Tuesday, April 17, 2018

Empirical aesthetics

Bill Idsardi

I'm in Germany this week, right now at the MPI in Frankfurt. So the discussion of SFP and EFP will be on hold for a week.

At æ yesterday David Poeppel was grilled about his career, and he revealed his secrets to success in science, which includes a bit of sleight of hand, as you can see in the picture.

Monday, April 16, 2018

The moral certainty of the existence of UG

This is a very long post. I was going to divide it in two and serve it tapas style, but I decided that this broke the flow. So instead I serve it up as a whole with the understanding that it is too long and probably too dense. To help you swallow what is on offer I spiced the serving up polemically (though I show unusual restraint in parts). To help you through it I thought I would offer up the two main punch lines.

First, it is virtually inconceivable that UG does not exist. It is as sure a bet as one can have and there are two reasons for this: (i) It is a very weak thesis and (ii) The data in its favor is both evident and overwhelming. So, does UG exist? No doubt yes

I address a second question and answer it.  Given that UG obviously exists why do so many think that this is a (at the very least a very very) controversial claim. The answer is that people who think that the UG hypothesis is either clearly false or at least way out there confuse two separate questions; the whether and the how. I explain what the difference is and argue that this confusion needs to be resisted. So properlyunderstood, that UG exists is a virtual certainty, though what exactly is in UG is rightly controversial. 

These are the main conclusions. Here is the long, very long, way too long, argument. 

Here is part 1.

0. The thesis and the argumentative roadmap

That humans are endowed with Universal Grammar (UG) is, I believe, true.  Indeed, I believe it is obviouslytrue. So obviously true that there are no reasonable intellectual grounds for denying it. Surprisingly (note bulge in cheek please), what I consider to be trivially and obviously the case is considered by many to be at best tendentious and at worst, false. But I assure you that I am right about this. In what follows, I will demonstrate that it is morally certainthat UG exists. 

Here’s the game plan.

First, I will review some robust facts about human linguistic facility and show how they together very strongly license the conclusion that something like UG must exist. I will observe that these facts’ robustness tracks their self-evidence.

Second, I will outline how something like UG is required to accommodate these facts. In the process, we will see that the claim that UG exists is actually quite weak, which is one reason why it is virtually certain to be true.

Third, I will identify a question that is often confused with the one that I take to be trivially true that is in fact notat all trivial and suggest that part of the contentiousness regarding UG’s existence revolves around confusing two related, yet very different, questions.

I will then end with a little sermonette.

1.The obvious facts

There are four big facts that motivate the generative enterprise:

1.     Species specificity: Nothing does language like humans do language, not even sorta kinda. 
2.     Linguistic creativity: “a mature native speaker can produce a new sentence on the appropriate occasion, and other speakers can understand it immediately, though it is equally new to them’ (Chomsky, Current Issues: 7). In other words, a native speaker of a given L has command over a discrete (and for all practical and theoretical purposes) infinity of differently interpreted linguistic expressions.
3.     Linguistic promiscuity: Any human child can acquire any language with native proficiency if placed in the appropriate speech community.
4.     Linguistic recency and stability: The human linguistic capacity to acquire a language with properties 2 and 3 is a relatively recent biological innovation (roughly 50-100 kya) and it its features have remained unchanged (indeed, linguistic promiscuity implies this).

These four facts have two salient properties. First, they (or at least the first three) are more or less obviously true. That’s why nobody will win a Nobel prize for “discovering” any of them. Here’s what I mean.

It is obvious that nothing does language remotely like humans do (humans are linguistically speaking qualitatively unique biologically). Humans are to language what fish are to swimming and birds to flying.  No other animals do it like we do. It’s not even close.[1]

Linguistic creativity in the sense of 2 is also self-evident. A moment’s reflection indicates that there is no apparent bound on the kinds of messages humans can linguistically express and there is no obvious upper bound on sentence complexity (length or depth). We regularly express and understand sentences never before encountered, and do so quickly and easily. There is for all practical and theoretical purposes an infinity of sentences over which a native speaker of a given language has command.

