Where does logical language come from? The social bootstrapping hypothesis

(Musings on the origins of logical language, inspired by work done in my lab by Ann Nordmeyer, Masoud Jasbi, and others).

For the last couple of years I've been part of a group of researchers who are interested in where logic comes from. While formal, boolean logic is a human discovery*, all human languages appear to have methods for making logical statements. We can negate a statement ("No, I didn't eat your dessert while you were away"), quantify ("I ate all of the cookies"), and express conditionals ("if you finish early, you can join me outside.").** While boolean logic doesn't offer a good description of these connectives, natural language still has some logical properties. How does this come about? Because I study word learning, I like to think about logic and logical language as a word learning problem. What is the initial meaning that "no" gets mapped to? What about "and", "or", or "if"?

Perhaps logical connectives are learned just like other words. When we're talking about object words like "ball" or "dog," a common hypothesis is that children have object categories as the possible meanings of nouns. These object categories are given to the child by perception*** in some form or other. Then, kids hear their parents refer to individual objects ("look! a dog! [POINTS TO DOG]"). The point allows the determination of reference; the referent is identified as an instance of a category, and – modulo some generalization and statistical inference – the word is learned, more or less.****

So how does this process work for logical language? There are plenty of linguistic complexities for the learner to deal with: Most logical words simply don't make sense on their own. You can't just turn to your friend and say "or" (at least not without a lot of extra context). So any inference that a child makes about the meaning of the word will have to involve disentangling that from the meaning of the sentence as a whole. But beyond that, what are the potential targets for the meaning of these words? There's nothing you can point to out in the world that is an "if," an "and," or even a "no."


For many folks this boils down to a classic argument from the poverty of the stimulus: there must be some innate logical concepts that underly the ability to acquire logical language. Let's call this idea "logical nativism." These innate logical concepts need not look like boolean primitives, but they should at least form some kind of basis for inducing a more complex semantics and making lexical mappings. To the extent that you can find evidence for logical reasoning in infants before they can talk*****, this would constitute evidence for the logical nativist perspective.

Others would deny this kind of innate structure. There are lots of reasons to be skeptical of strong nativist claims, whether because you think logic isn't the kind of thing that brains represent innately or because you believe such structures could be learned from input (relatedly, here's my take on "minimal nativism."). But if you make this sort of claim, then you are responsible for characterizing how children come to learn these words and use them correctly. Even if you skirt around Fodor's problem by assuming that children have access to a space of concepts expressive enough to discover these logical operators, you still might want to ask how they do so.

One possible learning theory is that children build the logical operators directly (perhaps through some kind of probabilistic induction). But I want to sketch the beginnings of a different acquisition theory here. On this theory – let's call it the social bootstrapping hypothesis – children begin by mapping logical words to speech acts with specifically social functions like rejection, offer, or threat. They then gradually generalize the broader logical functions of these words by noticing similarities between social uses of the words and other more abstract uses.

This post is a way of writing down my own speculations, and is not fully worked out. Probably someone has said something like this before – perhaps Liz Bates or Lois Bloom - I'm not sure, and that's why this is a blog post rather than a paper. That said, here are a couple of examples.

Negation

"No" is often one of children's very first words. (In some unpublished data, we even saw that this was especially true for second children – presumably they were saying to their sibling "don't DO that!") Consistent with this idea, early negation has been glossed as having the meaning "rejection" – something like "I don't want that" (lit review and up to date coding in this recent paper by Ann Nordmeyer and me). Some other early negations are used for nonexistence ("no cookies") which is a bit different, both syntactically – functioning as a determiner – and semantically. But it's been claimed that you see less early use of negation as what have been called "denials," where a proposition is being negated and the intended meaning is "it is not true that X."

Ann's study suggests that it's true you don't see these early propositional denials as often, but she did find more frequent denials for some  – often during book reading, where parents would ask polar questions like "is that a dog [pointing to a bird]?" and children would say "no!" It seemed like while these utterances were technically logical denials, they were more straightforwardly denying a name rather than a proposition. Further, they seemed like they made sense in those contexts and were being uttered by pretty young children.

More broadly: I wonder if the relevant target for initial mapping of "no" is essentially the social act of rejection – the head shake when a new food is offered, meaning "don't put that in my mouth." Then once this initial mapping is made, from a very salient and present social impulse (parents rejecting kid's behavior and kid rejecting parents' behavior), this meaning can be generalized to other cases. In particular, the trajectory from "no! don't do that" to "no! don't (you) say that" to "no! don't (you) think that" to "no! not true!" doesn't feel too implausible to me. This would especially be an easy conflation to make under a pre-theory of mind, naïve-realist viewpoint in which what I think is what you think is what is true of the world. It would also explain why the early denials that Ann saw were possible – they're very transparently instances of "rejection of a name" even though they look like "denial of a proposition" on the earlier analysis.

