, as well as the results of Kruskal-Wallis and Dunn’s tests (not reported here), suggest that parsable expressions of the LH type might indeed be conflated with compositional expressions of the LL type and parsable expressions of the HL type with non-analysable expressions of the HH type. The values of the transitional probabilities’ log ratios are, in contrast, normally distributed (Figure 3, right panel) as verified by the Kolmogorov-Smirnov test for goodness of fit in Table 1.

Figure 3 (Colour online) Densities of derivation to base family frequency ratios (left panel) and transitional probabilities’ log ratios (right panel), English and Russian.

Table 1 Results of the Kolmogorov-Smirnov test for standard normal distribution.

3.2. Experimental design

Both English and Russian experiments drew on the design proposed by Hay (Reference Hay2001). Subjects were presented with pairs of prefixed words and asked to provide intuitions about which member of the pair was more easily decomposable. Experiments participants were gathered via the Amazon Mechanical Turk crowdsourcing platform for English part and the Yandex Toloka crowdsourcing platform for Russian part. The experimental designs for both languages were identical. For English subjects, I repeated the instructions verbatim as they were given in Hay (Reference Hay2001), and for Russian participants, I simply translated them into Russian, having only changed the language examples.

Neither Amazon Mechanical Turk nor Yandex Toloka grants access to their workers’ personal data, but they do allow for some coarse-grained social stratification while assembling pools of users. Each word pair in my data was evaluated by 24 native speakers of each respective language, and each set of participants was constructed to conform to the matrix in Table 2.

Table 2 Participants’ matrix for each word pair.

Both experiments were completed online. Each participant was presented with just one pair of words sharing the same prefix (or pseudo-prefix coinciding with it in form) and asked to type in the word they thought was more complex. Task completion time was not limited (the average duration was 43 seconds for the English part and 53 seconds for the Russian part). Participants were explicitly urged to rely solely on their language intuition and not to consult with any online sources available to them, as there was no such thing as a ‘correct answer’ in this case.

Each word was paired with three of its counterparts of the same prefix and with one pseudo-affixed word with the initial element resembling this prefix. The order of presentation was randomised, so every word had an equal probability of being the first or the second member of the pair to appear. Overall, there were 1,920 English participants and 1,920 Russian participants:

$$ \left(\genfrac{}{}{0pt}{}{5}{2}\right) word\ combinations\cdot 8\hskip0.5em prefixes\cdot 24\hskip0.4em subjects=\mathrm{1,920}. $$

4.1. A probabilistic model of complex words’ perceived complexity

One useful way to investigate the relationship between two measures of complex words’ analysability is by thinking back to Hay’s original paper (2001), in which the idea of a derivation to base frequency ratio was initially proposed, and analysing the data therein. It is instructive to look at the table in Appendix 3 with the prefix stimuli from Hay’s article (2001). Here, the words in group A are more frequent than the bases they contain, and the words in group B are less frequent than the bases they contain. Thus, the hypothesis that Hay tested was that A words would be rated as less complex than B words. This suggestion was borne out as Hay observed that ‘among prefixed pairs, 65% of responses favoured the form for which the base was more frequent than the whole. Only 35% of responses judged forms that were more frequent than their bases to be more complex than their matched counterpart’ (Hay Reference Hay2001: 1049).

An interesting observation can be made if one looks at the transitional probabilities’ log ratios calculated for Hay’s experimental stimuli in the same way as I did before and at the construction types assigned to the words depending on these ratios. It turns out that most of the words in group A are of the HL rather than HH type and group B comprises mostly complex words of the LH and LL types rather than just the LL. Given that some non-analysable words are found in group B as well and that, as my experiment has shown, there seems to be no significant difference in the perceived complexity of English HL and LH constructions, one might wonder whether the two groups can indeed be reliably delineated with regard to their members’ morphological complexity.