Third, absent pathology, any human can acquire this kind of unbounded linguistic capacity and does so in more or less the same way regardless of the language being acquired. So native speakers of English, German, Japanese, Swahili, ASL etc. acquire their native “tongues” (i.e. their language specific Gs) at more or less the same rate, make more or less the same kinds of “errors” on the way and end up with proficiencies within their respective native languages that are more or less the same. Further, so far as we can tell, any child regardless of ethnic, geographical, social, or racial background is able to acquire any language with equal ease. In short, there are no apparent biologically relevant group distinctions that distinguish humans with respect to the capacity to acquire a natural language.[2]

Last of all (and the least secure of the four facts cited), the kind of linguistic facility we find in humans is a recentish innovation in biological terms (ok, this last point is more tendentious, or at least the first part of it is (see below)). Using cultural proxies as evidence for the rise of language in the species (unfortunately utterances have a very short physical shelf life), we find these en masse in only the last 50-100,000 years.[3]Moreover, so far as we can tell, a child’s capacity to acquire any language is the same across the species. Thus, whatever the capacity to acquire human language consists in, it has remained stable in humans since it arose (i.e. once having “evolved” it has not evolved any further in any biologically identifiable sub-groups to differentially facilitate G acquisition by different biological subgroups).[4]

To repeat, these four facts (or 3.5 of the four facts) about human language and human linguistic capacity are (for the most part) very robust and easily observed.[5]  And given this, one reasonable intellectual project is to explain howthese factscouldhold. The claim that UG is a biologically endemic human property is a step towards explaining these facts. More specifically, it is hard to see how one couldexplain them without making an assumption analogous to this. Let’s unpack this claim.

2. The facts and FL/UG, part 1.

The four facts above can usefully serve as boundary conditions on any adequate theory of language. Let’s consider how.

The first, species specificity, implies that there is something special about humans that allows them to be linguistically proficient in the unique way that they are. We can name the source of that proficiency: humans (and most likely onlyhumans) have a faculty of language (FL) capable of meeting the computational exigencies peculiar to language. Were humans biologically equipped with UG then the fact that humans are to language what flying is to birds and swimming is to fish (and you can add endlessly to this list (e.g. echo location is to bats, hopping is to kangaroos…) would follow. Note, this does notexplain why onlyhumans seem to be so linguistically proficient (nor does it explain how specifically the proficiency operates (see below)). We must make the further assumption that whatever the biological underpinnings that support human linguistic proficiency is developed uniquely richly in humans. So whatever FL is, whatever its properties, humans have one and other animals do not and were this so then it would explain why humans are uniquely linguistically proficient in the way they are. And given that humans areuniquely linguistically proficient (i.e. given that nothing does language like humans do) licenses the supposition that humans have FLs while other animals do not, though what exactly “having an FL” comprises is as yet undetermined. All the first fact entails is that there is somethinglinguistically relevant that is uniquely human, not that everything that pertains to language is the special property of humans.[6]This is a very weak claim, based firmly on a pretty evident fact.

Linguistic productivity implies that native speakers acquire Gs (recursive /procedures or rules) able to generate an unbounded number of distinct linguistic objects that native speakers can use to express their thoughts and to understand the expressions that other native speakers utter.[7]How does it imply this? 

Linguistic productivity is the observation that humans can use and understand an unbounded number of linguistic expressions (many/most of which are novel). In other words, a natural language comprises a discrete infinity of distinct expression and a native speaker of that language can effortlessly use and understand an unbounded number of them. This implies that a native speaker’s facility in a particular language involves acquiring rules that together can finitelyspecify the expressions of that language. Or to put this another way, the observed unboundedcapacity must rest on some finiterecursive capacity (i.e. rules that apply repeatedly) for unbounded capacities are not physically realizable in any other way. So, the physical realizability of an infinite capacity implies that what is actually physically realized is some finite specification of that capacity which underlies the unbounded manifestations of that capacity.

