Formal semantics of mass/count

The dominant view of mass/count in the current literature

• Count semantically encodes individuation
• Mass is compatible with (loose) individuation

We can't simply say mass = not individuated, because of object mass nouns (alt. fake mass nouns), e.g. furniture, equipment

Many different theories of Mass, e.g.

• Bale & Barner (2009); count = individuation; mass = no meaning
• Chierchia (1998, 2010, 2015, 2021): count = stable atoms; mass = vague atoms
• Landman (2016): count = disjoint, individuated 'base'; mass = overlapping 'base'

Flexible individuation

Based on nouns like wall and fence, it is suggested that the notion of individuation that count encodes must be sufficiently 'flexible' (Sutton & Filip 2016; Rothstein 2010, 2017)

(1)   The bedroom has a white wall and a wooden floor.
(2)   The bedroom has six walls.

E.g. Rothstein proposes that count has a context-sensitive denotation:

• For wall, it's easy to find contexts that non-trivially differ with respect to atoms
• For boy, it's much harder for extra linguistic reasons

The decompositional approach

A common feature of previous formal semantic theories of mass/count is decomposition

Nouns are syntactically decomposed. Roughly:

These sub-components are assigned model-theoretic denotations:

• $[\![$count$]\!]$ introduces flexible individuation
• $[\![$mass$]\!]$ is analysed differently by different theories
• $[\![$PL$]\!] = \ast$-operator (closure under $\sqcup$)

Limits of the decompositional approach

Assuming compositionality we want count and mass to have constant denotations that can apply to all nouns

But this has led to 'generalisation to the most bleached case':

• Count universally encodes flexible individuation (due to wall, fence)
• Mass universally has very weak meaning (due to object mass nouns)

We argue against the decompositional approach based on variation among hybrid nouns (alt. flexible nouns, dual-life nouns) in Midwest American English

Hybrid nouns

Hybrid nouns can be used as mass or count, e.g. stone, rope, apple, chocolate

Mass/count often has a clear interpretative effect for hybrid nouns

E.g., Barner & Snedeker's (2005) Comparative Task (see also Bale & Barner 2018)

(3)   Who has more stone_mass? ⇢ volume-based comparison
(4)   Who has more stones_count? ⇢ number-based comparison

We'll point out that there's more to the semantics of hybrid nouns

Our observations and claims

Key observations: Variety of count hybrid nouns

1. Some count hybrid nouns are not so flexible with respect to individuation (e.g. apple)
2. Some plural count hybrid nouns are compatible with no individuation (e.g. potatoes)

Proposal: Renounce decomposition in terms of count and mass, and characterise interpretations at the level of whole nouns

• N_count may encode an idiosyncratic manner of individuation   ⇒ 1.
• N_pl.count and N_mass have trivial interpretations, only if the other one is infrequent   ⇒ 2.

Flexible individuation: rope

Rope_count can describe all sorts of perceptually countable instances, similarly to wall

(5)   We've got some ropes here.

(6)   Let's cut one of the ropes into smaller ropes.

Specific individuation: apple

Apple_count can only describe whole apples, not sliced or diced instances

(7)   We have some apples here.

(8)   #Let's cut one apple into smaller apples.

Specific individuation: chocolate

Chocolate_count can describe a chocolate truffle, but not a bar or an arbitrary piece

(9)   I will give you a chocolate.

(10)   #Let's break a chocolate into smaller chocolates.

Interim summary

As expected, apple_count and chocolate_count cannot describe unindividuated instances

But they also cannot describe arbitrary pieces, even if clearly perceptually individuated

☞ These count nouns encode specific individuation (unlike wall and rope)

• $[\![$apple_sg.count$]\!] = \{x | x \textrm{ is a whole apple}\}$
• $[\![$chocolate_sg.count$]\!] = \{x | x \textrm{ is a chocolate truffle}\}$

• $[\![$wall_sg.count$]\!]^{c} = \{x | x \textrm{ is one wall in } c\}$
• $[\![$rope_sg.count$]\!]^{c} = \{x | x \textrm{ is a perceptually salient chunk of rope in } c\}$

(The encoded specific individuation is probably not completely random but can be idiomatic like chocolate)

An apple vs. a potato

A potato is similar to an apple

(11)   I'll give you an apple.			(12)   I'll give you a potato.

Apples

Given how an apple behaves, apples can be undesrtood in terms of the $\ast$-operator

(13)   I'll give you some apples.

(14)   Who ate more apples? ⇢ number-based comparison

Potatoes

Potatoes is not simply sums of singular potatoes: $[\![$potatoes$]\!] \neq \ast([\![$potato_sg.count$]\!])$

(14)   I'll give you some potatoes.

