What Does "Most" Even Mean? Generalized Quantifiers
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- เผยแพร่เมื่อ 26 ก.ย. 2024
- How can we tell what words like "few" and "many" do in our sentences? What's the right way to represent these words in our minds? This week, we talk about generalized quantifier theory: what the math for quantifiers should look like, what properties natural language quantifiers seem to all share, and what that means for how kids can learn them.
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Meaning Predicated on Logic: Predicate Logic - • What Makes a Sentence ...
Let's Talk About Sets: Set Theory and Adjectives - • How Do We Build Meanin...
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Sources:
Barwise, J. and R. Cooper. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 4:159-219.
von Fintel, K. (1994). Restrictions on Quantifier Domains. PhD Thesis, University of Massachusetts Amherst.
Hunter, T. and A. Conroy. (2009). Children's restrictions on the meanings of novel determiners. Proceedings of BUCLD 2008; file found here: www.tc.umn.edu/...
Westerståhl, Dag, "Generalized Quantifiers", The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), plato.stanford....
Looking forward to next week!
I love your channel
+LizBen@ Wow! I'm glad that you did so well, and I'm glad we were able to help. And yeah, we find when we watch educational videos, we appreciate the little summaries, so we make a point of sticking them in. That way, if you feel like you didn't learn those things, you can go back and review, too.
Thanks really for this message - we really like hearing this kind of thing. ^_^
+The Ling Space you're welcome :)
Gelf of the videos I'm watching are yours at this point. Super interesting stuff!
+Schindlabua Haha, great! Thanks. Good use of a Hartig quantifier there, too. Glad you're liking them. ^_^
Thanks for throwing me a reference bone, I've been a Dwarfer for what seems like 3+million years. And this channel is addictively fun.
Cool! Thank you very much for making this video!
Hi Moti, this is an interesting topic, how we humans appear to have an intrinsic understanding of expressivity in our heads. But with expressivity seems to come also ambiguity. Two different listeners might interpret "Most aliens are green" differently, but most would agree what "All aliens are green" means. However, there are of course some cases where even this is ambiguous. The US founders believed "All men are created equally", nevertheless they didn't seem to include slaves in the set of "all men".
Are all (or even some) human languages arbitrarily expressive? Or is every statement eventually ambiguous? And are there linguistic tools for measuring or quantifying expressivity?
I'm working on a PhD dealing with this question of Computer languages and programming paradigms. The question is how to balance performance vs expressivity in computer programming paradigms. There are many good ways to measure performance (speed, memory consumption, heat dissipation), but it is unclear how to measure expressivity. If there are tools for measuring this in the field of human language linguistics, I would like to try to apply these concepts to computer languages.
I hope you have some thoughts.
Another deep and interesting question! Hard to do justice to it in a TH-cam comment, but we'll give it a go! Addressing this gets kind of long, though. ^_^
To begin, it's worth separating out the different kinds of uncertainties that any given expression can introduce into a conversation. For one, people are often vague; if I say "my cousin is funny," it's left unspoken which of my many cousins I'm talking about, whether they're on my mother's or father's side, or even their age or gender. But this is what you might label indeterminacy -- what we say doesn't exhaustively specify every potentially relevant piece of information. And we don't usually think of this as being related to linguistic meaning, per se.
But, of course, expressions can be genuinely ambiguous too, in a few different ways. The word "funny" above has at least a couple of different senses: "funny ha-ha" and "funny strange"; this is what we might label lexical ambiguity, since a single word can have more than one meaning. And an entire sentence can have more than one meaning, too, not because of the words it contains but because of how they're put together. We find humour in a newspaper headline like "enraged cow injures farmer with axe" because that prepositional phrase at the end can easily be interpreted as attaching to either the noun phrase "farmer" or the entire verb phrase (meaning it was the cow that had the axe). Having seen our episode on the lambda calculus, you know that the scope ambiguities we discuss in our 8th episode are often thought of in this way. We also cover the topic of ambiguity in some depth in a couple of our more recent episodes, if you're interested.
Then there's context dependence, which applies to expressions that don't really refer to anything specific by themselves, and only take on a particular meaning when used in conversation. Pronouns (like "she" and "them") are good examples of context dependent words, since they pick out different people depending on who's using them, and how.
