Distributive/Collective/Cumulative reading scenario question

I’m still working on a sample and documentation for collective, distributive, cumulative readings and have a general (maybe rudimentary) linguistics question about how they work.

Imagine a world where:

  • Sasha is lifting table1 and table2 (together)
  • Francois and Bob are lifting table3 and table4 (together)

So, 3 people are lifting tables (in two groups), 4 tables are being lifted (in two groups).

In this world I say “3 people are lifting 2 tables”. While I might have to think a bit, I think I would say “yes that is true”.

I don’t think this is a distributive reading since table3/table4 and table1/table2 are each collective.
I don’t think this is a collective reading since there are more than one group of tables being discussed.
I don’t think it is a cumulative reading since there are 4 tables and my phrase is about 2 tables.

Am I just odd in responding “I agree” to the phrase (maybe I’m thinking about this too much)? Or am I misunderstanding how one of the reading modes work? or maybe it’s another class of scenario?

Distributivity is a property of a quantifier, not a sentence.

In that example, I think truth hinges on whether being involved in lifting a table counts as lifting, e.g. if Bob says “I’m lifting 2 tables!”, is that true? Maybe. I would find “supporting” easier to accept in this context.

In the reading you’ve given, “3 people are lifting 2 tables” is distributive with respect to quantifying over the people, since “lifting 2 tables” applies to each person.

I don’t think this example has a distributive/collective contrast for the tables – what would it mean to be simultaneously lifting each table without lifting both, or vice versa?


As for writing documentation, I would suggest using more natural examples where intuitions are clearer.

I’m not sure how detailed the documentation needs to be, given that MRS doesn’t indicate distributivity. Wouldn’t it be enough to give a couple of clear-cut cases and say that the distinction is not represented because it’s not syntactically marked?

Totally agree for the documentation. I am really using this as a way to see if the algorithm I’ve put together is “correct” or “correct enough based on best linguistics theories to date”. For the developer tutorial, I think there needs to be at least a base implementation/algorithm that shows how a developer could possibly handle collective/distributive/cumulative since it is such an essential part of how a phrase is interpreted. Otherwise, you can’t write a program that properly uses MRS against a world state.

I get that the MRS doesn’t indicate it, but I think developers will have to deal with it somehow, at least for certain types of applications. This is really just a “general linguistics” question I have about an edge case that I’m using to see how close my current implementation is to the best understanding…

OK, let me try to dial in a more precise version of my thought experiment.

According to my search engine, Mozart wrote 22 operas by himself, and Gilbert and Sullivan wrote 14 operas together.

Let’s stipulate that we’re not going to give either Gilbert or Sullivan credit for writing an opera on their own since they were a team. So “Gilbert (or Sullivan) wrote an opera” is false.

Someone says:

Gilbert, Sullivan and Mozart were amazing.
The 3 musicians wrote 14 operas!

To that I say “well, yeah. But, for the record, Mozart wrote even more than that!”. So, let’s also stipulate that the second phrase is true if the interpretation was that each group of musicians wrote at least 14 operas.

For this interpretation:

I don’t think the musician quantifier is distributive or collective since the only true values are: [Mozart], [Gilbert, Sullivan] which is neither.

I don’t think the musician quantifier is cumulative since I read it as “Mozart wrote (at least) 14 and [Gilbert and Sullivan] wrote (at least) 14 (other ones)” and that adds up to 28. A cumulative distribution would require that they add up to 14, right?

I’m trying to pin this down since my algorithm doesn’t handle this case and it’s either because I’m still confused about how these modes work or there’s another phenomena here that I’d like to at least point to if it is recognized.

I think I’m following your logic, but I think you’re mixing two issues, 1. group NPs (“Gilbert and Sullivan”) and 2. collective vs distributive vs cumulative readings. I’m going to set aside (1).

I found this explanation (Syrett 2018) of (2) that I found helpful. One of the examples presented is:

(16) Three boys are holding two balloons.