In order to test this, I built a probabilistic model that would predict a most likely winner in the complexity assessment contest for each of the 17 pairs of words in Hay’s data, drawing on the evidence obtained during my English experiment. The model, a fragment of which is visualised in Figure 7, is a Bayesian network with three types of nodes: (i) 11 prefix nodes, that is, nodes that encode how different prefixes encountered either in my or Hay’s experiment (con-, de-, dis-, en-, il-, im-, in-, out-, pre-, re-, and un-) affect complexity judgements; (ii) five construction type nodes, that is, nodes that encode how four construction types and one pseudo-affixed type (LH, HH, HL, LL, and PA) affect complexity judgements; and (iii) 110 contest nodes, that is, nodes that encode the likelihood of a word of a certain construction type being judged as more complex when paired with a word of a different type but the same prefix (un_HH_PA, dis_LH_HH, etc.).

Figure 7 (Colour online) Fragment of the Bayesian network for predicting the outcomes of words’ complexity contests.

Despite its fairly complicated global structure, on the local level, this Bayesian network is a very simple, state-observational model (Koller & Friedman Reference Koller and Friedman2009) that reproduces the same conditional probability distribution for each contest node given the values of its three parents: one prefix node and two construction type nodes. The prior probabilities in the marginal and joint distributions were specified in an uninformed, commonsensical way: (i) for both prefix and construction type nodes, the probability that they facilitate analysability was set as equal to the probability that they do not, and (ii) for contest nodes, the probabilities in the joint distribution simply reflected the (obvious) fact that if a word belongs to a construction type with a greater positive bearing on complexity judgements than the type of its adversary, then the former word is more likely to win the contest. Prefixes, however, might be expected to interact with construction-type pairings in an idiosyncratic manner, making differences between them either more pronounced or more attenuated.

The inference process was two-fold, based on both evidential and causal reasoning (Pearl, Glymour & Jewell Reference Pearl, Glymour and Jewell2016). First, I used the evidence for contest nodes obtained during my English experiment to infer posterior probability distributions for the prefix and construction type nodes. These distributions, unlike my non-informative priors, were conditioned on observed evidence and hence calibrated to be those under which the results of the experiment were most likely to occur. As a second step, I reverse-engineered the process and, using the updated prefix and construction type nodes’ probability distributions, inferred the most likely assignment of values for the contest nodes that corresponded to the 17 prefixed words’ pairings in Hay’s paper (2001).

4.2. Partial replication of Hay’s experiment (2001)

After the model had been trained, I used the learnt probabilities to predict the outcome of a hypothetical experiment where 24 participants would be asked to select a more complex word in each of the 17 pairs under investigation. In order to check the adequacy of the model’s predictions, I tested the same 17 pairs of words in an experimental setting identical to the one of my above-described English study. A total of 408 participants (24 × 17), none of which had taken part in the previous experiment, were assembled via the Amazon Mechanical Turk crowdsourcing platform so as to conform to the matrix in Table 2. Each subject was presented with just one pair of words and asked to decide which member of the pair was more complex.

The correlation between the predicted and observed proportions of success was found to be significant (r = 0.52, p = 0.03). Most importantly, in both hypothetical and real experiments, 55% of responses judged the words from group A to be more complex than their counterparts, and only 45% of responses chose the words from group B. The difference in the number of votes given to each group in the experimental setting was significant (M_{Group A} = 13, M_{Group B} = 11, t = 2.28, p = 0.02) and accurately predicted by the model (M_{Group A} = 13, M_{Group B} = 11, t = 2.72, p = 0.01).

Thus, my results are the opposite of what Hay reported: words in group A, although more frequent than the bases they contain, were rated more complex than words in group B, which are less frequent than the bases they contain. It is important to note that the two experiments are not directly comparable. Although the general design and instructions were the same, the number of participants was similar, and the prefixed stimuli were identical, there were two important divergencies. First, my experiment was completed online, and each participant worked with just one pair of words without seeing the whole list of stimuli. Second, I did not test suffixed words and used no filler word pairs.

These divergencies were likely to have one important consequence: my subjects were not primed to perform the same operation of segmenting out two base candidates and comparing them as free elements on each pair of words. I argue that people who have been previously asked to select a more complex word in a filler pair like family–busily, when confronted with the pair uncanny–uncommon, will be prone to mark uncanny (HL, group A) as less complex than uncommon (LH, group B). They will do so simply because they have been trained to directly compare canny with common without taking into account their interaction with the general meaning of the complex word. However, in a non-primed scenario, the reasoning patterns might be more complicated. I will elaborate on this in the following section.

