• theoretical & experimental pragmatics
  • natural language quantifiers
  • typicality of quantifier \(\textit{some}\)
• generalized linear model
  • types of dependent variables
  • predictors & link functions
• probabilistic model
  • gradient salience of alternative expressions
  • one predictor feeds two link functions

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some examples:

• \(\textit{no}\), \(\textit{some}\), \(\textit{all}\)
• \(\textit{most}\), \(\textit{many}\), \(\textit{few}\)
• \(\textit{three}\), \(\textit{at least 4}\), \(\textit{between 8 and 12}\)
• \(\textit{less than Jake drank}\), \(\textit{2 more than Jake could possibly drink}\)

• "No \(A\) is \(B\)" is true iff there is no \(A\) that is also a \(B\).
• "Some \(A\) is \(B\)" is true iff there is at least one \(A\) that is also a \(B\).
• "All \(A\) are \(B\)" is true iff there is no \(A\) that is not also a \(B\).

"None/some/all of the circles are black."

truth-value judgements for "Some of the circles are black."

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implicature

utterance meaning = semantic meaning + rational language use

Cooperative Principle

Make your contribution such as it is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged.

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Maxim of Quantity

1. Make your contribution as informative as is required for the current purposes of the exchange.
2. Do not make your contribution more informative than is required.

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Maxim of Relevance

Be relevant!

When would a cooperative speaker say: "Some of the 10 circles are black"?

upshot

The pragmatic felicity of a description \(m\) for a situation \(c\) is a measure of how adequate \(m\) is for a given purpose of talk relative to alternative descriptions.

pragmatic felicity depends on:

• purpose of conversation
• salience of alternatives

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pramgatic felicity is an elusive notion

ideally, we'd like to have a formal, even quantitative notion

• enter computational pragmatics

• replication/extension of previous work
  • van Tiel (2014), Degen & Tanenhaus (2015)
• 4 experimental variants:
  • binary truth-value judgements vs. 7-point rating scale
  • include filler sentences with \(\textit{many}\) and \(\textit{most}\) or not
• participants recruited via Amazon's Mechanical Turk

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• do binary and ordinal tasks measure the same thing?
  • one is about truth, the other about "goodness"
• is what either task measures influenced by presence/absence of alternatives?
• how would we answer these questions with standard statistical techniques?

data

head(cars) 

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

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model

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\[\text{logistic}(\eta, \theta, \gamma) = \frac{1}{(1 + \exp(-\gamma (\eta - \theta)))}\]

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• quantitative notion of pragmatic felicity \(F\) replaces predictor \(\eta\)
  • \(F\) is (function of) relative expected utility:
    • goodness of description (with "some") compared to alternative descriptions?
  • data-driven approach to infer gradient salience of alternatives
• \(F\) feeds into two link functions:
  • logistic model for binary truth-value judgements
  • thresholded-Phi model for rating scale judgements

• conditions \(c \in \{0, \dots, 10\}\): number of black balls
• degrees \(d \in \{1, \dots, 7\}\)
• messages \(m \in M = \{\textit{none}, \textit{one}, \textit{two}, \textit{three}, \textit{many}, \textit{most}, \textit{all}, \textit{some}\}\)
• semantics:

##       c=0 c=1 c=2 c=3 c=4 c=5 c=6 c=7 c=8 c=9 c=10
## none    1   0   0   0   0   0   0   0   0   0    0
## one     0   1   0   0   0   0   0   0   0   0    0
## two     0   0   1   0   0   0   0   0   0   0    0
## three   0   0   0   1   0   0   0   0   0   0    0
## many    0   0   0   0   0   1   1   1   1   1    1
## most    0   0   0   0   0   0   1   1   1   1    1
## all     0   0   0   0   0   0   0   0   0   0    1
## some    0   1   1   1   1   1   1   1   1   1    1

literal listener picks literal interpretation (uniformly at random):

\[ P_{LL}(c \mid m) = \text{Uniform}(c \mid \{ c' \mid m \text{ is true in } c' \} ) \]

utility for true \(c\) and interpretation \(c'\):

\[ U(c, c' \ ; \ \pi) = \exp(- \pi \ (c - c')^2 ) \]

expected utility:

\[ \text{EU}(m, c \ ; \ \pi) = \sum_{c'} P_{LL}(c' \mid m) \ U(c, c' \ ; \ \pi) \]

Gricean speakers choose maximally informative/useful messages:

\[ m \in \arg \max_{m' \in M} \text{EU}(m', c \ ; \ \pi) \]

scaled expected utility given set \(X\) of entertained alternatives:

\[ \text{EU}^*(c , X \ ; \ \pi) = \frac{\text{EU}(\textit{some}, c) - \min_{m \in X} \text{EU}(m, c)}{\max_{m \in X} \text{EU}(m, c) - \min_{m \in X} \text{EU}(m, c)} \]

salience of alternatives \(m \in M \setminus \{ \textit{some} \}\):

\[ s_m \sim \text{Beta}(1,1) \]

probability of entertaining \(X \subseteq M\) (crudely assume independence!):

\[ P(X \mid \vec{s}) = \prod_{m \in X} s_m \prod_{m \in M \setminus X} \ (1-s_m) \]

expected relative felicity:

\[ \text{F}(c \ ; \ \vec{s}, \pi) = \sum_X P(X \mid \vec{s}) \ \text{EU}^*(c , X \ ; \ \pi) \]

• model implemented in JAGS (Plummer 2003)
• 10,000 samples after 10,000 burn-in steps (2 chains, every second sample used)
• convergence checked visually and by \(\hat{R}\) (Gelman & Rubin 1992)

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## 256 sets to create.

• idea that truth-value and rating-scale task measure the same thing is tenable
• measure: scaled relative expected utitlity under variably salient alternatives
• this is influenced by presence/absence of alternatives
• theory-driven probabilistic modeling can advance methodological debate
• important to make explicit link functions part of full data-generating model