How many Optimality Theory grammars are there?

How many Optimality Theory (OT) grammars are there? I will assume first off that Con, the constraint set, is fixed and finite. I make this assumption because it is one assumed by nearly all work on the mathematical properties of OT. Given the many unsolved problems in OT learning, joint learning of constraints and rankings seems to add unnecessary additional complexity.¹ So we suppose that there are a finite set of constraints, and let $n$ be the cardinality of that set. If we put aside for now the contents of the lexicon (i.e., the set of URs), one can ask how many possible constraint rankings there are as a function of $n$.

Prince & Smolensky (1993), in their “founding document”, seem to argue that constraint rankings are total. Consider the following from a footnote:

With a grammar defined as a total ranking of the constraint set, the underlying hypothesis is that there is some total ranking which works; there could be (and typically will be) several, because a total ranking will often impose noncrucial domination relations (noncrucial in the sense that either order will work). It is entirely conceivable that the grammar should recognize nonranking of pairs of constraints, but this opens up the possibility of crucial nonranking (neither can dominate the other; both rankings are allowed), for which we have not yet found evidence. Given present understanding, we accept the hypothesis that there is a total order of domination on the constraint set; that is, that all nonrankings are noncrucial.

So it seems like they treat strict ranking as a working hypothesis. This is also reflected in the name of their method factorial typology, because there are $n!$ strict rankings of $n$ constraints. Roughly, they propose that one generate all possible rankings of a series of possible constraints, and compare their extension to typological information.² In practice, as they acknowledge, OT grammars—by which I mean theories of i-languages presented in the OT literature—contain many cases where two constraints $c, d$ need not be ranked with respect to each other. For example, the first chapter of Kager’s (1999) widely used OT textbook includes a “factorial typology” (p. 36, his ex. 53) in which some constraints are not strictly ranked, and this is followed by a number of tableaux in which he uses a dashed vertical line to indicate non-ranking.³

Prince & Smolensky also acknowledge the existence of cases where there is no evidence with which to rank $c$ versus $d$. The situation is important because any learning algorithm which was trying to enforce a strict ranking of constraints would have to resort to a coin flip or some other unprivileged mechanism to finalize the ranking. Such algorithms are in principle possible to imagine, but neither the final step nor the postulate are motivated in the first place.

Prince & Smolensky finally admit the possibility of what they call crucial non-ranking, cases where leaving $c$ and $d$ mutually unranked has the right extension, but $c lt d$ and $d lt c$ both have the wrong extension.⁴ Such would constitute the strongest evidence for viewing OT grammars as weakly ranked, particularly if such grammars are linguistically interesting.

I will adopt the hypothesis of weak ranking; it seems unavoidable if only as a matter of acquisition. If one does, the set of possible rankings is actually much larger than $n!$. For example, for the empty set and singleton sets, there is but one weak ranking. For sets of size 2, there are 3: ${a lt b; b lt a; a, b}$. And, at the risk of being pedantic, for sets of size 3 there are 13:

$a \lt b \lt c$
$a \lt b, c$
$a \lt c \lt b$
$a, b\lt c$
$a, b, c$
$a, c \lt b$
$b \lt a \lt c$
$b \lt a, c$
$b \lt c \lt a$
$b, c \lt a$
$c \lt a \lt b$
$c \lt a, b$
$c \lt b \lt a$

Already one can see that this is growing faster than $n!$ since $3! = 6$.

It is not all that hard to figure out what the cardinality function is here. Indeed, I think Charles Yang (h/t) showed this to me many years ago, and though I lost my notes and had to re-derive it, I’m reasonably sure he came up with is the same solution that I do here. This sequence is known as A000670 and has the formula

$$a(n) = \sum_{k = 0}^n k! S(n, k)$$

where $S$ is the Stirling number of the second kind. Expanding that further we obtain:

$$a(n) = \sum_{k = 0}^n k! \sum_{i = 0}^k \frac{(-1)^{k – i}i^n}{(k – i)!i!} .$$

This is a lot of grammars by any account. With just 10 constraints, we are already in the billions, and with 20 it’s between 2 and 3 sextillion. Yet this doesn’t look all that dramatically different than strict ranking when plot in log scale.

Clearly this is essentially infinite in the terminology of Gallistel and King (2009), and this ability to generate such combinatoric possibilities from a small inventory of primitive objects and operations is something that has long been recognized as a desirable element of cognitive theories. Of course, many of these weak rankings will be extensionally equivalent, and even others, I hypothesize, will probably be extensionally non-equivalent but extensionally equivalent with respect to some lexicon. None of this is unique to OT: it’s just good cognitive science.

Endnotes

Note that the constraint induction of Hayes & Wilson (2008), for example, only considers a finite set thereof and therefore there does exist some finite set for their approach. The same is probably true of approaches with constraint conjunction so long as there are some reasonable bounds.
Exactly how the analyst is supposed to use this comparison is a little unclear to me, but presumably one can eyeball it to determine if the typological fit is satisfactory or not.
As far as I can tell, he never explains this notation.
There is a tradition of using $\ll$ for OT constraint ranking but I’ll put it aside because $\lt$ is perfectly adequate and is the operator used in order theory.

References

Gallistel, C. Randy and King, A. P. 2009. Memory and the Computational Brain: Why Cognitive Science Will Transform Neuroscience. Wiley-Blackwell.
Hayes, B. and Wilson, C. 2008. A maximum entropy model of phonotactics and phonotactic learning. Linguistic Inquiry 39: 397-440.
Prince, A., and Smolensky, P. 1993. Optimality Theory: constraint interaction in generative grammar. Rutgers Center for Cognitive Science Technical Report TR-2.

Neural fossils

Neural network cognitive modeling had a brief, precocious golden era between 1986 (the year the Parallel Distributed Processing books came out) and maybe about 1997 (at which point the limitations of those models were widely known…though I’m little fuzzier about when this realization settled in). During that period, I think it’s fair to say, a lot of people got hired into the faculty, in psychology and linguistics in particular, simply because they knew a bit about this exciting new approach. Some of those people went on to do other interesting things once the shine had worn off, but a lot of them didn’t, and some of them are even still around, haunting the halls of R1s. I think something similar will happen to the new crop of LLMologists in the academy: some have the skills to pivot should we reach peak LLM (if we haven’t already), but many don’t.

Email discipline

There is a Discourse on what we might call email discipline. Here are a few related takes.

There are those who simply don’t respond to email at all. These people are demons and you should pay them no mind.

Relatedly, there are those who “perform” some kind of message about their non-email responding. Maybe they have a long FAQ on their personal website about how exactly they do or do not want to be emailed. I am not sure I actually believe these people get qualitatively more email than I do. Maybe they get twice as much as me, but I don’t think anybody’s reading that FAQ buddy. Be serious.

There are those who believe it is a violation to email people off-hours, or on weekends or holidays, or whatever. I don’t agree: it’s an asynchronous communication mechanism, so that’s sort of the whole point. I can have personal rules about when I read email and these depend in no way on my rules (or lack thereof) about when you send them. Expecting people to know and abide by your Email Reading Rules FAQ is just as silly.

I have an executive function deficit, diagnosed as a child (you know the one), and if you’re lucky, they teach you strategies to cope. I think non-impaired people should just model one of the best: email can’t be allowed to linger. If it’s unimportant, you need to archive it. If it’s important you need to respond to it. You should not have a mass of unopened, unarchived emails at any point in your life. It’s really that easy.

When LLMing goes wrong

[The following is a guest post from Daniel Yakubov.]