4.3. Semantic vector space of complex words

As discussed above, in analysable parsable words of two types, two different elements occupy slot positions and are likely to be parsed out during the semantic analysis. With LH words, the participants of the experiment had to assess whether a certain affix brings anything significant to the composite conceptualisation of the derived form once they have accounted for the contribution of the base. In contrast, with HL words, the participants had to evaluate the contribution of the base while holding the meaning of the affix fixed.

If we again use the machinery of semantic vector space modelling to reify the alleged difference in the processing of the words uncanny and uncommon from the previous example, the following set of operations will be needed: (i) subtract the vector of the filler element (canny in uncanny and un in uncommon) from the vector of the complex word $ \overrightarrow{W} $ , (ii) check whether the subtrahend vector $ \overrightarrow{S} $ encodes something meaningful and relatable to the meaning of the derivation, and (iii) check whether the difference vector $ \overrightarrow{D} $ encodes something meaningful and relatable to the meaning of the derivation. Operation (i) is straightforward. Operations (ii) and (iii) may be performed by finding the nearest neighbours of the vectors $ \overrightarrow{S} $ and $ \overrightarrow{D} $ and calculating the average cosine similarity of these neighbours’ embeddings to the vector representation of the complex word $ \overrightarrow{W} $ .

This may not be an inaccurate modelling of human reasoning. It seems plausible that the participants of the experiment, in order to come to a decision, first manipulated the filler element of a particular word and tried to assign some meaning to it. Next, they might have wanted to evaluate the general constructional meaning encoded by the fixed element with an empty slot free of any concrete lexical material. In this scenario, the search for nearest neighbours is a reasonable approximation of how people tend to semanticise language units using lexical paraphrases and synonyms (Wiegand Reference Wiegand1992; Mel’čuk & Polguère Reference Mel’čuk and Polguère2018).

In order to extrapolate the proposed logic of testing to the whole prefixed dataset in Hay’s study (2001), it is necessary to account for two other construction types. HH words that I believe to be non-analysable pose no challenge in this regard because they are more likely to be accessed directly without segmentation. With LL words, however, the meaning processing model is supposed to be different – not parsable as in LH and HL types, but compositional. The rationale behind this model implies arriving at a composite conceptualisation by means of combining the meanings of two distinct elements, so the set of vector operations should be different here: (i) obtain the vectors $ \overrightarrow{E} $ ₁ and $ \overrightarrow{E} $ ₂ of the first and second elements of the complex word $ \overrightarrow{W} $ , (ii) check whether the addend vector $ \overrightarrow{E} $ ₁ encodes something meaningful and relatable to the meaning of the derivation, and (iii) check whether the addend vector $ \overrightarrow{E} $ ₂ encodes something meaningful and relatable to the meaning of the derivation. Again, operations (ii) and (iii) may be performed by finding the nearest neighbours of the vectors $ \overrightarrow{E} $ ₁ and $ \overrightarrow{E} $ ₂ and calculating the average cosine similarity of these neighbours’ embeddings to the vector representation of the complex word $ \overrightarrow{W} $ .

I used the same FastText English model (Bojanowski et al. Reference Bojanowski, Grave, Joulin and Mikolov2017) as before. For each of the 34 prefixed words in Hay’s dataset (2001), I obtained its 20 nearest neighbours, applying in each case those vector modification operations which I outlined above for the respective construction types. Cosine similarity between each of the nearest neighbours and the target complex word was recorded. The average of these 20 similarities constituted the final measure of the word’s perceived complexity with a clear interpretation: the larger the value, the more likely the word to be judged complex.

Analysis of the results shows that if this measure were the only driver of choice, then out of 17 contests, words in group A would win 70% of the time – that is, much more often than what actually occurred during my experiment. The difference in the average cosine similarities between groups A and B was found to be large, although slightly above the conventional significance level (M_{Group A} = 0.72, M_{Group B} = 0.60, t = 1.87, p = 0.07). Notably, the differences in average cosine similarities between the paired words of the two groups were strongly positively correlated (ρ = 0.88, p < 0.001) with the differences in the number of votes cast for respective contestants, as predicted by my Bayesian network.