It is noteworthy that so far as we can tell only humans are linguistic creative. Thus, when we observe that nothing does language like humans do, part of what we are observing is that no other animal appears to have a system analogous to the linguistic system that humans have: one capable of effortlessly deploying a discrete infinity of expressions that couples a meaning to an articulation.[8]Given this, a key part of human linguistic proficiency consists in having a grammar (G) (i.e. rules that repeatedly apply to their outputs) of a particular language able to generate (i.e. specify) an unbounded number of discretely different linguistic expressions linked to a specific meaning and articulation (i.e. “sound”).[9]

If we combine this conclusion with fact (1), we get to the conclusion that part of what makes humans biologically unique is a species capacity to acquire Gs of the kind that we do. Or, to put this another way, linguistic creativity is a big part of what makes humans linguistically special. It’s (part of) what we do that other animals don’t do. As Gs undergird linguistic creativity, then the capacity to acquire Gs like ours is (part of) what makes humans linguistically special. Again: that nothing does language like we do is self-evident. That we are linguistically creative is as well. Together these two observations support the inference that (part of) what makes us special is the capacity to become linguistically creative, to acquire a G.

(3), linguistic promiscuity, implies that all human Gs have something in common; they are all acquirable by humans and any human can acquire anyone of them. This strongly supports the conclusion that there are some properties P that all humans have that allow them to acquire human Gs in the effortless reflexive way that they do. In other words, the capacity to acquire Gs is reasonably seen as the same across the species (just like the capacity to fly is seen to be the same across eagles and the capacity to swim is seen to be roughly the same across tuna) and a (indeed, the) standard way of accounting for such species wide capacities is to ground them in the biology of the species. If that is good enough for eagles and flying and tuna and swimming it is good enough for humans and language.[10]

So here is a reasonable (‘ineluctable’ is probably a better adjective here, but I am being very concessive so as not to annoy anyone (hah!)) conclusion from the enumerated obvious facts: humans (and only humans) come biologically equipped with a capacity to acquire human Gs (an FL) which (partially) undergirds native speakers’ capacities to produce and understand linguistic expressions creatively (i.e. novel expressions the native speaker has never before encountered). 

3. The facts and LF/UG, part 2.

Indeed, cursory inspection of the obvious facts allows us to say a bit more: (i) we know that the linguistic data available to the child during the course of G acquisition vastly underdetermines the kinds of Gs that we know humans can acquire thus (ii) it must be the case that some of the limits on the acquirable Gs reflect “the general character his [the acquirers NH] learning capacity rather that the particular course of his experience” (Chomsky, Current Issues; 112). (1), (2), (3) together with (i), (ii) very strongly license the conclusion that FL consists in part of language specific capacities that enable humans to acquire some kinds of Gs more easily than others (and, perhaps, some not at all). In other words, the class of humanly possible Gs is a subset of the class of logically possible Gs and FL is able to discriminate between the two classes.

Here’s another way of saying this. Assume that FL is a cognitive structure made up of various kinds of cognitive operations and general principles. Let’s call the linguo-centric aspect of FL, “UG.” More specifically, UG consists of those features of FL that are linguistically specific, in contrast to those features of FL that are part of human or biological cognition more generally. Note that this allows for FL to have features that are notUGish. What the above considerations support is the conclusion that there are somefeatures of FL that are linguistically proprietary. In other words, the UG part of FL is not null.[11]Or, somefeatures of FL are biologically special in humans, in particular (at least) whatever it is that circumscribes the restricted class of Gs that comprise the human Gs. 

