Events

The sample-space page answered “what could happen?” But most linguistic questions do not ask whether one exact form occurred. They ask whether the form has a property, such as third person, plural number, or accusative case. How do we represent such a question?

Basically, we collect every outcome that would make the answer “yes.” More specifically, an event is a subset of the sample space. Once a sigma-algebra \mathcal F has been specified, the measurable events are exactly the members of \mathcal F.

Begin with the twelve-form sample space

\Omega \equiv\{\textit{I},\textit{me},\textit{you},\textit{they},\textit{them}, \textit{it},\textit{she},\textit{her},\textit{he},\textit{him}, \textit{we},\textit{us}\}.

First ask which outcomes count as third person. Define the third-person event by

T_{12}\equiv\{\textit{they},\textit{them},\textit{it},\textit{she}, \textit{her},\textit{he},\textit{him}\}

and its complement by

T_{12}^c\equiv\Omega\setminus T_{12} =\{\textit{I},\textit{me},\textit{you},\textit{we},\textit{us}\}.

Now ask the corresponding case questions. The original case events are

A_{12}\equiv\{\textit{me},\textit{you},\textit{them},\textit{her}, \textit{him},\textit{it},\textit{us}\}

and

N_{12}\equiv\{\textit{I},\textit{you},\textit{they},\textit{she}, \textit{he},\textit{it},\textit{we}\}.

These two case events overlap:

A_{12}\cap N_{12}=\{\textit{you},\textit{it}\}.

Thus the surface form alone does not distinguish nominative from accusative uses of you or it, an instance of case syncretism. The sample also treats her only as a nonpossessive personal-pronoun token; the string her alone would not identify that restriction. These facts are properties of the chosen outcome representation.

The reduced four-form space below isolates number and case for the numerical probability calculations later in the module. It does not replace the original twelve-form construction.

Use the following stipulated sample space for the formal examples:

\Omega_4 \equiv\{\textit{he},\textit{him},\textit{they},\textit{them}\}.

The observation process selects one case-coded pronoun token from a population restricted to these four forms. Within this toy representation, he and him are coded singular, they and them plural, he and they nominative-form, and him and them accusative-form. Singular they is outside the stipulated population. These codings define the example; they are not claims about every use of these surface strings.

Defining the number and case events

In the smaller space, the linguistic questions are deliberately simple: is the form plural, and is it accusative? Each question determines one event.

Define the plural event by

P_4\equiv\{\textit{they},\textit{them}\}

and the accusative event by

A_4\equiv\{\textit{him},\textit{them}\}.

The outcome them belongs to both P_4 and A_4. Thus one outcome can make both events occur.

Intersection

The intersection contains the outcomes shared by two events:

P_4\cap A_4=\{\textit{them}\}.

Thus P_4\cap A_4 occurs exactly when the selected token is both plural and accusative under the stipulated coding.

Union

The union contains the outcomes belonging to at least one of two events:

P_4\cup A_4 =\{\textit{him},\textit{they},\textit{them}\}.

The connective is inclusive. The outcome them belongs to the union because it belongs to both events.

Complement

The complement is always relative to the declared sample space:

P_4^c \equiv\Omega_4\setminus P_4 =\{\textit{he},\textit{him}\}.

Thus changing \Omega can change P_4^c even when the members of P_4 remain fixed.

Event membership

For the outcome them,

\textit{them}\in P_4, \qquad \textit{them}\in A_4, \qquad \textit{them}\in P_4\cap A_4.

For the outcome him,

\textit{him}\notin P_4, \qquad \textit{him}\in A_4, \qquad \textit{him}\in P_4\cup A_4.

The limiting events

The empty event \varnothing contains no outcomes, whereas the full event \Omega_4 contains every represented outcome. Both are events because both are subsets of \Omega_4.

These two limiting cases answer useful checks. The impossible event never occurs under the representation, while the full event occurs for every represented outcome.

The outcome determines whether events can co-occur

Under this model, one outcome is one pronoun token. The nominative and accusative events are thus disjoint. If one outcome were instead an entire sentence containing several pronouns, the events “contains a nominative pronoun” and “contains an accusative pronoun” could occur together.

Mutual exclusivity follows from the event sets and the unit represented by one outcome. It does not follow from the labels alone.

The exact upshot is that an event turns a yes-or-no question about an outcome into a set. The next page asks which of those sets a particular probability model is allowed to measure.

Code

omega <- c("he", "him", "they", "them")
plural <- c("they", "them")
accusative <- c("him", "them")

intersect(plural, accusative)
union(plural, accusative)
setdiff(omega, plural)

Check your understanding

Define the nominative event N_4 and verify that N_4\cap A_4=\varnothing.
List P_4\cap A_4, P_4\cup A_4, and A_4^c.
Explain why them can make two events occur without counting as two outcomes.
Give a different outcome representation under which nominative and accusative events can co-occur.