One much-discussed example of early negation is the utterance "no mummy do it" (see Drozd, 1995), which means something like "I don't want mummy to do it." Drozd then presents the utterance "no Nathaniel a king," (Nathaniel is the kid here, who's speaking) which alternatively means something like "I don't want you to say that Nathaniel's [I am] a king" or "Nathaniel's not a king." You see how there is a pretty small step from rejecting an action to rejecting a proposition.

Related to this bootstrapping account is the persistent negativity of negation – in corpora, negative terms carry negative valence. To be fair, the account given in that paper notes that these effects may be pragmatic in nature. But the paper did lead me to a related hypothesis to my social bootstrapping idea, \namely that negation is “Learned early on with the association of ‘unpleasant feelings’” (from Bertrand Russell originally). I think that's probably right, although I'm arguing that the negativity of negation is not a direct affective mapping, it's instead a mapping to the social negativity of rejection.

Disjunction

In contrast to "no," "or" is a bit of a mess in acquisition. Children say "or" pretty early, but who knows what they mean? One big issue is that they hear disjunctions that seem to mean logical OR ("[waiter:] you can order dinner or drinks" - true if one is true, the other is true, or both), but they also hear some that appear to be XOR ("[waiter:] you can have dessert or the check" - true if one is true, or the other is true, but NOT both). What could be the target for mapping for this word?

Well, one part of the puzzle comes from Masoud Jasbi's paper, which is that these different uses have different prosody: the second one has a more distinctive rise/fall/rise pattern than the first. (Also, typically the disjuncts are logically inconsistent in XOR cases.) But there's a more general issue: how do you even think of OR and XOR as possible meanings?

Again, my suggestion is that the initial target is a social meaning: offer. Under this story, "X or Y" as a construction initially means "offer." Probably this comes up in the context of food offers, especially. The exclusivity of this offer (can you take both or only one) is then a secondary concern that can be worked out from context. But again, you can see the progression from "would you like carrots or string cheese" -> offer(X,Y) to "is john home or at school" -> offer(john at home, john at school). The key step is again, offering an action to offering a proposition.

Furthermore, as Masoud's dissertation uncovers, there are a host of other meanings for "or" that don't fit well at all with the basic boolean OR vs. XOR idea. For example, "I'm a wine-lover, or oenophile" (definitional disjunction) doesn't fit. And we constantly correct ourselves using disjunction, e.g. "I think it's in the closet. [observes that's not the case] Or under the piano." These broader meanings feel like they might be different classes of social meanings that map onto the lexical item in specific pragmatic and prosodic frames.

Implication (and Conclusion)

Before I wrap up I just want to mention "if," where I think there is possible story. Threat seems like a clear candidate as a target for mapping. "If you dump that out, you won't get any more" feels to me like a prototypical example of a child-directed utterance where the causal interpretation could eventually get generalized into whatever your semantics is for conditionals. Note here that in this case again there's a reversal. The Gricean pragmatics that is assumed on conventional accounts to be built out of a logical semantics actually becomes on this account the place where acquisition starts! So rather than causality being an implicature from the conditional, it's actually the starting point for mapping and generalization. I don't have data on this, but I'd be interested in investigating...

Hopefully, in this post, I've planted the idea that social meanings could be the roots of logical word learning. There are of course many obstacles to realizing this kind of account – first of all, specifying the relationship between the different semantic entities that can be acted on (from objects to actions to propositions). Further, it's not as clear how this would work for "and" or quantifiers like "some." But as I observe children's interactions and think about the way their pragmatic competence supports word learning, this is the sort of constructivist account that feels like the most plausible response to logical nativism.

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* Or invention. I won't get all philosophy of math on this right now.
** Of course, the logic of natural language is contaminated constantly with pragmatic inference – that's what I spend most of my time studying.
*** We'll ignore here both reciprocal effects of language on category formation and the difficulty of object recognition.
**** By "more or less" here I mean this is actually a major topic of study for a whole subfield. So there is a lot to learn. But at a high level this kind of social learning view is not terrible.
***** I have some criticisms of the inferences from this paper, but the experimental designs are extremely clever.

(Thanks very much to Chris Potts for helpful comments).