The fourth fact, that human linguistic capacity is a relatively recent biological innovation, implies more about the “size” of UG: (4) implies that the UG part of FL is rather restricted. In other words, though there are somecognitively unique features of FL that enable humans to acquire the Gs that they do (i.e. UG is not empty), FL contains many mental operations that it shares with other cognitive capacities and that are likely also not human specific (i.e. shared across species). In other words, though UG has content, muchof FL consists of operations and conditions not unique to FL.[12]

This forth fact highlights why the claim that UG exists should be treated as remarkably anodyne. The claim is veryweak. The claim is that there are somefewaspects of FL that are linguistically dedicated (i.e. UG) but that most of FL is comprised of operations, primitives and conditions that are cognitively/computationally general. Moreover, the weakness of the claim implies that its denial is correspondingly strong. In particular, the denial that UG exists must show thatallof FL is domain general and this requires addressing the host of phenomena that linguists have unearthed over the last 60 years and showing that they can be accounted for without invoking any linguistically proprietary features

Let me beat this point good and dead. To argue that UG does not exist  (i.e. that there is nothingcognitively special about human cognition that allows them to acquire and use the kinds of languages that have the specific propertiesthat they in fact have) requires demonstrating that no UG principles are required to explain why human Gs are restricted to having the special properties they have been empiricallyshown to have by legions of (well, many) linguists over the last 60 years. Nobody has come close to even addressing this issue (even the hand waving has been perfunctory), let alone showing that it holds. But absent such a demonstration, claims that UG is null (i.e. that there is nothing cognitively special about humans wrt language) is windy (and intellectually irresponsible) BS. That UG skepticism is considered a respectable intellectual position indicates more about the sad state of intellectual debate than anything about the plausibility of the arguments allegedly undergirding this skepticism. 

In sum, arguing that UG is null and that its apparent effects can all be accounted for using general cognitive and computational principles is a very tall order given the wealth of intricate, and subtle and specific(in the biologist’s sense) grammatical phenomena that Generative Grammar has unearthed over the last 60 years of linguistic investigation. Not surprisingly, nothing like this has been attempted, let alone successfully accomplished.

Nor should one be sanguine that it is possible. As noted, nothing does language like humans do and this lends prima faciesupport to the idea that there is somethingcognitively special about humans with respect to language that allows for this. What seems generally unappreciated is that this is a very weak claimgiven the basic evident facts and so showing that it is incorrect will prove to be very challenging. At the very least, isolating one or two linguistic phenomena and successfully demonstrating that they do not require specialized linguistic apparatus fails to come to terms with the challenge. And this is a logical point. Proving an existential statement to be incorrect takes a lot of work, virtually none of which has been tried, let alone successfully done.[13]

Note, that this has a somewhat surprising polemical consequence. UG skeptics like to think of themselves as defending common sense and eschewing a radical and highly implausible claim (linguistic nativism) that requires a lot of evidence to even be entertained. But on fact the state of play is the exact opposite. Skeptics are advancing a very bold claim given the obvious facts concerning humans and language. They are contending that there isnothingcognitively special about humans wrt this capacity. This is a logical possibility. But it is a very strong claim and requires a lot of hard work to establish. The weak claim is that there is something special about humans. The skeptics have it completely backwards, which, come to think of it, may account for the fact that they have done so little (nothing really) to demonstrate that their claim is even mildly plausible.

4.  The hard question is often confused with the easy one

The argument outlined above is often taken to be very controversial and highly speculative. As should be clear at this point, it isn’t. That humans have an FL with some unique UGish features is a trivial conclusion from very obvious facts. This should not be up for debate. What then should be? 