There are at least three different interpretations, categorized as:

distributive: three boys, six balloons, each boy holds two balloons
collective: three boys, two balloons, all three boys are holding both balloons
cumulative: three boys, two balloons, some combo of holding, e.g. two boys hold one balloon each, one boy has no balloon

Let’s briefly consider a simpler example like 3. “two men ate a pizza.” The readings of this sentence are traditionally as follows (and as laid out in Syrett 2018):

distributive: two men, two pizzas, each man eats one pizza
collective: two men, one pizza, each man eats some portion of the pizza

The cumulative readings of this sentence would be either indistinguishable from the collective or, crucially, seem less relevant, such as:

cumulative 1: two men, one pizza, each man eats some portion of the pizza
cumulative 2: two men, one pizza, one man eats one pizza, one man doesn’t eat pizza

I feel like what you’re asking about in your example with “The 3 musicians wrote 14 operas!” is similar to this cumulative 2 reading. For instance, you might observe two men sitting at a table at a restaurant and observe both of them eating along with an almost empty pizza platter and surmise “two men are eating a pizza.” But, you only find out later that one of the men didn’t eat any of the pizza. Were you wrong? One could argue so, but probably not. There were two men, they were both eating, so cumulatively they ate the pizza. It doesn’t matter that one of them didn’t participate. Similarly, in your example, Mozart wrote 22 operas, not 14, but it’s not incorrect to suggest that he wrote 14.

This is where MRS and compositional semantics bumps up into pragmatics. I would say that it’s not that it isn’t accurate to say these examples in these conditions, but rather that it isn’t relevant to say them. This is captured in Grice’s Maxims, which are a whole can of worms in and of themselves. The relevant maxim is the Maxim of Relation:

one should ensure that all the information they provide is relevant to the current exchange; therefore omitting any irrelevant information

Then, the idea of when someone flouts or violates a Grician Maxim is that they’re indicating some additional information only available in the discourse. To bring this back to the pizzas and operas, imagine a twist in a murder mystery where the detective discovers that the victim had lunch not by himself but with a mystery man who didn’t eat any pizza, now “two men ate a pizza” seems like a reasonable thing to say.

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Hi @EricZinda, what algorithm? What is the goal? Given an MRS, what are you planning to produce?

I tried to send this on the 14th by replying to emails, which clearly isn’t working:

Apologies that I am just coming into this conversation (because I am officially on vacation and getting bored with my usual work email) and may have missed context. But I want to suggest that being very careful about the terminology and the phenomena we’re talking about would be useful.

Distributivity is a property of a quantifier, not a sentence.

I disagree. The causes of distributivity are multi-faceted and I think too many things are being conflated in the discussion.

I think we can use the term distributivity in a precise way, to mean making an inference about an individual entity that is part of a group of entities, or indeed about a subgroup of individuals that are part of a larger group. In an extended way, we can also use it to talk about making an inference about a subpart of a mass term or making an inference about a subevent. There are a number of things that are relevant to whether this inference is valid:

  1. Lexical semantics and context - obvious but not something the ERG is trying to capture. According to me (and no doubt many others but I can’t remember anyone at the moment), there are no inherently distributive or collective predicates. There are predicates like “smile” which have a meaning that predominantly relates to an individual human and predicates like “meet”, where multiple individuals need to be involved. We can see this with examples like:

The groups met.

there could be a big collective meeting, but it could be each group meeting separately. I think the same is true of “diverse” in Guy’s latest example.

  1. The semantics of individual quantifiers - e.g., “every” versus “all the”. (I note here that “every” is a fairly recent word in English and that it used to be “every each” (various spellings) - arguably, the emphasis on “every” in formal semantics is a mistake, because it’s weird …)

It is worth emphasizing that there don’t need to be any (classical) quantifiers or plural NPs in examples:

Kim and Sandy lifted Lee and Abrams.

  1. Things like “each” in adverbial position (which Dowty, I think, calls floated quantifiers, but I think that’s misleading). Note that there are lots of other adverbials that can affect the distributivity inference in specific examples - it’s just that “each”, “together” etc always do that, so formal semanticists do want to figure out how they work. I think this is the only role for the ERG really in the discussion - making sure that the adverbials have access to the right bits of the situation so the inference can be made.

  2. Quantifier scoping - this is something that puzzled me for ages, so I will give a (traditional) example:

a. Two boys ate three pizzas.

This has a reading where there are a total of two boys and three pizzas in the situation. It also has a reading where there are six pizzas (i.e., the boys eat three each - either they are greedy or this is a longish time period). It doesn’t seem to have a reading where there are six boys, and this is the puzzle because if it were to do with subjecthood, we’d have to try and capture this in the ERG. But I eventually convinced myself that:

b. Two pizzas satisfied three boys.

has a six pizza reading. Hence we can assume that it’s not a structural effect and the MRS is OKish. (Phew!)