You’ve probably noticed that industries have been jumping to adopt some vague notion of “AI” or peacocking about their AI-powered something-or-other. Unsurprisingly, the scrambled nature of this adoption leads to a slew of issues. This post outlines a fact obvious to technical crowds, but not business folks; even though LLMs are a shiny new toy, LLM-centric systems still require careful consideration.

Hallucination is possibly the most common issue in LLM systems. It is the tendency for an LLM to prioritize responding rather than responding accurately, aka. making stuff up. Considering some of the common approaches to fixing this, we can understand what problems these techniques introduce.

A quick approach that many prompt engineers I know think is the end-all be-all of Generative AI is Chain-of-Thought (CoT; Wei et al 2023). This simple approach just tells the LLM to break down its reasoning “step-by-step” before outputting a response. While a bandage, CoT does not actually inject new knowledge into an LLM, this is where the Retrieval Augmented Generation (RAG) craze began. RAG represents a family of approaches that add relevant context to a prompt via search (Patrick et al. 2020). RAG pipelines come with their own errors that need to be understood, including noise in the source documents, misconfigurations in the context window of the search encoder, and specificity of the LLM reply (Barnett et al. 2024). Specificity is particularly frustrating. Imagine you ask a chatbot “Where is Paris?” and it replies “According to my research, Paris is on Earth.” At this stage, RAG and CoT combined still cannot deal with complicated user queries accurately (or well, math). To address that, the ReAct agent framework (Yao et al 2023) is commonly used. ReAct, in a nutshell, gives the LLM access to a series of tools and the ability to “requery” itself depending on the answer it gave to the user query. A central part of ReAct is the LLM being able to choose which tool to use. This is a classification task, and LLMs are observed to suffer from an inherent label bias (Reif and Schwarz, 2024), another issue to control for.

This can go for much longer, but I feel the point should be clear. Hopefully this gives a more academic crowd some insight into when LLMing goes wrong.

References

Barnett, S., Kurniawan, S., Thudumu, S. Brannelly, Z., and Abdelrazek, M. 2024. Seven failure points when engineering a retrieval augmented generation system.
Lewis, P., Perez, E., Pitkus, A., Petroni, F., Karpukhin, V., Goyal, N., …, Kiela, D. 2020. Retrieval-augmented generation for knowledge-intensive NLP tasks.
Reif, Y., and Schwartz, R. 2024. Beyond performance: quantifying and mitigating label bias in LLMs.
Wei, J. Wang, X., Schuurmans, D., Bosma, M., Ichter, B., Xia, F., Chi, E., …, Zhou, D. 2023. Chain-of-thought prompting elicits reasoning in large language models.
Yao, S., Zhao, J., Yu, D., Shafran, I., Narasimhan, K., and Cao, Y. 2023. ReAct: synergizing reasoning and acting in language models.

Snacks at talks

The following is how to put out a classy spread for your next talk; ignoring beverages and extras, everything listed should ring up at around $50.

The most important snack is cheese. Yes, some people are vegan or lactose-intolerant, but cheese is one of the most universally-beloved snacks world-wide. Most cheeses keep for a while with refrigeration, and some even keep at room temperature. Cheese is, as a dear friend says, one of the few products whose quality scales more or less linearly with its price, and I would recommend at least two mid-grade cheeses. I usually buy one soft one (Camembert, Brie, and Stilton are good choices) and one semi-hard one (Emmental or an aged Cheddar for example). The cheese should be laid out on a cutting board with some kind of metal knife for each. The cheese should not be pre-cut (that’s a little tacky). Cheeses should be paired with a box of Carr’s Water Crackers or similar. Estimated price: $15-20.
Fresh finger vegetables are also universally liked. The easiest options are finger carrots and pre-cut celery sticks. If you can find pre-cut multi-color bell peppers or broccoli, those are good options too. You can pair this with some kind of creamy dip (it’s easy to make ranch or onion dip using a pint of sour cream and a dip packet, but you need a spoon or spatula to stir it up) but you certainly don’t have to. Estimated price: $10-20.
Fruit is a great option. The simplest thing to do is to just buy berries, but this is not foolproof: blueberries are a little small for eating by hand; raspberries lack structural integrity, and where I live, strawberries are only in season in the mid-summer, and are expensive and low-quality otherwise. In Mid-Atlantic cities, there are often street vendors who sell containers of freshly-cut fruit (this usually includes slices of pineapples and mangos and bananas, and perhaps some berries) and if this is available this is a good idea too. Estimated price: $10-15.

This, plus some water, is basically all you need to put out. Here are some ways to potentially extend it.

Chips are a good option. I think ordinary salty potato chips are probably the best choice simply because they’re usually eaten by themselves. In contrast, if you put out tortilla chips, you need to pair them with some kind of salsa or dip, and you need to buy a brand with sufficient “structural integrity” to actually pick up the dip.
Nuts are good too, obviously; maybe pick out a medley.
Soda water is really popular and cheap. I recommend 12oz cans. It should always be served chilled.
A few bottles or cans of beer may go over well. With rare exceptions, should be served chilled.
A bottle of wine may be appropriate. Chill it if it’s a varietal that needs to be chilled.

If the talk is before noon, coffee (and possibly hot water and tea bags) is more or less expected. There is something of a taboo in the States of consuming or serving alcohol before 4pm or so, and you may or may not want your event to have a happy hour atmosphere even if it’s in the evening.

And here are a few things I cannot recommend:

In my milieu it is uncommon for people to drink actual soda.
I wouldn’t recommend cured meats or charcuterie for a talk. The majority of people won’t touch the stuff these days, and it’s pretty expensive.
I love hummus, but mass-produced hummus is almost universally terrible. Make it at home (it’s easy if you have a food processor) or forget about it.
Store-bought guacamole tastes even worse, and it has a very short shelf life.

Introducing speakers

The following are my (admittedly normative) notes on how to introduce a linguistics speaker.

The genre most similar to the introduction of speaker is the congratulatory toast. An introduction should be brief, and the lengthy written introduction should be scorned. The speaker is already making an imposition on the audiences’ time, and for the host to usurp more of this time than necessary is a further imposition on both host and audience.
The introduction is not an opportunity for the introducer to demonstrate erudition, but it can be an opportunity to show wit.
The introduction should be in the introducer’s voice. For this reason, a biography paragraph provided by the speaker should not be read as part of the introduction.
The introduction should be exemporaneous. The introducer can prepare brief notes, but they should fit on a notecard or their hand, and the notes should never be “read”.
Polite humor, brief personal anecdotes (e.g., when the introducer first met the speaker or became aware of their work), and heart-felt superlatives or compliments (one of the nicest introductions I ever received stated that I was “in the business of keeping people honest”) are to be encouraged.
The introduction should state the speakers’ current affiliation and title, if any, but need not list their full occupational or educational history unless it is judged relevant.
Introducers may feel an urge to read the title of the talk when concluding their introduction, but should resist this urge. There is no real need—the audience already has seen the talk title in the program or other announcements, and they can read the slide—and the speaker normally feels the need to read it out loud regardless.
The introduction should conclude with the speaker’s name. In one common style, which I consider elegant, the introducer is careful not to say the speaker’s full name until this conclusion, and uses epithets like “our next speaker” or “our honored guest” earlier in the introduction.