What is controversial, and rightly so, is what exactly UG consists in.What exactlyis the linguistically special cognitive content of UG? This is quite definitely NOTobvious and this is what linguists (and others interested in language and its cognitive and biological underpinnings) are (or should be) trying to figure out. I would argue that linguists have a pretty good working (i.e. effective) theory of FL/UG and have promising leads on its fundamental properties, though this is not the topic of this post. But, and I really want to emphasize this, even if many/most of the details are wrong the basic conclusion that humans have an FL with some UGish touches is virtually certain to be right. To repeat,thatFL is a human biological endowment with some proprietary linguistic features is (or should be) uncontroversial, even if whatFL consists in and what its UGish linguistic proprietary features are isn’t.[14]  

In a rational world (and boy do I hope that there is one that is reasonably accessible from our own), there should also be consensus around a second point: how to go about refuting the claim that UG exists. Given that the claim is weak (indeed veryweak) and (pretty directly) reflects some very basic evident facts, proving it wrong requires getting deep into the details of specificclaims and showing that they can be derived without UGish assumptions. So, for example, if someone (e.g. Chomsky) argues that hierarchical recursion or structure dependence is a UG property of FL then arguing that it is not requires showing how to reduce these specificproperties of human grammars to generalfeatures of cognition.[15]Details will matter. Absent these, arguments against UG should carry little weight. Recall, there are obvious facts that point to somethingspecial about language in humans. Arguing against this must establish that there is nothing cognitively special about language and this means that allof its specific properties can be reduced/explained in more general terms. In my opinion, once this truism is embraced the conclusion that humans are biologically endowed with an FL with language specific UG properties will quickly become the conventional wisdom (and posts like this one will seem incomprehensible to our intellectual descendants (I pray for that day!)).[16]

5.  The sermonette

Let me end with a small nod to 17thand 18thcentury discussions about skepticism. Thinkers of this era distinguished logical certainty from moral certainty. Something is logically certain iff its negation is logically false (i.e. only logical truths can be logically certain). Given this criteria, not surprisingly, virtually nothing is certain. Nonetheless, we can and do judge many propositions more or less certain that are neither tautologies nor contradictions. Those things that enjoy a high degree of certainty but are not logically certain are morallycertain. In other words, it is worth a sizable bet with long odds given. My claim is the following: that FL with UG touches exists is morally certain. That there is a species specific dedicated capacity based on someintrinsic linguistically specific computational capacities is as close to a sure thing as we can have. We can even be more specific: some of the “specialness” will relate to the distinctive computational properties that human Gs have. Of course, this might, logically speaking, be wrong, but only in the way that our bet that birds are built to fly might be wrong, or fish are built to swim might be. In other words, though it is a logical possibility that it is wrong, there is a moral certainty that it is not. 

Here, in other words, is the current state of play. Maybe there is nothing special about birds that allow them to fly (maybe as Chomsky once wryly suggested, eagles are just very good jumpers). Maybe fish swim like I do only more so (i.e. fish are to Michael Phelps what Phelps is to me). Maybe. And maybe you are interested in this beautiful bridge in NYC that I have to sell you. That FL/UG exists is a moral certainty. The interesting question is what’s in it, not if it’s there.

Why do I mention this? Because in my experience, discussions in and about linguistics often tend to run the whether/that and the what/howquestions together. This is quite obvious in discussions of the Poverty of Stimulus (PoS). It is pretty easy to establish that/whethera given phenomenon is subject to PoS, i.e. that there is not enough data in the PLD to fix a given mature capacity. But this does not mean that any given solution for that problem is correct. Nonetheless, many regularly conclude that becausea proposed solution is imperfect (or worse) that there is no PoS problem at all and that FL/UG is unnecessary. But this is a non-sequitur. Whether something has a PoS profile is independent of whether any of the extant proposed solutions are viable. 

Similarly with evolutionary qualms regarding rich UGs: that something like island effects fall under the purview of FL/UG is, IMO, virtually uncontestable. What the relevant mechanisms are and how they got into FL/UG is a related but separable issue. I want to walk this back a bit: that some proposal runs afoul of Darwin’s Problem (or Plato’s) is a good reason for re-thinking it. But, this is a reason for rethinking the proposed specific mechanism, it is not a reason to reject the claim that FL has internal structure of a partially UGish nature. Confusing questions for one another leads to baby/bathwater problems, so don’t do it!

