Overview

One of the central technical notions that we’ll rely on in this course is that of a model. A model is, in short, a way of associating the most basic expressions of whatever language we are studying with interpretations. It can be useful to think of a model as encoding a way that things could be, or—in the parlance of possible world semantics—a “possible world”. For example, if we are assigning meanings to a language that has the verb runs as one of its basic expressions, then we might associate this verb with a set—specifically, the set of people (or other entities) that run. In one model, Jo and Bo might be the ones who run (so that the set of people who run is $\{\ct{j}, \ct{b}\}$); in another, the interpretation of runs might be the set containing Bo and Mo ($\{\ct{b}, \ct{m}\}$); in yet another, maybe nobody runs ($\varnothing$); and so on.

Technically, a model $\mathcal{M}$ is a structure that consists of a pair of a domain $D$ and an interpretation function $I$—something which takes atoms (i.e., the most basic expressions, whatever those are) of the language onto elements of a set $M$ (‘$M$’ for “meanings”):

Definition:
a model $\mathcal{M}$ is a tuple $⟨D, \{\True, \False\}, I⟩$ of a non-empty set $D$, the set of truth values $\True$ and $\False$, and a function $I : atom → M$.

In (1), $atom$ is just whatever the set of atoms is chosen to be.

Exactly how $M$ is chosen will vary by case—but in general, it is something “constructed” out of $D$, the domain. For example, it might be $D$ itself; or it might be $D ∪ 2^{D}$ (the set containing as members both all of the elements of $D$ and all of the subsets of $D$); or it might be something else more complicated.

Further, the way the set $atom$ is chosen will also vary by case. For example, if we are defining a language $L$ over some alphaet $Σ$, $atom$ might be chosen to be $Σ$ itself—or perhaps some subset of $Σ$. The way we decide what $atom$ is depends on how we decide we want to construct the interpretations of more complex expressions of the language out of the meanings of more basic expressions.

--- title: "Overview" bibliography: ../../semantics.bib format: html: css: ../styles.css html-math-method: mathjax mathjax-config: loader: {load: ['[tex]/bussproofs','[tex]/bbox','[tex]/colorbox']} tex: packages: {'[+]': ['bussproofs','bbox','colorbox']} --- ::: {.hidden} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\True{\ct{T}} \def\False{\ct{F}} $$ ::: One of the central technical notions that we'll rely on in this course is that of a model. A model is, in short, a way of associating the most basic expressions of whatever language we are studying with interpretations. It can be useful to think of a model as encoding a *way that things could be*, or---in the parlance of possible world semantics---a "possible world". For example, if we are assigning meanings to a language that has the verb *runs* as one of its basic expressions, then we might associate this verb with a set---specifically, the set of people (or other entities) that run. In one model, Jo and Bo might be the ones who run (so that the set of people who run is $\{\ct{j}, \ct{b}\}$); in another, the interpretation of *runs* might be the set containing Bo and Mo ($\{\ct{b}, \ct{m}\}$); in yet another, maybe nobody runs ($\varnothing$); and so on. Technically, a model $\mathcal{M}$ is a structure that consists of a pair of a *domain* $D$ and an interpretation function $I$---something which takes *atoms* (i.e., the most basic expressions, whatever those are) of the language onto elements of a set $M$ ('$M$' for "meanings"): (@ex-model-def) Definition: <br> a model $\mathcal{M}$ is a tuple $⟨D, \{\True, \False\}, I⟩$ of a non-empty set $D$, the set of truth values $\True$ and $\False$, and a function $I : atom → M$. In (@ex-model-def), $atom$ is just whatever the set of atoms is chosen to be. Exactly how $M$ is chosen will vary by case---but in general, it is something "constructed" out of $D$, the domain. For example, it might be $D$ itself; or it might be $D ∪ 2^{D}$ (the set containing as members both all of the elements of $D$ and all of the subsets of $D$); or it might be something else more complicated. Further, the way the set $atom$ is chosen will also vary by case. For example, if we are defining a language $L$ over some alphaet $Σ$, $atom$ might be chosen to be $Σ$ itself---or perhaps some subset of $Σ$. The way we decide what $atom$ is depends on how we decide we want to construct the interpretations of more complex expressions of the language out of the meanings of more basic expressions.