Libfix report: -gler and what we can learn from it

The libfix -gler is an interesting case that seems to illustrate Zwicky’s hypothesis that blends lead to libfixes. Patient zero is clearly Googler, which is corporate’s preferred term for Google employees; it is is widely used in-group as well. This is an ordinary example of the relatively productive -er suffix that creates agent nominals (backbencher, J6er), with a connotation that the agent does something habitually (e.g., pickleballer) or as an occupation (cartographer). That a Googler is someone who works at Google, and not just a habitual user of the search engine, is slightly notable but not shocking.

The next best-established forms look more like blends based on Googler. First and most saliently, there is noogler ‘new Google employee’, where the word-onset has been replaced to create a blend with new. Members of an official affiliational group (/listserv) for Jewish Googlers call themselves Jewglers, with a similar single-phoneme substitution. Since both new and Jew share the /Cuː-/ initial of Google(r)—the only adaptation is changing the place and manner of the initial C—this still looks like blending rather than recutting. Xoogler ‘former employee of Google’ is presumably pronounced ex-oogler (I’ve never heard it said out loud myself); the term is commonly used by entrepeneurs in their fund-raising (cf. xoogler.co); this could be a blend or just regarded as a one-off truncation of ex-Googler.

Cats and cows (and ball pythons) are not permitted at Google’s offices, but I have seen mewgler and moogler for cat- and cow fancier Google employees. However, Googlers are permitted to bring their (well-behaved, vaccinated) dogs to work, and employees who do so call themselves dooglers. (Then again, my colleague LeeAnn says the dogs themselves are the dooglers!) My intuition is that this is pronounced [duː.glɚ] and not *[dɒ.glɚ], and thus this is less blend-like than any of the aforementioned examples, because dog and the base Googler have a different vowel (albeit both back vowels). The same is true for Zoogler for Google employees based out of the Zurich office since the first vowel in the US English pronunciation of Zurich is [ʊ], and even we get a seemingly more dissimilar-to-base front-gliding diphthong in gaygler, the term for members of the company-internal LGBT affiliation group (/listserv).

In my analysis, doogler, Zoogler and gaygler strongly suggest that we have gone from a blend with Googler as its base to an incipient liberated affix -gler denoting agents associated with Google.

The biggest puzzle about English libfixes, for me, one not answered in any of the prior work, is why the recutting occurs where it does. It is perhaps not surprising that the rather-homophonous -er has not been given yet another sense, but why is it -gler and not -0ogler (which would give us the not-preposterous, but unattested *gayoogler) or -ler (*gayler)? While Zwicky’s cline hypothesis does not answer this, here is one possible way to operational this: recutting is blend reanalysis, with the source morphemes like new and Jew are parsed maximally in noogler and Jewgler, with the remainder giving us the new affix -gler.

I know of many other libfixes consistent with this analysis. For example, from the blend fursona (fur + persona) we have –sona (e.g., catsona, puppysona); from glitterati (glitter + literati) we have -rati (e.g., Twitterati, technorati), from funtastic (fun + fantastic) we have -tastic (e.g., chavtastic, shagtastic), from telethon (telephone + marathon) we have -(a)thon (e.g., saleathon, mathathon).

This is of course not the full story. Most other libfixes in my corpus seem to arise at a prexisting morpheme boundary, with one or the other piece reinterpreted as a productive affix with new lexical semantics based on the full form. Some examples include cran- from cranberry (e.g., crantini), -gate from Watergate (e.g., Troopergate), -mare from nightmare (e.g., editmare), and -berg from iceberg (e.g., fatberg). In many cases, the morpheme boundaries are abstract ones mirroring the segmentation of Latin or Greek complex word borrowings as in -(i)verse from the Latin-based universe (e.g., Buffyverse) or -(o)nomics from the Greek-based economics (e.g., the brand name Chemonomics). My corpus also includes Franken– from the German compound name Franken-stein (e.g. Frankenfood) and -nik from the Russian derivative s-put-nik (e.g., peacenik). In the above cases, the segmentation is more or less the same as in the donor language.

In other cases, though, the segmentation is different than that in the donor language, as in -ohol(ic) ultimately from Arabic, which retain a little less of the base than one might expect (here al- is the Arabic definite prefix), and -copter from neo-Greek and -nado from Spanish, which both retain a little more. While I don’t think it’s that controversial to posit that cranberry, economics or universe are represented as complex nouns (our understanding of the morphophonology of English seems to depend on this conclusion, and virtually all behavioral research on word processing supports this), it is perhaps not shocking that ordinary English speakers are unfamiliar with the morphology of the etyma of alcohol, helicopter, and tornado in the ultimate donor languages. These more ad-hoc recuttings don’t necessarily line up with syllable boundaries—though hypothetical *-cohol or *-ler would have—so I suspect there is just some inherent stochasticity to how this final type of recutting proceeds.

Tutorial on Substance-Free Logical Phonology

A few days ago, we (myself with the help of graduate student Rim Dabbous and professor Charles Reiss) gave a detailed tutorial on Substance-Free Logical Phonology (LP) at the LSA meeting in Philadelphia. I was pleasantly surprised with how many people showed up (I printed about 30 handouts, and we ran out) and how engaged they were. If the question period is any indication, our colleagues understood the theory quite well. The handout is now on LingBuzz and we have a lot more material coming for you soon.

I’ll try to summarize how LP fits into theories of exceptionality in a separate post soon.

Postscript

I also took the liberty of creating a simple archive for our Logical Phonology papers, the Logical Phonology Archive (LOA for short, pronounced [lwa] like French loi ‘law’ or Kreyòl loa ‘vodou spirit’). Technically, this site is sort of interesting: other than some static text, the table containing papers is an HTML iframe, dynamically generated from a Google Sheet using a template populated by server-side Javascript using the Google Apps Scripts platform; hat tip to my colleague Rivka Levitan for making me aware of this very handy possibility.

Underspecification: the Inkelasverse

Before I continue the series on theories of lexical exceptionality, I need to pause for a minute and review some work by Sharon Inkelas and colleagues: Inkelas and Cho 1993, Inkelas 1995, Inkelas and Orgun 1995, Inkelas et al. 1997. Inkelas and colleagues’ work has recently been generalized by Charles Reiss and colleagues (e.g., Bale et al. 2014, Reiss 2021, and Gorman & Reiss 2024, the latter under review and available on LingBuzz), turning Inkelas’s original idea into a generalized theory of exceptionality, and I’ll review that before long, in a separate post. I don’t consider the Inkelas(verse) approach to represent a complete theory of exceptionality and so I am not presenting them as such.

Preliminaries

A few other caveats are in order before I begin. First, underspecification is hardly a novel idea: it wasn’t even a new one when it first became a major subject of consideration in the 1980s. The American structuralists espoused the axiom of biuniqueness, which in some sense presages underspecification. At a high level, biuniqueness holds that there is a one-to-one correspondence between the phonemic and allophonic (“phonetic”) level. In practice, this rules out the correspondence modern phonologists term neutralization, in which the allophones of two distinct phonemes overlap. Formally, neutralization occurs when two phonemes /x, y/ (where x ≠ y) both have an allophone [z] (regardless of whether z ∈ {x, y} or not). A well-known example of neutralization in many dialects of English (Joos 1942) is the intervocalic [ɾ] in writer and rider. On the basis of write and ride, one would be tempted to set this up as an allophone of /t, d/, respectively, but this neutralization violates biuniqueness.¹

This axiom is generally consonant with the overall late structuralist search for a deterministic discovery procedure. One is tempted to liken the late structuralists to alchemists, a workable discovery procedure to the philosopher’s stone, and their failed search to the magnum opus.