So what’s the take home message: we can know that something is so without knowing how it is so. We know that FL has a UGish component by considering very simple evident facts. These simple evident facts do not suffice to reveal the fine structure of FL/UG but not knowing what the latter is does not undermine the former conclusion that it exists. Different questions, different data, different arguments. Keeping this in mind will help us avoid taking three or more steps backwards for every one or two steps forward.

[1]A recent vigorous discussion of this uniqueness can be found in Laland 2017.
[2]Note that this is the case even ifwe allow (which we shouldn’t, btw) that some language specific Gs are not recursive. What is at issue is not whether every human G is recursive but whether every human has the capacity to acquire a recursive G. The fact that some G that has been acquired is not so is not evidence that the underlying capacity does not exist. Everett has catapulted himself to stardom on the basis of this elementary confusion. Quite amazing if you think about it.
[3]The real problem with the claim that FL arose recently is that it is unclear how closely the cultural proxies used supervene on a linguistic capacity like ours. Is unbounded recursion the secret sauce behind elaborate burial rituals or fancy jewelry, or is it something else (e.g. displacement (Everett) or communicability (see here)). We really do not know. What we know is that the kinds of cultural artifacts that we (strongly) believe supervene on something like our linguistic practices are of relatively recent vintage. We also know that some big shots who study this think that it can be traced to a pretty sudden emergence of “language” about 50-100 kya. That’s it. And this is a pretty weak reed upon which to ground interesting claims. But that’s what we have and so we do what we must.
[4]For reasons that elude me this fact seems to be little discussed in the EvoLang literature. Say you believe that the capacity to acquire a G (aka UG) did slowly and gradually evolve over millennia. Then the fact of Promiscuity implies that it stoppedevolving before humans went their separate ways (after leaving Africa). But why should it have stopped? Why slow evolution until time X and nothing further after? Did the forces making for the evolution simply disappear? Perhaps. The game then would be to identify the relevant pressures and then argue that these pressures evaporate at time X for whatever reason. Nobody to my knowledge has attempted this. Indeed, nobody to my knowledge has even addressed the issue. But clearly ifpromiscuity is right (and there is no currently persuasive reason to think that different kids acquire different Gs qualitatively differently (though this is logically possible)) then the gradualist have a Lucyesque obligation (“some esplaining to do”) to address this fact. Otherwise promiscuity constitutes a prima facie good reason for thinking that gradualism is problematic.
[5]Interestingly, the least robust fact is the one that underlies the program that tries to reducethe linguistic specificity of FL (see discussion below). There is a curious tension between two assumptions that critics of UG like to simultaneously hold: (i) UG doesn’t exist and all that we think of as linguistically specific can be traced to the operation of mechanisms of general cognition/computation and (ii) FL has gradually emerged over a very long time span. Why are these in tension? Because the main reason for the long time span is to explain how the obvious idiosyncrasies of the human capacity for language could have gradually evolved despite its specific uniquenesses. But (i) denies that there is anything really unique. But if so, the long time span is not required and we expect to see inchoate “language” in animals (and plants?) all around us. But we really don’t. Not even a little. So either language really is different and maybe a long gradual mode of evolution might explain how this might have happened (though there is NOTHING on the market today that offers any enlightening suggestions) or there is nothing really cognitively different about language and so Darwinian gradualism is not really required to explain what we see. At any case, it seems odd that opponents of UG like to grab onto to both ends of a (mildly) inconsistent stick.
[6]Note that we have not concluded here that FL is linguistically proprietary. We return to this question below. 
[7]Observe the claim is that acquiring a G is necessary to linguistic competence, not that it is sufficient. Factors other than G competence are causally relevant to our linguistic proficiency. 
[8]We can go further: we have no evidence that other animals have anything like the grammatical capacities evident in humans even abstracting away from the meaning with “sound” pairing. This, despite the fact, that animals have some pretty fancy computational capacities (dead reckoning, navigation, probability estimation) that implicate some pretty fancy (perhaps even recursive) computational powers. 
[9]The scare quotes are there to remind the reader that natural languages can be signed as well as spoken.
[10]Note that pointing to uniformity of capacity as a species property does not deny that some humans might be more linguistically adept than others, any more than saying that eagles/tuna capacity to fly/swim implies that all eagles are equally adept fliers or all tuna equally accomplished swimmers.
[11]Note that this is a very weak claim. I return to this below.
[12]Even those features that are shared across cognitive domains and with other species likely have been retrofit to “deal with language” in bespoke ways. So the claim is that FL contains features that are qualitatively unique (call these UG) and those that are qualitatively similar to what we find in other animals (i.e. general principles of computation and cognition). 
[13]I should point out, that in my opinion there have been virtually no successful demonstrations showing that key features that linguists have argued to be properties of UG are in fact explicable using non-linguistically proprietary assumptions. But even if this assessment is incorrect, the logic still stands. The very likely existence of UG rests in part on what a modest claim it actually is. It is the UG skeptics that are making strong baseless assertions, not UG proponents. I suspect that this point will be considered wildly counterintuitive which is why I am writing this post.
[14]This is very like the mathematical distinction between an existence proof vs a constructive proof. We often have proofs that something is the case without knowing what that something is.
[15]So too with other purported UG features (e.g. the Binding Theory, the ECP, Island effects, etc. See ( for a list of some plausible UGish characteristics of FL).
[16]Again, this does not mean that most of FL is UGish, only that some part is. The Minimalist Program (MP), a program that has generated remarkable hostility from those hostile to the claim that UG exists aims to reduce what is linguistically proprietary to FL to a minimum. That at least is the ambition. Thus, if MP succeeds then the “size” of UG will be pretty small. In my view, MP has been quite a successful program, but there are still many plausible UG features that remain unexplained except by deploying linguistically specific cognitive powers.  That said, I am personally optimistic that what we know about FL can be reduced so that only a smallish UG core remains. If I were interested in showing that UG does not exist and that FL is entirely comprised of cognitively/computationally general operations and principles, I would become a devotee of MP for this, if successful, would reduce the target requiring explanation in non-linguistically specific terms. 