  1. events and subevents - e.g., in many contexts “lift” can be referring to multiple subevents of completed lifting - e.g., “Kim lifted those weights for an hour yesterday.” This isn’t just about the events - it means you can’t make necessarily valid inferences about Kim’s ability to benchpress 100kg or whatever if the weights are 100kg in total.

  2. involvement - this is a bit tangential perhaps, but if we define distributivity in terms of inference, it is relevant:

The reporters asked questions during the press conference. (again Dowty example, I think)

does not allow one to infer that an individual reporter asked a question. Again, essentially this comes back to 1, but demonstrates that regarding a verb like “ask” as inherently distributive isn’t going to work in all circumstances.

Conversely “meet” isn’t always fully collective - there are examples where you can get distribution down to subgroups. See also 1.

All best,



So this seems like quite a contrived example, but it illustrates that we can’t just talk about fully distributive vs collective and cumulative because you can get distribution to salient subgroups. There’s other cases of that in the literature, although I’m afraid I can’t recall references right now. Let’s ignore the “at least” part, which isn’t really relevant to the distributivity issue. You’re talking about a reading where what’s going on is:

For each salient subgroup of musicians, there exists a group of operas such that the subgroup of musicians wrote those operas.

If you want to handle this computationally, you need to assume an oracle that can give you the salient subgroups.


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I will concede that I was not careful in my terminology here (I should not have focused on the quantifier). To be more precise: distributivity can apply to one property of one group, without applying to another property or another group, even in the same sentence. In other words, a sentence can have a reading which is distributive with respect to one property or group but collective with respect to another.

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Overall, I’m trying to describe a relatively comprehensive approach for evaluating the truth of an MRS against a world state. I feel like the lack of one written down (that I’ve been able to find, at least) is going to block developers from building systems with MRS. They simply won’t know where to start. I’m just focused on the mechanics of predication evaluation, not on how to code up the underlying semantics of a given predication. I’ll definitely post links to the different parts as I write them up.

Given an MRS and a world state, the algorithm produces all of the solutions to the MRS for that world state, in a stream (so you can stop after one for yes/no type questions). Solutions are simply assignments of set based values to all the variables in the MRS.

The algorithm I’ve been developing is the part of the system that deals with properly handling plurals. A really brief summary of the main theory I’m working through is:

The fully resolved tree for an MRS can be evaluated against a world state in two stages:

Stage 1: Remove all the numeric determiner semantics (“many”, “2 or more”, “all”, “some”, “the”, etc.) from the fully-resolved tree and solve it. This involves literally removing the numeric adjective determiners and their modifiers (e.g. card(2,e,x) or much-many_a(e8,x3)) and converting the numeric quantifier determiners (e.g. _all_q(x3,RSTR,BODY) or _the_q(x3,RSTR,BODY)) to udef_q. This creates a set of “undetermined solutions”.

Stage 2: Create groups out of the undetermined solutions that satisfy the first determiner and run each group recursively (left to right) through the rest of the numeric determiners in order. The groups that succeed are solutions. Forward and reverse readings happen via different fully-resolved trees.

So far, this approach has simplified the logic for dealing with plurals and allows for some interesting optimizations.

I’ve written up a much more detailed description of it. Love any feedback on the approach.

I’m reasonably sure it will work in theory (so far I’ve tested various scenarios across the, a, a few, card(N), together, only (as in “only a few”, “only 2”, etc)), but it clearly has some challenging combinatorics, so now I’m working through seeing if I can make it work in practice and tame the combinatorial explosion.

For the combinatorics, it turns out complicated phrases like “2 files are in two folders with 2 other files” really only need to find one true case to stop and say “correct!” so that makes one whole class of scenarios easier. If you add “which” on the front, then you need to find all the cases, and that’s when things get expensive.

Update: Also, I should put my standard disclaimer up front: I’ve really tried to properly use linguistics terms when appropriate (and only make up new ones when I can’t find an existing one), but I’m sure I’ve failed on both counts in various areas. I always appreciate corrections!

Thanks @trimblet ! Yes, I found that Syrett 2018 as well, and have been using it to generate test cases.