Now, nailing down exactly what's going on with the meaning of "most" can be tricky. For sure, we keep it simple in our episode, and define it to mean exactly "more than 50%". But, say we had 4 people and 3 of them were eating toast; it seems right to say "most people are eating toast". Now what if we had 1000 people, and exactly 501 were eating toast? Is it okay to use "most"? What if the set were even larger? Or infinite? If it's not as clear, here, that the word "most" is appropriate in these contexts, maybe the word is actually context dependent. Or if people disagree about when it's okay to use it, then maybe we're dealing with a bit of lexical ambiguity. One thing's for sure: no natural languages that we know of explicitly rule out these kinds of uncertainties.
As for whether increases in the expressive power of a language introduce the potential for ambiguity, that definitely can be true. If English were a Type-3 language, as we defined the idea in our episode about the Chomsky hierarchy, it could generate the sentence "enraged cow injures farmer with axe", but it could only ever assign it one unambiguous structure (the one where the farmer has the axe). We know English is at least a Type-2 language, in part because we know that this sentence has at least two different possible structures. So, rules that allow for greater syntactic expressivity may necessarily introduce the possibility of ambiguity. Conversely, the introduction of ambiguity could be argued to allow for greater expressivity, since it enables comedy, poetry, and other kinds of linguistic creativity that we might not otherwise. For more on that idea, see: itre.cis.upenn.edu/~myl/languagelog/archives/001719.html.
As for how to measure expressivity: beyond applying something like the Chomsky hierarchy to natural language, we can also ask about what kinds of concepts a language is able to express; in other words, we can talk about semantic expressivity. Now, like asking where exactly language fits into the Chomsky hierarchy, the question of how semantically expressive human language is is going to be somewhat open-ended. But, to give you some idea, when we asked a semanticist and friend of the channel about this, he suggested that at least some claim natural language can be fully characterized using "two-place monadic quantificational logic," which is a fancy way of saying that in order to capture the meaning of at least some sentences, we need to dip into second-order logic. That's a step above what we talked about in our episode on predicate logic, and it allows us to talk about both individuals and sets of individuals.
For instance, the episode above argued that a word like "most" really needs to be thought of as comparing two sets, and that makes it a part of second-order logic. First-order logic, which is restricted to talking only about individuals, just can't do this. It's also been argued that sentences like "some critics admire only one another" are nonfirstorderizable (see en.wikipedia.org/wiki/Nonfirstorderizability), which means we need something more powerful than first-order logic to get a grip on their meanings. Exactly how high up we need to go, though, is an open empirical question, and may have no definite answer! There's no principled upper limit to how complex formal logic can get, so it's always possible we'll find some language out there that needs more than what we've been working with so far.
Now, even though none of us are computer scientists, it's undoubtedly interesting to think about the sorts of trade-offs that programmers have to deal with for expressive power (like check en.wikipedia.org/wiki/Expressive_power_(computer_science)). But, honestly, it isn't really the sort of thing that theoretical linguistics has to worry all that much about, compared to other more practical fields. A lot of modern semantics is done using lambda abstraction, which we talked about in one of our more recent episodes, and its design allows us to express any order of logic we need to. There's not necessarily a lot of time spent worrying about the upper limits of language, perhaps because we don't have to deal with the same kinds of restrictions that others do. In the absence of external constraints, the concepts developed within linguistics tend to be limited only by the imagination! ^_^
Talking toaster, Mister Flibble. You, Moti, are apparently a Red Dwarf fan. :) Is the squid you refer to the "Despair Squid"?
+Cornelius Sneed More our director and our scriptwriter for this episode than me myself, but yeah. ^_^
We did originally have "Despair Squid" in there, but we try to keep it so that the references aren't distracting for people who don't know the material, and we felt that it was kind of too much - people who know Red Dwarf would get it anyway. And it appears that we were right! ^_^
On 5:53, I am not very clear what you mean by "a conservative version" of ”gleed" and "a non-conservative version" of "gleed". Can you give me some concrete examples to elaborate it?
I know I'm difficult, but I don't think of "most" as "more than half". That's a simple majority. "Most" feels like "WAY more than half, but short of almost all".
Still, I'm down with this.
Most, by definition, is anything over half. That seems most intuitive to me anyway. I would consider it synonymous with 'the majority of'
Thank you so much!
Is there a video on quantifying abstract nouns as opposed to quantifying concrete nouns?
They've been very bad Mr. Flibble, what are we going to do with them?....We can't possibly do that. Who'd clean up all the mess?
Technically its the squid's ink
Slow down please😢
In addition to set theory one can also employ topological spaces where one has a family of sets.
Where do you come up with these example sentences...??