Wednesday, April 11, 2018

Hot Cross Buns

Bill Idsardi & Eric Raimy

In a comment Fred asked why we think the EFP structure is a directed multigraph, rather than just a simple digraph. The answer is because we allow self-loops and multiple parallel edges. We'll give linguistic examples later, but in the meantime, here's a fun "blast from the past".

In the book Rules, constraints, and phonological phenomena edited by Bert Vaux and Andrew Nevins, we had a chapter where we discussed various aspects of reduplication. One non-linguistic case that we found quite striking was Jeanne Bamberger's discussion of children's tune-building with Montessori bells in chapter 7 of The Mind Behind the Musical Ear.

Here's the tune for "Hot Cross Buns" in standard musical notation: 

And here's our reinterpretation of the child's encoding of the tune using events, features and precedence, following Bamberger's discussion quite closely:

Bamberger's chapter makes for very interesting reading because the Montessori bells have particular physical characteristics that she is able to  creatively employ to interpret the children's tune-building in very insightful ways. Unlike a piano, there are multiple bells with the same pitch. And the bells themselves are all the same size, so you cannot tell the relative pitch of the bells by visual inspection, you must strike them and hear the note. The bells were deliberately designed in this way, and are part of the Montessori sensorial materials. This allowed Bamberger to make interesting observations about which notes the children felt were the "same" or "different". For instance, the two separate "C" events in our diagram were executed on separate bells, and the same is true for the two "D" events.