Yes, this is an example of the kind of edge case I’m trying to get my head around. Partially, it is just so that I use the terminology right, and partially it is because if the accepted linguistics models don’t cover a scenario I worry that I’ve got a bug.

I’ll note that this exact kind of scenario was just pointed out by @AnnC in her post below with

“The reporters asked questions during the press conference.” does not allow one to infer that an individual reporter asked a question

“the reporters” may be 5, but only 4 asked questions. Although, on the other hand, it might not be the same scenario, given that I’ve seen the act as a selector of “the thing that works in the rest of the sentence” before, and I’ve had to model it that way. For example, in a world where there are 50 frogs around me, 4 of which are in a hole, I could say “the frogs in the hole are scared”, and the is only capturing the “the frogs in the hole”, not “all the frogs”. I guess that’s one way to model the reporters as well: the captures “the reporters that asked questions”, not all the reporters in the room. Interpreted that way, it’s just a special case of the magic of the, and not saying anything about plural semantics.

I haven’t encountered Grice yet, thanks for the pointer! You’re providing some good terminology to describe what I’m doing here. I am trying to make the system be accurate, whether it is relevant is a whole 'nother level of system machinery I’m definitely not tackling for the moment. But: I do agree it seems like this is where some of my confusion is. Even in the (admittedly very contrived) composer thought experiment, I said I’d have to pause a bit before I answered which was an admission that this really isn’t a common scenario.

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Yes, agree it is very contrived, I tried searching for a more natural one but couldn’t find it, which I guess might indicate that it really is an edge case if it ever gets spoken at all.

Agreed. So, in terms of terminology, would you call this a … “mixed” scenario? The salient groups are not wholly distributive (N groups of 1) or collective (1 group of N) so that makes me want to say it is a cumulative scenario … but: the way the groups behave over operas seems like the distributive behavior. So maybe (making up terms) “impurely distributive” or “subgroup distributive”… or maybe there just isn’t a term for this.

Really, I’m just trying to use the terminology correctly and make sure I understand the models. Maybe this is such an edge case that there isn’t a term?


Reading through your other post, and given that distributivity can cover a subgroup, maybe my reading really is a distributive reading? And cumulative really only comes into play when you need to “add up” the contributions of two groups to get the number described by the next determiner (e.g. 14), which isn’t the case here.

@AnnC, If I understand you correctly, you are saying that it doesn’t seem to have a natural reading for speakers? The ERG does generate an MRS that has a fully resolved tree where the distributive reading of “there are 3 pizzas and 6 boys” has 6 boys (and maybe you’re saying this would never be read that way by a real speaker?):

                         ┌── _pizza_n_1(x10)
             ┌────── and(0,1)
             │             └ card(3,e16,x10)
                  │                          ┌── _boy_n_1(x3)
                  │              ┌────── and(0,1)
                  │              │             └ card(2,e9,x3)
                  └─ udef_q(x3,RSTR,BODY)
                                      └─ _eat_v_1(e2,x3,x10)

This is a case where that same (reversed with respect to word order) tree does actually make sense to a real speaker? And thus this ERG interpretation really does make sense in some scenarios?

I’m just trying to make sure I’m understanding right …

Yes, I believe that’s Ann’s point. The two sentences are structurally equivalent, so the naturalness of the intepretation strongly depends on the lexical semantics and context, rather than something structural like subjecthood.

Yes. (I still find the reading unnatural, but I would analyse it as distributive if forced.) As Ann said above, distributivity is an inference about the individual entities in a group. But what counts as individual entities? At the very least, they should be disjoint and they should together make up the whole group (more formally, their “mereological sum” is the group entity, and each pair has no “mereological overlap”). For a group of people, usually we would the individual entities to be the people. But formally speaking, we could consider larger entities (like “Gilbert and Sullivan”) or smaller entities (like Gilbert’s left leg).

Let me try pushing your intuitions even further: imagine a world where Gilbert and Sullivan collaborated with Kim and Sandy to write operas. None of them ever worked alone, and they only ever worked in pairs. Each pair wrote five operas together. So each person collaborated on fifteen operas (five with each other person), and there were thirty operas between the four of them. Would you accept “the four musicians wrote five operas”?

OK, thank you @AnnC and @guyemerson. That’s a very clear definition of distributive grouping for this context.