The structuralists imposed no biuniqueness constraint on a more abstract, poorly-theorized “morphophonemic” level which would, apparently, contain generalizations like English post-tonic flapping. The morphophonemic level was poorly theorized so how exactly this ought to work is a little unclear. The archiphoneme, popularized by the Prague School structuralists Nikolai Trubetzkoy and Roman Jakobson, suggests one possibility. An archiphoneme is a sort of morphophonemic symbol not yet specified for features which are to be later neutralized. By convention they are denoted in derivations with capital letters not otherwise used for IPA symbols. In the case of post-tonic flapping, one might write the flap character as the arciphoneme /D/, and propose allomorphic or morphophonemic rules which insert it into writer and rider in place of /t, d/. In the Inkelasverse, archiphonemes are simply a way to annotate underspecification (here presumably of the Voice feature that distinguishes /t, d/), and their convention of using capital letters for underspecified segments calls back to this tradition.

Given the archiphonemic “escape hatch” and noting that many late structuralists also published influential morphonemic sketches (e.g., Bloomfield 1939, Chomsky 1951, Hamp 1953), one might reasonably question how closely held the biuniqueness axiom really was at the time. Chomsky and Halle (e.g., Halle 1959, Chomsky 1964, Chomsky & Halle 1965) argue that biuniqueness introduces a great deal of duplication, with morphophonemic rules reprising the same generalizations as phonemic ones. They thus propose to reject biuniqueness and to merge the phonemic and morphophonemics into a single level of description. Yet we see that Chomsky and Halle got a lot of pushback (see cited papers) from late structuralist éminences grises, so their rejection of biuniqueness was not uncontroversial at the time.

Secondly, there are a rather-distinct set of arguments, not considered in any detail by Inkelas and colleagues, for surface underspecification. Keating (1988), for example, provides phonetic arguments for the use of underspecification. If one permits underspecification on the surface, it is not obvious why it should not also be permitted in earlier stages of the derivation.

Jumping forward a bit, underspecification theory received a lot of attention in the late 1980s and early 1990s, but interest seems to have waned after Donca Steriade’s chapter in the first Blackwell Handbook of Phonology (1995), which reviews a number of theories of underspecification, taking a dim view of the overall enterprise. Steriade’s review is very good—I would recommend you read it if you haven’t—but as I remember it, the critiques mostly do not apply to the Inkelasverse.

Inkelas and Cho 1993

Inkelas and Cho (1993; henceforth I&C) propose a theory in which underspecification (or its absence, prespecification) is used to derive a number of interesting effects. Their slogan is: prespecification is inalterability; one might add that thus underspecification is mutability.

I&C begin with a common-place observation that geminates are often immune to processes which singleton segments undergo. They review attempts to derive this effect using autosegmental representations and various rule-application conventions, but conclude that no set of representations and conventions work in general. I&C propose that it is not geminate status per se but rather a more general convention of prespecification that is at play.

For example, in certain dialects of Berber, singleton consonants spirantize (and voice if they are pharnygealize), but geminates are always voiceless stops. Note that this is not merely a static generalization: the singleton and geminate forms are morphologically related, because these languages have templatic morphology and certain inflectional and/or derivational forms call for consonants to be geminated. Noting that Continuancy and Voice are generally non-contrastive for consonants, I&C propose:

(1) Berber consonants (after I&C, §5.2.1):
a. consonants, either singleton or geminate, are not underlyingly specified for Continuant or Voice.
b. a geminate-specific feature-filling rule inserts [−Continuant], and if [+Pharyngeal], [-Voice].²
c. a later feature-filling rule inserts [+Continuant], and if [+Pharyngeal], [-Voice].

Crucially, rule (1b) bleeds rule (1c) by virtue of being feature-filling, and thus geminates are exempt. This opens up a few new avenues to consider.

First, they note that some of the relevant dialects of Berber have apparent lexical exceptions: voiced, continuant geminates […ɣɣ…]. They propose that these apparent exceptions can be prespecified [+Voice, +Continuant]. Given the condition that rules (1bc) are feature-filling, these “exceptions” are not really exceptional at all! I&C note (§5.7) that this is a highly-restrictive theory of lexical exceptionality compared to earlier approaches but claim some advantages as well.

In a footnote (p. 556, fn. 26), I&C note that they cannot handle cases in which “exceptionality takes the form of failure to trigger, rather than failure to undergo, a rule” which “remain a problem for us until they can be resolved in a representational fashion”.

Secondly, they observe that the account given in (1) is in no way specific to geminate status. It requires, of course, that (1b) can target geminates to the exclusion of singletons, but this follows from their assumption that geminates are prespecified with two moras and a standard interpretative procedure; i.e., if a rule specifies that a target is linked to two moras, a segment linked to only one is not a target.³ Thus, this account generalizes to instances of “inalterability” which are not related to geminacy. For example, exceptions to vowel harmony in Turkish may simply reflect pre-specified Back and Round features which are “inalterable” with respect to structure-filling harmony rules.

Third, they note that this sort of account cannot be applied to structure-changing rules, and find no counterexamples. Indeed, they suggest, as many others have, that so-called structure-changing rules are decomposed into a feature-delinking rule followed by feature-insertion (or spreading) rule.

I note that a few examples given by I&C, such as Klingenheben’s Law in Hausa, are of no obvious relevance to synchronic phonology. For example, this law can be thought of as a diachronic change, which may or may not have had a synchronic analogue in the past, or as a static phonotactic constraint deriving (diachronically) from said change, which may not have any synchronic force.

I&C take their proposal to be an instance of Kiparsky’s notion of radical underspecification (RU), according to which features are binary and only marked feature values are present underlyingly. Indeed, their account of inalterability as prespecification is a good argument against privativity and thus for binarity. For instance, if Voice is privative (and thus equivalent to [+Voice] in a binary system), then there is no way for (1b) to block (1c). However, in my opinion RU does not acquit itself well in I&C’s account. They note some cases where it conflicts with their assumptions about feature markedness. Furthermore, it’s not clear to me what the presence of prespecified faux-exceptions I&C posit like the […ɣɣ…]s in Berber dialects means for RU.

Inkelas 1995

Inkelas (1995) attempts to relate some of I&C’s earlier insights to ideas in Optimality Theory (OT).

Inkelas begins with some now-famous data from Turkish. There are three kinds of consonant-final roots: those in which the final consonant is consistently voiceless (e.g., sanat/sanatı ‘art’; the second form is the accusative), those which are consistently voiced (etüd/etüdü ‘etude’), and those which alternate (e.g., kanat/kana[d]ı ‘wing’). Short of root suppletion, it is hard to imagine how one would handle this three-way distinction without making use of underspecification. Inkelas proposes, naturally, that the alternating kanat is underlyingly /kanaD/, where /D/ denotes a coronal stop underspecified for Voice, and the non-alternating final consonants are prespecified for Voice. This is once again an instance of prespecification as inalternability, because presumably the processes which insert final voice specifications are feature-filling and thus do not apply to fully-specified root-final /t, d/.

Inkelas considers but rejects an alternative using morpheme-level exceptionality, an idea pursued in some detail in SPE and subsequent work and still in vogue in OT circles. She suggests this is no different than introducing morpheme-specific multiple grammars, and that with this tool in place, it is hard to know when to stop proliferating grammars. For instance, while it might be sensible to treat one of the three voicing patterns in Turkish as exceptional, giving us two grammars, one could just as well derive all three patterns using the same underlying stem-final /d/ and three different grammars. In my opinion, one might equate “exceptionality” with the lack of productivity, and thus use standard diagnostics (e.g., perhaps child product errors, wug-tests, or evidence from loanword adaptation) and/or theories (e.g., Tolerance) to address this issue, and one might also argue against grammar proliferation on evaluation-metric grounds. But I agree that the morpheme-level theory is radically underconstrained without additional theoretical commitments along these lines.