In Figure 8, the dots represent the complex words from group A (they are labelled selectively to avoid overplotting), the X-axis coordinates correspond to the differences between these words’ average cosine similarities and those of their counterparts from group B, and the Y-axis coordinates correspond to the differences between the number of people (out of 24 hypothetical participants) who, according to my model, would choose these words as more complex and the number of people who would prefer their counterparts from group B.

Figure 8 (Colour online) Differences (group A – group B) in cosine similarities and predicted votes.

These results suggest, first, that the cosine similarity-based measure might be a reliable predictor of English complex words’ degrees of complexity and, second, that this particular dataset serves as a useful illustration of how relying solely on relative frequency calculations can lead to the conflation of different construction types and thus obfuscate the difference between two meaning processing models, one based on the principle of compositionality and the other on the principle of parsability.

The choice and pairing of 34 prefixed stimuli in this dataset were such that, given the probabilities of success presented in Table 3, words in group B were more likely to win in only seven contests out of 17. As for the other 10, the chances were either approximately equal or in favour of words from group A. Especially telling is the comparison of three HH words that made their way in group B due to low derivation to base frequency ratios with their matched counterparts from group A, which belong to the HL type (Table 6). The Bayesian network predicted that all of these HL words would be considered more complex, which was, to a large extent, borne out in the experiment with human participants. There is an almost perfect positive correlation of values in the votes (predicted), votes (observed), and cosine similarity columns as well as a less than ideal and unexpectedly positive correlation of the same values with those in the derivation/base ratio column.

Table 6 Statistics of HH words in group B and their HL counterparts in group A.

4.4. Two types of complexity and two models of meaning processing

The presentation of the examples in Table 6 is intended to show that HL words, which constitute the majority of Hay’s group A, are in fact not as simple as they might look, although they are quite distinct from LL words. The main point here is that there are two different types of complexity corresponding to the two meaning-processing models – parsable and compositional. The distinction between them, as has been shown, is imprinted in the semantic vector space and can be explained as follows.

LL words like inadequate or reorganise strongly overlap in semantics and distribution with their bases. We can easily replace inadequate with not adequate and reorganise with organise again. However, uncouth is not so easily replaceable with not couth, or reiterate with iterate again. Here, as the nearest neighbours of these words’ modified vectors suggest, some general sense is encoded by the construction as such while the meaning of the base is only used for concretisation. Consider, for instance, the variability of specific lexical meanings in the set of words aligned with uncouth’s vector $ \overrightarrow{D} $ : uncivilized, insolent, amoral, unprofessional, disrespectful, irresponsible, and obnoxious.

To put it simply, inadequate is conceptualised as [– ADEQUATE] while uncouth is conceptualised as [(DEPRIVED OF DESIRABLE QUALITY) & (THIS QUALITY BEING COUTH)]. The distinction between two models of meaning processing does not necessarily pertain to different relative frequencies of derivations and bases. For example, the derivation to base family frequency ratios of the HL words in Hay’s dataset range from 0.01 (immodest) to 4.43 (uncanny) and the derivation to base family frequency ratios of the LL words range from 0.003 (immoderate) to 0.95 (illegible). There is a lot of variability in these values, which makes it difficult to separate two construction types using only relative frequency criterion. On the other hand, taking into account the discrepancy in probabilities of transition from base to affix and from affix to base, one can correctly predict, in most cases, the model at work.

To provide an example of a different type of affix from a different language, consider two German nouns of the same stem: Büchlein ‘small book’ and Bücherei ‘library’. Their respective frequencies in German Web 2020 corpus (detenten20_rft3, Sketch Engine) are 47,121 and 67,174 tokens, which, given the frequency of Buch ‘book’ (5,536,382 tokens), forces one to conclude that both words should be equally analysable (and probably indistinguishable under relative frequency account). However, whereas the meaning of Büchlein is undoubtedly compositional [BOOK + SMALL], the meaning of Bücherei can hardly be expressed as *[BOOK + PLACE WHERE IT IS KEPT]. Rather, it is conceptualised as [(PLACE WHERE PEOPLE DEAL WITH CERTAIN OBJECTS PROFESSIONALLY) & (THIS OBJECT BEING BOOK)] (cf. Käserei ‘cheese factory’, Mosterei ‘firm producing must, cider, or perry’, etc.).

