Setting the stage

To round out this set of notes, we give a brief overview of the main desiderata which we aim to have PDS satisfy. These are the following:

compositionality: models of inference should be derived compositionally from semantic grammar fragments.
modularity: factors affecting inference judgments should be theorized about independently and combined.
abstraction: models of meaning and inference should be statable abstractly, without reference to implementation.

We say a little more about these here, in turn.

Compositionality for models

What could it mean for models of linguistic inference (e.g., as represented in a judgment dataset) to be compositional? Our basic basic strategy is to build the distributional assumptions associated with, e.g., a mixed-effects model, into the semantics. This way, when basic meanings compose, so do these distributional assumptions.

Indeed, it is reasonable to ask what such distributional assumptions represent. Our answer is uncertainty; specifically, uncertainty about which particular inferences are licensed on any given occasion of language use. Importantly, these kinds of assumptions may be combined when determining the meanings of complex expressions from the meanings of the more basic expressions they contain. Thus while the meaning of a sentence such as Jo laughs might be determined compositionally as

\[ ⟦\textit{jo laughs}⟧ = ⟦\textit{laughs}⟧ ▹ ⟦\textit{jo}⟧ = laughs(j) \]

within a traditional semantic framework, PDS, instead, compositionally associates this sentence with a probability distribution.

\[⟦\textit{jo laughs}⟧ = ⟦\textit{laughs}⟧ ▹ ⟦\textit{jo}⟧ = \begin{array}[t]{l} j ∼ JoDistr \\ laugh ∼ LaughDistr \\ Return (laugh(j)) \end{array} \]

This distribution is a distribution over truth values: it assigns some probability $p$ to True and $1 - p$ to False. Moreover, it is determined by certain sampling statements (whose interpretations we will formally define tomorrow). Informally, the meaning of Jo takes some distribution over entities, while the meaning of laughs takes some distribution over functions from entities to truth values. These distributions may then be combined to yield a distribution over values gotten by applying such functions applied to such entities; i.e., a distribution over truth values. Remarkably, the distributions over the meanings of such basic expressions end up corresponding exactly, within PDS, to the parameters of some hierarchical Bayesian (e.g., mixed-effects) model which may be used to fit human inference judgment data.

Modularity

We also want theories constructed within PDS to be modular. Specifically, we want the factors affecting inference to be able to be theorized about independently and combined. These include:

lexical and compositional semantics
world knowledge
response behavior: how does someone use a testing instrument (e.g., slider scale)?

An upshot of this feature is that PDS can have different uses. For example, one could swap out a model of response behavior for a model of likely utterances (perhaps, $S_{1}$).

Abstraction

Finally, we want such theories to display a certain amount of abstraction. That is, we should be able to state models of inference judgment data that:

describe probability distributions,
do not concern themselves with how distributions are computed.

There are a couple useful consequences of this feature. First, it allows traditional semantic theories to be plugged into PDS rather seamlessly. Second, it allows separation between theories stated within PDS and model stated within those thoeries. This second consequence allows:

Allows flexibility about implementation.
Allows the theory to be simpler.
Allows seamless integration between formal semantics and probabilistic semantics. (More tomorrow!)

--- title: "Setting the stage" bibliography: ../../pds.bib format: html: css: ../styles.css --- To round out this set of notes, we give a brief overview of the main desiderata which we aim to have PDS satisfy. These are the following: - [compositionality]{.gb-orange}: models of inference should be derived compositionally from semantic grammar fragments. - [modularity]{.gb-orange}: factors affecting inference judgments should be theorized about independently and combined. - [abstraction]{.gb-orange}: models of meaning and inference should be statable *abstractly*, without reference to implementation. We say a little more about these here, in turn. #### Compositionality for models What could it mean for models of linguistic inference (e.g., as represented in a judgment dataset) to be *compositional*? Our basic basic strategy is to build the distributional assumptions associated with, e.g., a mixed-effects model, *into the semantics*. This way, when basic meanings compose, so do these distributional assumptions. Indeed, it is reasonable to ask what such distributional assumptions represent. Our answer is *uncertainty*; specifically, uncertainty about which particular inferences are licensed on any given occasion of language use. Importantly, these kinds of assumptions may be combined when determining the meanings of complex expressions from the meanings of the more basic expressions they contain. Thus while the meaning of a sentence such as *Jo laughs* might be determined compositionally as $$ ⟦\textit{jo laughs}⟧ = ⟦\textit{laughs}⟧ ▹ ⟦\textit{jo}⟧ = laughs(j) $$ within a traditional semantic framework, PDS, instead, compositionally associates this sentence with a *probability distribution*. $$⟦\textit{jo laughs}⟧ = ⟦\textit{laughs}⟧ ▹ ⟦\textit{jo}⟧ = \begin{array}[t]{l} j ∼ JoDistr \\ laugh ∼ LaughDistr \\ Return (laugh(j)) \end{array} $$ This distribution is a distribution over truth values: it assigns some probability $p$ to True and $1 - p$ to False. Moreover, it is determined by certain *sampling statements* (whose interpretations we will formally define tomorrow). Informally, the meaning of *Jo* takes some distribution over entities, while the meaning of *laughs* takes some distribution over functions from entities to truth values. These distributions may then be *combined* to yield a distribution over values gotten by applying such functions applied to such entities; i.e., a distribution over truth values. Remarkably, the distributions over the meanings of such basic expressions end up corresponding exactly, within PDS, to the parameters of some hierarchical Bayesian (e.g., mixed-effects) model which may be used to fit human inference judgment data. #### Modularity We also want theories constructed within PDS to be modular. Specifically, we want the *factors affecting inference* to be able to be theorized about independently and combined. These include: - lexical and compositional semantics - world knowledge - response behavior: how does someone use a testing instrument (e.g., slider scale)? An upshot of this feature is that PDS can have different uses. For example, one could swap out a model of response behavior for a model of likely utterances (perhaps, $S_{1}$). #### Abstraction Finally, we want such theories to display a certain amount of abstraction. That is, we should be able to state models of inference judgment data that: - *describe* probability distributions, - do not concern themselves with how distributions are computed. There are a couple useful consequences of this feature. First, it allows traditional semantic theories to be plugged into PDS rather seamlessly. Second, it allows separation between theories stated within PDS and model stated within those thoeries. This second consequence allows: - Allows flexibility about implementation. - Allows the theory to be simpler. - Allows seamless integration between formal semantics and probabilistic semantics. (More tomorrow!)