At the same time, Inkelas also claims the use of underspecification to account for such phenomena as unconstrained. She suggests that the notion of lexicon optimization (LO), slightly revised, provides an appropriate constraint.

She points out that LO, as originally defined, is intended to pick out an “faithful” UR for cases of pure allophony, and proposes a variant of LO which is sensitive to the presence of alternations (her 16). Basically, the principle is to pick as the UR of a morpheme M the form which is most harmonic with respect to all allomorphs of M. While I understand the motivation here, this does not seem to be formal enough to implement. For instance, let us suppose S is the set of surface forms of M, and let I be the set of candidates for the UR. Note also that S and I may be disjoint (i.e., elements of I need not be observed surface forms), an assumption which will become important in a moment. Let us assume that, somehow, the grammar is already known. Finally, suppose that the child has computed, for each hypothetical UR in I, a tableau for surface forms s with respect to the grammar; there will be |S| × |I| such tableaux. Then, for candidate UR i, how exactly is the child to combine the harmony scores of |S| tableaux so that it can be compared to the scores for candidate UR j? In the cases Inkelas considers, |I| = 3 (because she considers, e.g., in the Turkish case, /t, d, D/), and the winning i harmonically bounds the other candidate URs, but it is not clear to me that this is generally true. However, I put these big issues aside for sake of argument.

Inkelas argues that, in the case that there are predictable alternations, this revised form of LO will select a UR with the underspecified segment with respect to the alternating features, and the UR will be fully specified otherwise. For instance, if Turkish in fact has high-ranked constraints motivating word-final plosive devoicing, it will select Voice-underspecified /kanaD-/ for kanat, whereas non-alternating sanat and etüd will be fully specified as /t, d/ respectively. This, she shows, is also the case in simpler two-way alternations, like ATR harmony in Yoruba: she argues that the revised form of LO will leave prefix vowels which undergo ATR harmony underspecified for ATR.

A key idea here is that underspecification is not driven by universal principles (as in earlier markedness-based approaches) but is quite intimately tied to the grammar. Indeed, her form of LO depends on the child having already converged on a grammar before setting (or perhaps “optimizing”) the URs.

This paper concludes with a number of implications for the proposal. She compares and constrasts her proposal with the Prague school archiphoneme, which shares some important similarities. More importantly, she notes that underspecification which gives rise to ternarity is an argument against privativity, since there is no way to derive ternarity in a privative feature system. I would add that, since there does not seem to be any principle that governs which features can be underspecified nor any evidence suggesting that some features cannot, there is no evidence that any features are privative.

Inkelas’s proposal is an dialogue with (her advisor) Kiparsky’s notion of absolute neutralization. Kiparsky (1973) introduces this notion to critique SPE‘s roundly-criticized notion that nightingale contains an underlying, never-surfacing velar fricative /x/ which blocks trisyllabic shortening, and what he actually critiques is positing an underlying /x/ which is deleted in all surface forms. Here the “absolute” part of “absolute neutralization” could easily be referring to the fact that /x/ is not merely neutralized, but entirely deleted. However, his proposed remedy, the alternation condition, is stated more broadly:

(2) Alternation Condition (Kiparsky 1973:65): neutralization processes cannot apply to all occurrences of a morpheme.

This makes it reasonably clear to me that he intends “absolute” to refer to the fact the neutralization (which just happens to be deletion in this case) occurs in all cases (i.e., across the board).

It seems to me that Inkelas’s proposal not only conforms to (2), at least in simple cases, it actually provides a glimpse into how the child builds URs that conform to (2). This is a marked improvement.

While Inkelas seems to view her proposal as closely tied to OT and its assumptions, I do not necessarily agree: I would argue that any coherent theory of phonology will have some analogue to lexicon optimization, a point already anticipated in the SPE-era evaluation metric.

Inkelas and Orgun 1995

Inkelas and Orgun (1995; henceforth I&O) attempt to merge the previous proposals about underspecification into a larger theory of morphological structure. Their object of study is a partial sketch of Turkish morphophonology, and as such, interacts with Inkelas and colleagues’ earlier discussion of various phonological phenomena in Turkish.

While it is not all that germane to the Inkelasian theory of underspecification, I should briefly summarize the novel elements of their morphological theory. First, I&O propose that morphemes are “prespecified” for the levels in which they are introduced. Secondly, I&O propose that these levels may have different phonology. Neither of these proposals is controversial, and echo, e.g., Siegel (1979 [1974]). More interestingly, though, they propose that (partial) forms are subject only to the phonology of the levels at which they derive, a principle they call level economy. Level economy means that, for example, a form which lacks level 3 morphemes is not subject in any way to level 3 morphology, and when a level 2 morpheme is inserted, the resulting (possibly partial) form is not subject to level 1 morphology, and so on.

One case of relevance to underspecification is their new take on Turkish ternary voicing (I&O:776f.), discussed earlier. They claim that monosyllabic roots are all of the non-alternating variety: e.g., at/atı ‘horse’ or ad/adı ‘name’. Their analysis of this restriction is novel: they propose that a hypothetical /aD-/ would be identical to at/atı because of the interaction a between final-consonant invisibility, a minimality condition which prevents this from applying to monosyllables, and level prespecification to exempt certain stems from minimality and thus preventing final-consonant invisibility.

Later in the paper (op. cit.:787f.) it is revealed that this generalization has many exceptions. I&O search a Turkish dictionary for plosive-final monosyllables, and filtering out those not known to the second author, a native speaker of Turkish. In total, there are 4 monosyllables ending in a plosive which are voiced and non-alternating, 77 which are voiceless, and 17 which are alternating. I&O propose that the 17 alternating stems are prespecified as level 1 morphemes, which ultimately prevents them from undergoing devoicing at level 1.

I for one am not convinced there are any meaningful generalizations here. First, these frequencies could very well be historical accident (a point made in a different context by Inkelas et al. 1997:402), and there is no evidence children acquiring Turkish attend to them. Indeed, the other elements of their analysis (minimality and final-consonant invisibility) are either motivated by other exception-filled generalizations (e.g., their analysis of velar drop), and some of these are static conditions on URs which may also be ignored by children. Secondly, it is not clear why I&O derive the “exceptionality” of alternating final plosives from their morphological prespecification, whereas non-alternating voiced stems, which are much rarer, are instead treated phonologically. I&O write that in their analysis, “[n]onalternating voiced final plosives are exceptional among monosyllables…we account for them through the prespecification of [+voice]”. But by their analysis alternating final plosives are in the same sense “exceptional” in lacking a voice specification: their use of level prespecification solves a problem that I&O have themselves introduced by making feature-changing devoicing as a level 1 process.

In conclusion, I much prefer the simpler analysis of ternary voicing provided by Inkelas (1995), which does not make mention of final-consonant invisiblity, minimality, or level prespecification.

Inkelas et al. 1997

In our final paper, Inkelas et al. (1997) provide a detailed critique of morphemic exceptionality, which they call cophonology. Much of their discussion is based on ternary voicing in Turkish, and a proposed static generalization known as labial attraction; the second is the ternary voicing alternation.

You have already heard what both Inkelas and colleagues and I have to say about ternary voicing; it is interesting in this regard that they cite, but do not rehearse, the more complex (and IMO problematic) analysis proposed by I&O two years earlier.