It is possible to predict which model – compositional or parsable – is more likely to be chosen in each case by comparing two words’ transitional probabilities. For Büchlein,

and for Bücherei,

Given that 0 < ε_Büchlein < 1 < ε_Bücherei, one concludes that Büchlein is a compositional expression of the LL type, a free combination of morphemes that do not constitute a conceptual unity. Bücherei, on the other hand, belongs to the HL type and has high chances of being treated as a parsable, collocation-like item (Sinclair Reference Sinclair1991; Mason Reference Mason and Kirk2000; Lindquist Reference Lindquist2009) in which an element that tells us less about its counterpart (suffix -erei in this example) activates general constructional meaning and an element that has a greater predictive power (base Buch) serves as a filler for the construction’s empty slot.

5.1. The contributions of parsable words to their affixes’ productivity

Distinguishing between two types of analysable complex words – compositional and parsable – plays an important role in how we understand and describe the mechanics of morphological productivity. One influential theory claiming that the relationship between affixes’ productivity and analysability is that of strong positive correlation was first formulated by Hay & Baayen (Reference Hay, Harald Baayen, Booij and van Marle2002). As a way to evaluate affixes’ productivity, they used Baayen’s hapax-based measure (Baayen Reference Baayen, Booij and van Marle1991, Reference Baayen, Booij and van Marle1992, Reference Baayen1994, Reference Baayen, Lüdeling and Kyto2009; Baayen & Lieber Reference Baayen and Lieber1991; Baayen & Renouf Reference Baayen and Renouf1996; Plag Reference Plag, Aarts, McMahon and Hinrichs2021), and to evaluate affixes’ analysability, they proposed the notion of parsing ratio. For each affix, its parsing ratio gives us the probability that a certain word with this affix will be decomposed by a language user during access (Hay & Baayen Reference Hay and Harald Baayen2003). Mathematically, parsing ratios are defined as the proportions of forms (types or tokens) that fall above the so-called parsing line given by the following equation: log(base frequency) = 3.76 + .76 * log(derivation frequency) (Hay & Baayen Reference Hay, Harald Baayen, Booij and van Marle2002).

Using this set of measures, Hay and Baayen found i) a significant inverse relationship between token frequency and the proportion of tokens that are parsed and ii) a significant positive relationship between the proportion of tokens that are parsed and Baayen’s productivity measure. Based on these results, authors claimed that i) ‘the more often you encounter an affix, (…) the less productive that affix is likely to be’ and that ii) ‘the more often we encounter an affix (…), the less likely we are to parse words containing it’ (Hay & Baayen Reference Hay, Harald Baayen, Booij and van Marle2002: 219). Thus, their main result was linking analysability and productivity together.

The notion of parsing ratio builds upon the logic of relative frequency account of analysability. While this approach seems perfectly justified for words of the HH type (which are non-analysable and thus cannot bring anything to the productivity of their affixes) and words of the LL type (which are compositional and hence bear witness to their affixes’ wide applicability), the picture is not as clear with parsable words of the LH and HL types. As already discussed, the derivation to base frequency ratio, whether calculated for stand-alone bases or for morphological families, can unpredictably conflate these words either with their non-analysable or compositional counterparts. For example, among my English experimental stimuli, both decrease (LH) and deforest (LL) made it above the parsing line while both debunk (HL) and describe (HH) fell below it.

Most importantly, it remains unclear what contribution LH and HL words really make to the overall morphological productivity of their affixes. On the one hand, the derivational elements in HL multi-morphemic words or multi-word expressions in German, Russian, and English, being fixed by construction, are often called semiproductive in the literature (Jackendoff Reference Jackendoff2002) in the sense that they have input limitations, that is, do not accommodate every base that is semantically compatible with the preverb, prefix, or particle (McIntyre Reference McIntyre, Dehé and Wanner2001; Blom Reference Blom2005). On the other hand, as observed in the Russian part of my experiment, derivational elements that fill in empty slots of LH words, although easily analysable, can be completely disregarded by speakers if the meaning of the whole construction significantly overlaps with the meaning of its base. Taking all of this into account, I would expect a high proportion of parsable words among all derivations with a certain affix to be indicative of this affix’s limited morphological productivity.