(15)   Let's cut one of these potatoes into smaller potatoes.
(16)   Who ate more potatoes? ⇢ both volume and number comparisons possible

• Flexible individuation, $\ast$-plural (rope-class)
  rope, paper, brick, wire, cable, string, hair, (sub)sandwich, fibre, oat, plank, board, pipe, steak, talk, meeting, exercise

• Specific individuation, $\ast$-plural (apple-class)
  apple, chocolate, lemon, banana, eggplant, artichoke, hamburger, pizza, cake, beard, chicken, duck, mango, candy, song, show, movie, bone

• Specific individuation, mass-plural (potato-class)
  potato, strawberry, carrot, leaf, french fries, chickpea, pea, potato chip, lentil, tomato, noodle, blueberry, pebble

• Flexible individuation, mass-plural (cloud-class)
  cloud, wind, detail

Key observations

Two observations about count hybrid nouns:

1. Specific individuation
   N_sg.count may encode a specific manner of individuation (e.g. apple, chocolate)

2. Plural count nouns with mass denotations
   N_pl.count may have a mass-like denotations (e.g. potatoes, clouds)

1.   Specific individuation

All singular count nouns encode individuation, but some encode specific manners

☞ A generic individuation function for 'Spelke objects' will be too permissive; a Natural Unit function will be too restrictive

Idea: The interpretation of N_sg.count is not compositionally derived from $\sqrt{N}$ and count

• $[\![$apple_count$]\!] = \{x | x \textrm{ is a whole apple}\}$
• $[\![$potato_count$]\!] = \{x | x \textrm{ is a whole potato}\}$
• $[\![$chocolate_count$]\!] = \{x | x \textrm{ is a chocolate truffle}\}$

• $[\![$wall_sg.count$]\!]^{c} = \{x | x \textrm{ is one wall in } c\}$
• $[\![$rope_sg.count$]\!]^{c} = \{x | x \textrm{ is a perceptually salient chunk of rope in } c\}$

2.  Plural count nouns with mass denotations

Potatoes can describe all sorts of instances, unlike apples

NB: potato_mass is also available

(17)   There's some (mashed) potato on your shirt.

Proposal: What differentiates potato and apple is the relative frequency of N_mass

• Grimm & Wahlang's (2021) corpus data (350 million words) with Allan's (1980) diagnostic for countability

• Potato is mostly marked count (84.3%) vs. apple (61.4%)

• 10 most skewed hybrid nouns: pickle, dumpling, sprout, carrot, bouquet, leaf, pea, potato, beet, egg

Frequency effects and RSA with Lexical Uncertainty

Idea: When N_mass is relatively infrequent, N_plural.count can be used to mean the same thing

We use a Rational Speech Act (RSA) model with Lexical Uncertainty (Bergen, Levy & Goodman 2016, Scontras & Goodman 2017)

• N_pl.count has two potential denotations, Plural: $\ast([\![$N_sg.count$]\!])$ vs. Mass: $[\![$N_mass$]\!]$
• When N_pl.count is used, the listner probabilistically guesses what the intended interpretation is, based on cooperativity
• The relative frequency of N_mass (implemented as speaker cost) and the frequency of intended meaning affect the reasoning
• If N_mass is not costly, cooperative speaker avoids ambiguous N_plural to mean mass
• Details in Appendix

Further directions

1. Symmetric frequency effects of mass/count

• If N_mass is infrequent, N_plural can mean the same thing    ☞ potatoes
• If N_plural is infrequent, N_mass can mean the same thing    ☞ object mass nouns

2. We expect cross-linguistic/dialectal variation and language change over time

• Frequencies change for many reasons
• Very specific idiosyncratic meanings of singular count nouns can fail to be acquired

"Spearman, and others [...], carried out many ingenious researches using mental tests and guided by his 'two factor' hypothesis."
(Godfrey Thomson, 1947, "Charles Spearman, 1863–1945", doi.org/10.1098/rsbm.1947.0006)

Summary

Key observations: Count hybrid nouns show

1. Interpretive variation with respec to individuation

• Flexible individuation: rope, string, paper
• Specific indiivduation: apple, chocolate, potato

2. Unexpected mass meaning in the plural for some: potatoes, clouds, lentils

Proposal

• Singular count nouns may encode compositionally opaque individuations
• The denotations of plural count nouns are computed via compeition with mass

The observations pose challanges for the traditional decompositional approach

RSA with Lexical Uncertainty

We assume that N_plural has two potential interpretations, plural and mass

We implement this ambiguity in terms of a parameter $p$ on $[\![\cdot]\!]$ (Bergen, Levy & Goodman 2016, Scontras & Goodman 2017)

$$\begin{array}{lcl} [\![N_{\textrm{plural}}]\!]^p & = & \ast([\![N_{\textrm{sg.count}}]\!])\\\\ [\![N_{\textrm{plural}}]\!]^m & = & [\![N_{\textrm{mass}}]\!] \end{array}$$

RSA with Lexical Uncertainty

• The literal listner $L_0$ infers what interpretation $i$ is intended based on the utterance $u$ and a parameter $x$ on $[\![\cdot]\!]$

$$L_0(i | u, x) \propto P(i) \cdot [\![u]\!]^x(i)$$

• The $n$th speaker $S_n$ picks out the best message for $i$ given $x$, relative to $L_{n-1}$

$$S_n(u | i, x) \propto e^{\lambda \cdot (\log L_{n-1}(i | u, x)-\textrm{cost}(u))}$$

• Infrequent expressions are more costly

• The $n$th listener $L_{n}$ decodes $u$ relative to $S_n$

$$L_n(i, x | u) \propto S_n(u | i, x)\cdot P(i) \cdot P(x)$$

Simulation