I think the definitions of distributive/collective/cumulative, when talking about distribution of groups across two or more x variables, need to talk about the second variable as well. Part of what has been so confusing about the space is the way the terms “collective” and “distributive” are used, they:

  • sometimes mean just the grouping of x
  • sometimes mean the grouping of x and how to count y against those groups

Introducing “cumulative” into the mix forces you to get clear about it. So, here’s my attempt at definitions:

For “two firefighters carried two hoses” (just to mix things up!):

When referring to a fully-resolved tree (which means that forward and reverse readings with respect to word order are just different trees and aren’t included in the definition):

  • “The distributive reading”
    • Firefighters: 2 or more subgroups, each of size > 0, where every individual in question is in exactly one group (the definition above).
    • Hoses: All subgroups must have two hoses each. Individual hoses may be repeated in subgroups.
  • “The collective reading”
    • Firefighters: Exactly 1 subgroup that contains the entire set of individuals in question
    • Hoses: Identical to distributive
  • “The cumulative reading”
    • Firefighters: Identical to distributive
    • Hoses: The total of unique individual hoses across all subgroups adds up to two.


  • Distributive and collective group the first variable differently, but do the same math problem across the group(s) for the second variable.
  • Cumulative does the same first variable grouping as distributive, but a different math problem across the groups for the second variable.

I believe these definitions cover all the scenarios we’ve been discussing (and some contrived ones that might be accurate but not relevant, using @trimblet’s terms).

There is one overlapping case left (only one, I believe) that is both distributive and cumulative by those definitions: when the two hoses per firefighter group just happen to be the same:

x=[firefighter1], y=[hose1, hose2]
x=[firefighter2], y=[hose1, hose2]

The problem is in the definition of distributive: “… Individual hoses may be repeated in subgroups”. But removing that ends up with cases that aren’t classifiable, but make sense, like:

x=[firefighter1], y=[hose1, hose2]
x=[firefighter2], y=[hose1, hose3]

You could try to fix that by making cumulative cover more scenarios by adding to its “hoses” rule something like “…or: each subgroup has a 2 individuals but at least one individual is duplicated somewhere” but that seems unsatisfying and really makes the mechanisms murky.

So, I’m left with the conclusion that the definitions above are good ones, even if one edge case overlaps.

Bringing it all back, this would say (as I think we’ve agreed) that:

  • “3 people are lifting 2 tables” (the original example) is distributive
    • 2 or more subgroups means distributive or cumulative
    • Total per subgroup (not across all subgroups) is 2 → distributive
  • “The 3 musicians wrote 14 operas!” is distributive
    • 2 or more subgroups means distributive or cumulative
    • Total per subgroup (not across all subgroups) is 14 → distributive
  • “the four musicians wrote five operas” (@guyemerson’s last scenario) is distributive
    • 2 or more subgroups means distributive or cumulative
    • Total per subgroup (not across all subgroups) is 5 → distributive

And to answer @guyemerson’s questions about whether I’d accept “the four musicians wrote five operas”, I think its another “accurate but not relevant” scenario. In real life, I’d think about it for 1/2 a second and say “what do you mean?” since working out the math would be too hard, but I’d agree “in principle” after Guy explained it.

I think there’s a lot in here to respond to, but I’m confused/concerned about the characterizations like this one:

“3 people are lifting 2 tables” (the original example) is distributive

As discussed in this thread, there are a lot of scenarios in which a sentence could have a single reading, but most of the examples we’ve been discussing are ambiguous, including these. But, it sounds like you’re saying there is only one valid reading.

Like a lot of valid ambiguity, it can be difficult to imagine a context in which some of the readings are felicitous, but of course if you’re in those contexts, it may be difficult to imagine it any other way.

To use the “3 people lifting 2 tables” example, both the collective and distributive readings seem reasonable to me:

  • distributive: three people, six tables
  • collective: three people, two tables

For instance, the distributive reading might make sense if people are carrying folded up tables whereas the collective reading might make sense with teenagers doing silly stunts (or performers doing silly stunts).

I was only referring to its use in my original example where:

  • Sasha is lifting table1 and table2 (together)
  • Francois and Bob are lifting table3 and table4 (together)

Agree that in general it could have all possible readings depending on the world state.

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@EricZinda, I thought no one could possibly accept that example. I suppose life is full of surprises! :wink:


Hahaha! Maybe it is a sign I’ve spent too much time with this stuff…