5.2. Data collection

In order to analyse the relation between the analysability of English and Russian prefixed LH and HL words and the productivity of their prefixes, I used the following two measures. The parsability ratio of a prefix was calculated as the proportion of words for which the absolute difference between P (affix | base) and P (base | affix) is greater than 1% (as a threshold value suggested by my experimental stimuli) among all words with this prefix. The English data, obtained from WordNet (Fellbaum, Reference Fellbaum1998), comprised a total of 25,816 words with the following 24 prefixes: anti-, con-, counter-, cross-, de-, dis-, em-, en-, fore-, im-, in-, inter-, mid-, mis-, non-, out-, over-, pre-, re-, sub-, super-, trans-, un-, and under-. (Some authors might not view elements like over- or super- as prefixes but as combining forms; however, I trod here a conventional path, relying on the authoritative opinion of the Oxford English Dictionary.) The Russian data, obtained from Tikhonov’s morphemic dictionary (Tikhonov Reference Tikhonov1985), comprised 9,018 words with the following 27 prefixes: de-, diz-, do-, iz-, na-, nad-, niz-, ob-, pere-, pre-, pro-, po-, pod-, pred-, pri-, raz-, re-, s-, so-, o-, ot-, u-, v-, voz-, vz-, vy-, and za-. The numbers for calculating transitional probabilities were gathered from the same internet corpora of English and Russian that I used while collecting data for the experiments.

As for the linguistic productivity of a prefix, I did not want to use Baayen’s hapax-based measure, because, as has been pointed out in the literature, this measure is ill-suited for the comparison of affixes with different token numbers (Gaeta & Ricca Reference Gaeta and Ricca2006; Pustylnikov & Schneider-Wiejowski Reference Pustylnikov and Schneider-Wiejowski2010; cf. Bauer Reference Bauer2001). Calculating the ratio of the number of hapax legomena with a given affix to the total number of tokens with that affix is likely to result in overestimating the productivity values of less-frequent constructions, which is undesirable for the purposes of this study. Instead, I used the algorithm suggested by Monakhov (Reference Monakhov2023b) and evaluated the morphological productivity of the prefixes in my data as their probability to combine with a random base.

The productivity values for English and Russian prefixes to be found in Table 7 (alongside parsability ratios) stand to reason. As a sanity check for the English data, I did the following. First, 995 random content words (nouns, verbs, and adjectives) without prefixes were sampled from the ententen18 internet corpus, their frequencies ranging from 48,421,599 to 54. Each of 24 English prefixes on my list was coupled with each of those 995 bases so that the bases remained the same for all prefixes. The raw frequencies of all constructed derivations were then queried in the same corpus. The proportions of non-zero hits among all requests were found to be almost perfectly correlated with the probabilistic productivity measures of the respective prefixes in Table 7 (r = 0.94, p < 0 .001). Therefore, it appears that the estimated probabilities are not too far off the mark.

Table 7. Parsability ratios and productivity values for English and Russian prefixes.

5.3. Analysis of English results

The results presented in Table 7 for the English data are plotted in Figure 9. The observable distribution of dots here has a characteristic U shape and is reasonably well modelled by a polynomial regression with two terms. This suggests that parsability ratio, calculated as I propose in this paper, is related to productivity in a very special way. To better understand what is going on, it is important to remember exactly what this ratio signifies: it describes how many lemmas with a certain prefix comprise elements of which one is more or less fixed and another is more or less free to vary. It logically follows that if a prefix is highly productive, the proportion of such lemmas in its output will be low, because the majority of lemmas will be constructions of the LL type with comparably low transitional probabilities.

Figure 9 (Colour online) English prefixes’ parsability and productivity.