But before I tell you what they have to say, I should briefly discuss labial attraction. As it happens, chapter 3 of my dissertation (Gorman 2013) actually discusses labial attraction in some detail. This is in the context of a larger critique of phonotactic theory and its relation to ordinary phonology: I am in this work claiming that many phonotactic patterns derive directly from the occulting effect of phonological processes, and I argue that others are not necessarily internalized by speakers. I first perform a simple statistical analysis of labial attraction in Turkish using a digital lexicon of the language (Inkelas et al. 2000). In the analysis, I compare the frequency of disharmonic …aCu… to harmonic …aCı… sequences, conditioned on whether the intervening consonant is labial or not. As it happens, there is significantly more disharmony when the intervening consonant is non-labial, suggesting the proported lexical trend is not a trend at all.⁴ Secondly, I statistically reanalyze the results of a nonce word judgment task conducted by Zimmer (1969).⁵ This reanalysis finds that Zimmer’s Turkish speakers fails to find a significant effect of labial harmony: that is, they do not prefer disharmonic, labial-attraction nonce words like pamuz over harmonic nonce words like pamız. Pace Lees and Zimmer, there is no evidence that labial attraction exists, and subsequent theorizing based on it is fruit of the poisonous tree. Indeed, Inkelas et al. are also skeptical at various points about labial attraction. For instance, they write that “because of Lexicon Optimization, the lexicon of Turkish will look the same whether or not Labial Attraction is actually part of the grammar” (op. cit.:412; emphasis theirs), and provide tableaux establishing this point. I agree.

Let us put this all aside for sake of argument. What do Inkelas et al. propose?

First, they again endorse the use of underspecification for ternary alternations proposed by I&C and Inkelas (1995), and the use of underspecification for predictable alternations and prespecification elsewhere as proposed by Inkelas (1995). They express some concern whether prespecification can be made consistent with certain types of synchronic chain shifts.

Secondly, Inkelas et al. also tentatively endorse the notion that “structure-changing” rules ought to be decomposed into structure-deleting and structure-inserting rules; this idea, which has a long precedent, plays a major role in our recent work on Substance-Free Logical Phonology (LP). They express some concern as to whether this reintroduces “conspiracies” where structure-deleting and structure-inserting rules have similar contexts. I for one disregard this concern because I don’t regard Kisseberth’s (1970) notion of conspiracy to be sufficiently formal, and phonologists after Kisseberth have engaged in a sort of motivated cognition in which their intuitions about what is or is not a conspiracy are unduly informed by what kinds of redundancies they are prepared to excise. Let us, for sake of argument, formalize this notion as related to the decomposition proposed by Inkelas et al., using the tools of LP. A definition follows. Let us suppose that there exist two rules of the following form, where A, C, and D are natural classes and B is a set of feature specifications.

(4) A ∖ B / C _ D
(5) A ⊔ B / C _ D

Then, if (4-5) are both present in some grammar and there is no evidence to establish that some other rule applies after (4) but before (5), we will say that (4-5) are in a conspiracy. Our experience applying LP to real data suggests such conspiracies do occur but are not the norm. For instance, consider Gorman & Reiss’s (2024) analysis of Cervara metaphony. First, I repeat Gorman & Reiss’s rules (20-21) as (6-7); these are clearly in a conspiracy as defined above.

(6) [-Low, +ATR] ∖ {-High} / _ C₀ [+High]
(7) [-Low, +ATR] ⊔ {-High} / _ C₀ [+High]

The same is true of Gorman & Reiss’s rules (24-25). But this is not always the case; rules (19), (23) and (26-28) are all unification rules which are dissimilar to any subtraction rule. And it should be noted that Gorman & Reiss’s analysis is essentially a translation of an earlier analysis (Danesi 2022) into LP, and was not engineered to reach this conclusion.

Third, Inkelas et al. rehearse, in longer form, Inkelas’s (1995) earlier argument against cophonologies, and adds in a problem I discussed in an earlier post: cophonologies have little to say about cases in which one segment in a morpheme is exceptional with respect to some process but another is not. Their example, Spanish diphthongization, is not the best one, but Rubach’s discussion of yer-deletion in Polish, discussed in that post, is more germane.

Endnotes

Interestingly, as far as I can tell, Joos, a structuralist in good standing at the time, does not seem to have noticed the apparent violation of biuniqueness nor does he draw attention to the “morphophonemic” nature of flapping forced by biuniqueness.
I grant, sake of argument, that this “if pharyngeal, voiceless” sub-condition, reminiscent of the largely-forgotten SPE-era angle-bracket notation, can be encoded within a single rule, but I would probably prefer to split (1b) into two rules: one for spirantization and one for pharyngeal voicing, and similarly for (1c).
I&C adopt moraic theory, but I suspect that one could do something quite similar with X-theory. I&C also note that intervocalic geminates occupy both coda and onset position and this can have a similar effect. This works so long as syllable position is determined (i.e., syllables are parsed) before the relevant rules apply; it doesn’t require that it be prespecified.
There are other comparisons I could have done here, but I thought this was the most salient comparison. This comparison controls for the frequency at which a is followed by a [+High, +Back] vowel in the following syllable, and thus directly focuses attention on labial attraction as a source of exceptions to harmony.
My reanalysis was necessary because while Zimmer reports that he conducted a statistical analysis of the data, he provides no useful details about his statistical procedures.

References

Bale, A., Papillon, M., and Reiss, C. 2014. Targeting underspecified segments: a formal analysis of feature-changing and feature-filling rules. Lingua 148: 240-253.
Bloomfield, L. 1939. Menomini morphophonemics. Travaux de Cercle Linguistique de Prague 8: 105-115.
Chomsky, N. 1951. Morphophonemics of Modern Hebrew. Master’s thesis, University of Pennsylvania.
Chomsky, N. 1964. Current Issues in Linguistic Theory. Mouton.
Chomsky, N. and Halle, M. 1965. Some controversial questions in phonological theory. Journal of Linguistics 1: 97-138.
Danesi, P. 2022. Contrast and phonological computation in prime learning: raising vowel harmonies analyzed with emergent primes in radical substance free phonology. Doctoral Dissertation, Université Côte d’Azur.
Gorman, K. 2013. Generative phonotactics. Doctoral dissertation, University of Pennsylvania.
Gorman, K., and Reiss, C. 2024. Metaphony in Substance Free Logical Phonology. Submitted. URL: https://lingbuzz.net/lingbuzz/008634.
Halle, M. 1959. The Sound Pattern of Russian. Mouton.
Hamp, E. 1953. Morphophonemes of the Keltic mutations. Language 27: 230-247.
Inkelas, S. and Cho, Y.-M. 1993. Inalterability as prespecification. Language 69: 529-574.
Inkelas, S., 1995. The consequences of optimization for underspecification. In Proceedings of the 25th Meeting of the North East Linguistic Society, pages 287-302.
Inkelas, S. and Orgun, C.O., 1995. Level ordering and economy in the lexical phonology of Turkish. Language 71, 763-793.
Inkelas, S., Orgun, C.O., and Zoll, C. 1997. The implications of lexical exceptions for the nature of grammar. In I. Roca (ed.), Derivations and Constraints in Phonology, pages 393-419. Clarendon Press.
Inkelas, S. Küntay, A., Orgun, C.O., and Sprouse, R. 2000. Turkish Electronic Living Lexicon (TELL). Turkic Languages 4: 253-275.
Joos, M. 1942. A phonological dilemma in Canadian English. Language 18(2): 141-144.
Keating, P. 1988. Underspecification in phonetics. Phonology 5:275-292.
Kiparsky, P. 1973. Phonological representations: Abstractness, opacity and global rules. In O. Fujimura (ed.), Three Dimensions of Linguistic Theory, pages 57-86. TEC.
Kisseberth, C. W. 1970. On the functional unity of phonological rules. Linguistic Inquiry 1(3): 291-306.
Lees, R. B. 1966a. On the interpretation of a Turkish vowel alternation. Anthropological Linguistics 8: 32-39.
Lees, R. B. 1966b. Turkish harmony and the description of assimilation. Türk Dili Araştırmaları Yıllığı Belletene: 279-297.
Siegel, D. 1979. Topics in English Morphology. Garland Publishing.
Steriade, D. 1995. Underspecification and markedness. In J. Goldsmith (ed.), The Handbook of Phonological Theory, pages 114-174. Blackwell.