--- title: "Events" --- The [sample-space page](possibilities-and-events.qmd) answered “what could happen?” But most linguistic questions do not ask whether one exact form occurred. They ask whether the form has a property, such as third person, plural number, or accusative case. How do we represent such a question? Basically, we collect every outcome that would make the answer “yes.” More specifically, an [**event**](https://bruno.nicenboim.me/bayescogsci/ch-intro.html) is a subset of the sample space. Once a sigma-algebra $\mathcal F$ has been specified, the measurable events are exactly the members of $\mathcal F$. Begin with the twelve-form sample space $$ \Omega \equiv\{\textit{I},\textit{me},\textit{you},\textit{they},\textit{them}, \textit{it},\textit{she},\textit{her},\textit{he},\textit{him}, \textit{we},\textit{us}\}. $$ First ask which outcomes count as third person. Define the third-person event by $$ T_{12}\equiv\{\textit{they},\textit{them},\textit{it},\textit{she}, \textit{her},\textit{he},\textit{him}\} $$ and its complement by $$ T_{12}^c\equiv\Omega\setminus T_{12} =\{\textit{I},\textit{me},\textit{you},\textit{we},\textit{us}\}. $$ Now ask the corresponding case questions. The original case events are $$ A_{12}\equiv\{\textit{me},\textit{you},\textit{them},\textit{her}, \textit{him},\textit{it},\textit{us}\} $$ and $$ N_{12}\equiv\{\textit{I},\textit{you},\textit{they},\textit{she}, \textit{he},\textit{it},\textit{we}\}. $$ These two case events overlap: $$ A_{12}\cap N_{12}=\{\textit{you},\textit{it}\}. $$ Thus the surface form alone does not distinguish nominative from accusative uses of *you* or *it*, an instance of [case syncretism](https://babel.ling.upenn.edu/~beatrice/syntax-textbook/ch8.html). The sample also treats *her* only as a nonpossessive personal-pronoun token; the string *her* alone would not identify that restriction. These facts are properties of the chosen outcome representation. The reduced four-form space below isolates number and case for the numerical probability calculations later in the module. It does not replace the original twelve-form construction. Use the following stipulated sample space for the formal examples: $$ \Omega_4 \equiv\{\textit{he},\textit{him},\textit{they},\textit{them}\}. $$ The observation process selects one case-coded pronoun token from a population restricted to these four forms. Within this toy representation, *he* and *him* are coded singular, *they* and *them* plural, *he* and *they* nominative-form, and *him* and *them* accusative-form. Singular *they* is outside the stipulated population. These codings define the example; they are not claims about every use of these surface strings. ## Defining the number and case events In the smaller space, the linguistic questions are deliberately simple: is the form plural, and is it accusative? Each question determines one event. Define the plural event by $$ P_4\equiv\{\textit{they},\textit{them}\} $$ and the accusative event by $$ A_4\equiv\{\textit{him},\textit{them}\}. $$ The outcome *them* belongs to both $P_4$ and $A_4$. Thus one outcome can make both events occur. ## Intersection The intersection contains the outcomes shared by two events: $$ P_4\cap A_4=\{\textit{them}\}. $$ Thus $P_4\cap A_4$ occurs exactly when the selected token is both plural and accusative under the stipulated coding. ## Union The union contains the outcomes belonging to at least one of two events: $$ P_4\cup A_4 =\{\textit{him},\textit{they},\textit{them}\}. $$ The connective is inclusive. The outcome *them* belongs to the union because it belongs to both events. ## Complement The complement is always relative to the declared sample space: $$ P_4^c \equiv\Omega_4\setminus P_4 =\{\textit{he},\textit{him}\}. $$ Thus changing $\Omega$ can change $P_4^c$ even when the members of $P_4$ remain fixed. ## Event membership For the outcome *them*, $$ \textit{them}\in P_4, \qquad \textit{them}\in A_4, \qquad \textit{them}\in P_4\cap A_4. $$ For the outcome *him*, $$ \textit{him}\notin P_4, \qquad \textit{him}\in A_4, \qquad \textit{him}\in P_4\cup A_4. $$ ## The limiting events The empty event $\varnothing$ contains no outcomes, whereas the full event $\Omega_4$ contains every represented outcome. Both are events because both are subsets of $\Omega_4$. These two limiting cases answer useful checks. The impossible event never occurs under the representation, while the full event occurs for every represented outcome. ## The outcome determines whether events can co-occur Under this model, one outcome is one pronoun token. The nominative and accusative events are thus disjoint. If one outcome were instead an entire sentence containing several pronouns, the events “contains a nominative pronoun” and “contains an accusative pronoun” could occur together. Mutual exclusivity follows from the event sets and the unit represented by one outcome. It does not follow from the labels alone. The exact upshot is that an event turns a yes-or-no question about an outcome into a set. The [next page](event-spaces.qmd) asks which of those sets a particular probability model is allowed to measure. ```{r} #| eval: false omega <- c("he", "him", "they", "them") plural <- c("they", "them") accusative <- c("him", "them") intersect(plural, accusative) union(plural, accusative) setdiff(omega, plural) ``` ## Check your understanding 1. Define the nominative event $N_4$ and verify that $N_4\cap A_4=\varnothing$. 2. List $P_4\cap A_4$, $P_4\cup A_4$, and $A_4^c$. 3. Explain why *them* can make two events occur without counting as two outcomes. 4. Give a different outcome representation under which nominative and accusative events can co-occur.