This, however, explains only the downward trend in Figure 9. The U-shape pattern suggests that there must be at least one other variable, besides parsability ratio and productivity measure, that influences the distribution of dots in this plot. One cannot but notice that the curve is pulled upwards by the prefix in-, specifically by its prepositional variant (e.g. in expressions like in-place running or in-text citation). Hence, one can hypothesise that the third variable of interest is the frequency of respective prepositions or particles.

Indeed, an ordinary least squares regression model with a prefix’s productivity as the response regressed on two interacting independent variables, namely the prefix’s parsability ratio and the log-transformed frequency of the respective preposition or particle, accounts for a considerable amount of the total variation (Table 8). The obtained coefficients show that for the prefixes that have no free counterparts or correspond to relatively low-frequency prepositions/particles (over, log-transformed frequency of 16.99; out, 17.38), the lower the parsability ratio, the greater the linguistic productivity. In contrast, for the prefixes that have high-frequency free counterparts (in, 19.87), the higher the parsability ratio, the greater the linguistic productivity.

Table 8. Regression model summary (English prefixes).

Note: F(3, 20) = 4.74, p = 0.01, R ² = 0.41.

5.4. Analysis of Russian results

The Russian language provides many more possibilities for this type of analysis. Of the 27 prefixes in my data, 17 not only are historically related to prepositions but also have prepositional counterparts in modern Russian: v- (v ‘in, at’), do- (do ‘to, before’), za- (za ‘for, behind’), iz- (iz ‘from, out of’), na- (na ‘on’), nad- (nad ‘over, above’), o- (o ‘about’), ob- (ob ‘about’), ot- (ot ‘from’), po- (po ‘along, by’), pod- (pod ‘under’), pred- (pered/pred ‘before, in front of’), pri- (pri ‘by, at’), pro- (pro ‘about, of’), s- (s ‘with’), so- (so ‘with’), and u- (u ‘from, by’). The second group of prefixes, which have no prepositional counterparts in modern Russian, encompasses morphemic borrowings, prefixes of non-prepositional origin and prefixes derived from prepositions that are no longer part of the Russian language.

The methodology of calculating parsability ratios and productivity measures for Russian prefixes was exactly the same as that for English. The results provided in Table 7 are visualised in Figure 10. The picture, overall, bears a remarkable resemblance to the U-shape distribution of English prefixes observed in Figure 9. Again, this distribution can be reasonably well approximated by a polynomial regression with two terms.

Figure 10 (Colour online) Russian prefixes’ parsability and productivity.

More telling is the distribution of prefixes if one takes into account their prepositional or non-prepositional natures. For non-prepositional prefixes only, a clear negative linear trend is observable: the lower the parsability ratio, the greater the linguistic productivity. This means that the bewildering U-shape pattern is created solely by prepositional prefixes.

As before, a regression model with a prefix’s productivity as the response regressed on two interacting independent variables – the prefix’s parsability ratio and the log-transformed frequency of the respective preposition – explains a significant amount of the total variation (Table 9).

Table 9 Regression model summary (Russian prefixes).

Note: F(3, 23) = 15.9, p < 0.001, R ² = 0.67.

Thus, Russian prefixes, corresponding to low-frequency prepositions, behave exactly like non-prepositional prefixes: the lower the parsability ratio, the greater the linguistic productivity. On the other hand, those prefixes whose prepositional counterparts are frequent remain highly productive, even though the proportion of lemmas with unilaterally fixed elements in their overall output is significant.

To assess how cognitively plausible this is, let us consider two different prefixes. One is pred- ‘before’, which corresponds to the relatively infrequent (15.7) preposition pred. The prefix pred- has a parsability ratio of 0.4, indicating that 40% of the words with this prefix are of the LH or HL types. The productivity measure of this prefix, on the other hand, is only 0.16, which means that out of 100 random bases, it can be expected to combine only with 16. In comparison, the prefix o- ‘about, around’ has an even higher parsability ratio of 0.59, yet its linguistic productivity is 0.81.

I would argue that the difference here is because the corresponding preposition o is 6.75 times more frequent than the preposition pred. Thus, even though one would expect the prefix o- to be unproductive given its high parsability ratio, the presence of a free element coinciding with it in form and partially overlapping in meaning may facilitate the production of new items.