Languages with two i’s

[I’ve edited this after the fact to add a new case study. -KBG.]

Doing some research on underspecification and its use as an exceptionality-generating device for our work on Logical Phonology, I have encountered an common pattern: there are quite a few languages, of wide genetic and geographic distribution, whose description suggests their phonological inventory contains two underlying segments ordinarily realized as [i(ː)] but showing different phonological behaviors. The following is a brief catalog followed by some discussion.

Barrow Inupiaq

I have already discussed this well-known case, which is discussed by Dresher (2009:§7.2.1), who in turn has the data from Kaplan (1981:§3.22). In a number of dialects of Eskimo-Aleut, a four-phoneme vowel system *i, *u, *ə, *a has been simplified to an ordinary three-vowel system by the merger of *ə into *i. However, in Barrow Inupiaq, there is a “strong i” (< *i) which triggers palatalization of a following coronal consonant whereas “weak i” (< *ə) does not.

(1) Barrow Inupiaq (Kaplan 1981:§3.22, his 27-29):

a. iglu ‘house’, iglulu ‘and a house’, iglunik ‘houses’
b. ini ‘place’, inilu ‘and a place’, ininik ‘places’
c. iki ‘wound’, ikiʎu ‘and a wound’, ikiɲik ‘wounds’

Presumably, the stem-final i in (1b) is weak and the one in (1c) is strong. One possible way to derive this (slightly different from what I said in my earlier post) is to treat palatalization as triggered by [−Back, +High] strong i, leave weak i underspecified for Back, and then use the following late redundancy rule to fill in the missing specification for weak i.

(2) [+High] ⊔ {−Back}

Given the definition of unification, this will non-vacuously apply to weak i, vacuously apply to strong i, and unification with [+High, +Back] /u/ will fail; /a/ does not meet the structural description because it is [−High].

Blackfoot

Charles Reiss (personal communication) brings my attention to a case in Blackfoot. Frantz (2017:§6B) describes the situation clearly:

The initial vowel of stem ipii ‘enter,’ unlike the initial vowel of itsiniki ‘tell a story,’ always causes a preceding k to be replaced by the affricate ks. We will speak of this phenomenon as breaking of k, and of the i which is involved as a breaking i. For any morpheme which begins with i we need to know whether that i is a breaking i or not; if it is a breaking i, then if it immediately follows a morpheme ending in k we know that the k will be replaced by ks.

Naturally enough, Frantz proposes that there are two i‘s in Blackfoot. Thus far, one would expect the analysis to be parallel to our analysis of palatalization in Barrow Inupiaq, with “catalytic” (breaking), fully-specified /i/ and “quiescent” (non-breaking) underspecified segment (which Frantz writes as /I/). But there is there is one wrinkle; a few pages later Frantz (op. cit.:§6D) reports that the second person prefix k- is “impervious to breaking”. Thus, it seems like we also need two k‘s: one “mutable” (breaking), underspecified segment we can write as /K/, and one “inalterable” (non-breaking), fully-specified /k/. Breaking then occurs only in the sequence /…K-i…/.

Czech

Anderson & Browne (1973; henceforth AB) discuss several vocalic alternations in two registers of Czech. I will focus on the literary language (“LL” for AB; henceforth just Czech) and put aside additional complications associated with their analysis of the common spoken register, which makes some possibly problematic simplifying assumptions for sake of argument.

The surface front vowels of Czech are [i, iː, ɛ, ɛː].¹ However, specific instances of these three vowels either trigger or do not trigger palatalization of the preceding consonant.² For example, the masculine animate nom.pl. -i [-i] is palatalizing, but the feminine nom.pl -y [-i] and the gen.pl. -ych [-ix] are not:

(3) sestřin [sɛstr̝in] ‘sister’s’, sestřini [sɛstr̝iɲi] masc.anim. nom.pl.,
sestřiny [sɛstr̝ini] fem. nom.pl., sestřinych [sɛstr̝inix] gen.pl.

The same is true of [ɛ]-initial suffixes. For example, the -ě [-ɛ] prep.sg. suffix is palatalizating, but the -ech [-ɛx] prep.pl. is not (note the -o suffix in the citation form is the neut. nom.sg.):

(4) okno [okno] ‘window’, okně [okɲɛ] prep.sg., oknech [oknɛx] prep.pl.

Czech [i, iː] are reflexes of both Old Czech i(ː) and ɨ(ː), the latter two merging into the former two respectively. This distinction is preserved in the writing system as [i, iː] from earlier ɨ(ː) is written <y, ý> respectively. AB adopt the sensible proposal that this distinction is also preserved phonologically. That is, they suggest non-palatalizing <y, ý> are underlyingly [+High, −Low, −Round, +Back] and palatalizing <i, í> are [+High, −Low, −Round, −Back]. While they do not sketch out a full analysis, presumably <y, ý> are fronted after palatalization has applied. Alternatively (and more in tune with the approach of Gorman and Reiss 2024), one might instead propose that <y, ý> are [+High, −Low, −Round] and underspecified for [Back]. Palatalization will be triggered by a following [−Back] vowel, and then the following unification rule will (non-vacuously) front <y, ý>:

(5) [−Low, −Round] ⊔ {−Back}

As written, this will also also handle the parallel behavior of non-palatalizing <ě>, which gives it the same featural representation as palatalizing <e> [ɛ]. Under either account, though, Czech has two i‘s (in long and short variants) and two e‘s too.

Hungarian

Charles Reiss (p.c.) draws my attention to the existence of a roughly 60 Hungarian noun stems (Siptár & Törkenczy 2000:§3.2.2; henceforth ST) which are apparent exceptions to harmony in that they have a front vowel but select for back-vowel allomorphs. For instance, “regular” víz ‘water’ has the harmonic dative víznek and ablative víztől, and so on. But there are some exceptions, which ST call “antiharmonic”.

(6) Antiharmonic stems (after ST:68, their 33):
a. híd ‘bridge’, hídnak dat.sg. (*hídnek), hídtól abl.sg. (*hídtől)
b. szid ‘scold’, szidhat ‘may scold’, szidó ‘scolding’
c. héj ‘crust’, héjnak dat.sg., héjtól abl.sg.

According to ST, most antiharmonic stems are in i or í as in (6ab), and just a few are in é as in (6c). While these are traditionally understood as lexical (morpheme-level) exceptions, one possible account is to treat antiharmonic í, i, or é as having a different underlying specification than harmonic í, i, and é.

There are a number of possible ways this might work out. According to ST, í, i, and é are not so much “harmonic” but rather “neutral”. They claim the front-harmonic suffix allomorphs (like –nek and –től) are the default allomorphs. Indeed, stems consisting of a non-neutral vowel like á [aː], a [ɒ], o [o] or ó [oː] followed by í, i, and é select the back allomorphs, and do so without exception, suggesting í, i, and é are simply transparent to harmony.

(6) Back-neutral stems (after ST: loc. cit., their 34b):
a. papír ‘paper’, papírnak dat.sg., papírtól abl.sg.
b. dózis ‘dose’, dózisnak dat.sg., dózistól abl.sg.
c. kávé ‘coffee’, kávénak dat.sg., kávétól abl.sg.

If we adopt ST’s perspective, then one possibility is that harmonic-neutral í, i, and é are specified [−Low, −Round] and antiharmonic vowels are [−Low, +Back, −Round]—keeping them distinct from underlying u [u], ú [uː], and ó [oː]—and become [−Back] later in the derivation. The following context-free rules, applied in order after harmony has been computed, ought to suffice.³

(7) [−Low, −Round] ∖ {+Back}
(8) [-Low, -Round] ⊔ {−Back}

(7) gives harmony-neutral and antiharmonic front vowels the same representation, applying non-vacuously only to the latter, and (8) insures that both are realized as [−Back] on the surface. The [-Round] condition is essential to exempt front round vowels—namely ü [y], ű [yː], ö [ø], and ő [øː]—and the [-Low] condition exempts low vowels, which are back-harmonic.

Kashaya

Buckley (1994) describes four unusual rules in Kashaya, given below in extensional (i.e., non-featural) notation.

(9) i -> a / m _
(10) i -> u / d _
(11) V -> a / q _
(12) V -> o / q^w _

While (9-10) appear to be quite regular processes, he notes that 3 out of 21 i-initial suffixes (the inchoative -ibic and the reflexives -iyic’ and -ic’) fail to undergo them. Similarly, most i‘s undergo (11-12) but not the previously mentioned i-initial suffixes. With (11) the result is […ki…] instead of *[…qa…]; with (12) the result is […qo…] instead of *[…q^wo…].

Buckley proposes that Kashaya has two i‘s—mutable /i/ and inalterable /î/—which are surface-identical except with respect to (9-12). He proposes that /î/ is [+High] whereas /i/ is underlyingly underspecified for it. This analysis can be easily translated into Logical Phonology. For instance, (9) can be respecified as follows.⁴

(13) [+Vocalic] ⊔ {-High, +Low} / [+Labial, +Nasal, …] _

(13) applies non-vacuously to only one segment: underspecified /i/. Both High and Low must be specified since Kashaya also has mid vowels /e, o/. Then a context-free redundancy rule can then be used to “merge” /i, î/, non-vacuously applying to any underspecified /i/ remaining after (13) and the Logical Phonology analogues of (10-12).

(14) [+Vocalic] ⊔ {+High, -Low}

Spanish

The final case concerns the so-called raising verbs of the Spanish 3rd conjugation. This is a well-known case and you can read my our take on it in Gorman & Reiss 2024 (§4) here. Briefly, we argue that the “raising” verbs have an underlying vowel underspecified for High which is realized as i except when it is followed by a [-Back, +High] vowel in the following syllable. Verbs with the underspecified i-like vowel, like the pedir-pido ‘ask for (s.t.)’ are thus distinct from verb stems with non-alternating i (e.g., vivir-vivo ‘live’) or e (sumergir–sumerjo ‘submerge’).

Discussion

Summarizing our analyses:

Barrow Inupiaq has two i‘s, one “catalytic” and one “quiescient” with respect to palatalization.
Blackfoot as two i’s, one catalytic and one quiescent with respect to breaking (and also two k‘s, one mutable and one inalterable).
Czech has two i‘s (ignoring phonemic length; and two e’s, also ignoring phonemic length), one catalytic and one quiescient with respect to palatalization.
Hungarian has two i‘s (ignoring phonemic length; and two long e’s), one catalytic and one quiescient with respect to vowel harmony.
Kashaya has two i’s, one catalytic and inalterable with respect to several processes, and one quiescent and mutable with respect to those same processes.
Spanish has two i‘s, one mutable and one inalterable with respect to i-dissimilation.

The approach to underspecification developed by Sharon Inkelas and colleagues, whose slogan is “prespecification as inalterability”, is well-equipped to handle Kashaya and Spanish, where i‘s are underspecified to generate mutability and prespecified to generate inalterability. However, it has nothing to say about Barrow Inupiaq, Blackfoot, Czech, or Hungarian, a point which Inkelas & Cho (1993:556, fn. 26) concede. One of the innovations of Logical Phonology is that it generalizes their treatment of targets to triggers, adding a new slogan—”prespecification as catalysis”—allowing for all five cases to receive a more-uniform, purely phonological treatment.

Is there something about i, and perhaps e and a in some occasions, which might account for its apparent tendency to either mutate or resist mutation, or to either trigger or fail to trigger mutation? Let us call these tendencies duality. First, it should be said that this generalization is based on a handful of reasonably-well characterized languages, and it is not clear that it is a meaningful tendency. But, granting the significance of this tendency for sake of argument, one might be tempted to derive the duality of i from some phonetic—acoustic-auditory or articulatory—property of high front vowels. One can imagine many just-so stories. For instance, /i/ is very commonly as the default and/or epenthetic vowel, and I have been told that epenthetic segments are longer on average than non-epenthetic segments. Perhaps its default or epenthetic status, or an associated shorter temporal duration, accounts for its duality, though it is not clear how. Or perhaps duality derives from something about its phonetic cues. Logical Phonology, however, is a strictly substance-free approach, and as such, such explanations, however interesting, must be extra-grammatical, of the sort studied by Juliette Blevins and John Ohala among others.

Endnotes

I have taken the liberty of adapting AB’s transcriptions into IPA. I also follow standard conventions in transcribing the mid front vowels as [ɛ(ː)] rather than their [e(ː)], though nothing depends on this.
I put aside two details here. First, there appears to be a separate rule of velar palatalization which is not conditioned on the palatalizing/non-palatalizing contrast under discussion. Secondly, palatalization triggered by palatalizing e is realized as [j] insertion when the preceding consonant is a plain consonant (e.g., /v/) lacking a palatal allophone.
In Logical Phonology, feature-changing processes are always expressed as a deletion rule followed by a unification rule, so the apparent redundancy in (7-8) is intentional. This separation allows us to dispense with the distinction between feature-filling and feature-changing rules, and has other desirable properties discussed elsewhere.
In this rule I am using an ellipsis for the specific features of /m/ since the exact specification of this segment is unimportant here.

References

Anderson, S. A., and Browne, W. 1973. On keeping exchange rules in Czech. Papers in Linguistics 6: 445-482.
Buckley, E. 1994. Prespecification of default features: the two /i/’s of Kashaya. In Proceedings of the 24th Annual Meeting of NELS, pages 17-30.
Dresher, B. E. 2009. The Contrastive Hierarchy in Phonology. Cambridge University Press.
Franz, D. G. 2017. Blackfoot Grammar. 3rd edition. University of Toronto Press.
Gorman, K., and Reiss, C. 2024. Metaphony in Substance Free Logical Phonology. Submitted. URL: https://lingbuzz.net/lingbuzz/008634.
Inkelas, S. and Cho, Y.-M. 1993. Inalterability as prespecification. Language 69: 529-574.
Kaplan, L. D. 1981. Phonological Issues in North Alaskan Inupiaq. Alaska Native Language Center.
Siptár, P., and Törkenczy, M. 2000. The Phonology of Hungarian. Oxford University Press.