Kaplanian semantic model

A model of semantics must capture how even what might be called literal meaning might depend on context. For instance:

(1)
He went to the bank yesterday.

This has the context-dependent "he" and "yesterday" as well as the homonymous "bank".

One might distinguish "character" and "content". For instance, these sentences have the same character but different contents:

(2)
Mary: I am hungry.
John: I am hungry.

So sentences have a character, a content/intension, and an extension. Given a character, you resolve references, fix indexicals, and disambiguate to get the content/intension, which is a function from possible worlds to truth values. Then depending on the possible world that is assumed, predicates will have different extensions.

Write out a formal model with these features.

Below is a sketch of a Kaplan‐style formal framework (inspired by David Kaplan’s “Demonstratives” and related work) that distinguishes character, content (or intension), and extension, while also accommodating context‐sensitivity and ambiguity. This kind of framework captures how an expression’s literal meaning can depend on features of the context (e.g., speaker, time, disambiguation parameters), how one gets a full “content” (a proposition, in the case of sentences) once those parameters are fixed, and how that content then evaluates to a truth value in each possible world.

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1. The Ingredients

1.1. Worlds and Domains

1. Let $W$ be a nonempty set of possible worlds.
2. For each world $w\in W$, let $D_w$ be the domain of individuals (or objects) in that world.
   • Depending on one’s semantic framework, one may either assume a single domain $D$ across all worlds, or allow $D_w$ to vary with $w$. We will treat the more general case where $D_w$ can vary.

1.2. Contexts

A context $c$ provides all the extra‐linguistic parameters needed to interpret context‐sensitive expressions:

$$c=⟨\text{Speaker}(c),\text{Time}(c),{\text{World}}^{\ast }(c),\text{Location}(c),\dots ,\text{Disambiguation}(c)⟩.$$

For example:

• $\text{Speaker}(c)$ is the individual speaking in context $c$.
• $\text{Time}(c)$ is the time of utterance in context $c$.
• ${\text{World}}^{\ast }(c)$ is sometimes called the “world of the context” or the “actual world” as per that context.
• $\text{Location}(c)$ may specify the location of utterance, if relevant.
• $\text{Disambiguation}(c)$ can capture which sense of a homonymous word is intended (e.g., “bank” as a financial institution vs. “bank” as a river bank).

Let $C$ be the set of all possible contexts.

1.3. Character, Content (Intension), and Extension

We will distinguish:

1. Character: A function from contexts to contents.

   • Intuitively, this is the “linguistic meaning” plus rules for how to use context‐sensitive items (indexicals like “I” or “yesterday,” pronominal references like “he,” and so on).

2. Content (or intension, in the case of sentences): A function from worlds (and possibly times, if we treat time as part of the index) to extensions.

   • For a sentence, the content will be a function from worlds to truth values ($\{\mathrm{\top },\mathrm{\perp }\}$)—i.e., a proposition.
   • For a noun phrase, the content might be a function from worlds to individuals (its referent in each world).
   • For a predicate, the content might be a function from worlds to sets of individuals (its extension in each world).

3. Extension: The result of evaluating the content at a particular world (and possibly time). For a sentence, the extension is a truth value; for a singular term, the extension is the referent (an individual) in that world, etc.

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2. A Formal Model

We can wrap these pieces up into a single framework as follows.

2.1. The Model Structure

A Kaplanian Model $\mathcal{M}$ can be given by a tuple:

$$\mathcal{M}=⟨W,(D_w{)}_{w\in W},C,\mathcal{I},\mathcal{Char}⟩,$$

where

1. $W$ is a nonempty set of possible worlds.

2. For each $w\in W$, $D_w$ is the domain of that world.

3. $C$ is a set of possible contexts.

4. $\mathcal{I}$ is an interpretation function that assigns basic (non‐context‐sensitive) expressions and constants to their contents (e.g., to individuals, sets of individuals, etc., in each world).

5. $\mathcal{Char}$ is a function that assigns characters to (possibly complex) expressions. Each character is itself a function:

   $$\mathcal{Char}(e):C\to (W\to \text{ExtensionSpace}),$$

   where “$\text{ExtensionSpace}$” depends on syntactic category (for instance, $\{\mathrm{\top },\mathrm{\perp }\}$ for sentences, or $\bigcup _{w\in W}{D}_w$ for singular terms, etc.).

In more familiar (less abstract) terms, for each expression $e$, $\mathcal{Char}(e)$ is the mapping: $$\mathcal{Char}(e):\text{context }c\mapsto \text{content }[[e]{]}_c.$$ And then each content $[[e]{]}_c$ is a function: $$[[e]{]}_c:W\mapsto \text{extension }[[e]{]}_{c,w}.$$

Hence, for each context $c\in C$ and world $w\in W$, we can speak of the extension $[[e]{]}_{c,w}$.

2.2. Illustrating the Process

1. Character ${\chi }_e$: Given an expression $e$, its character ${\chi }_e$ says how to use context to determine the content.

   • For example, the character of “I am hungry” is a function that takes a context $c$ and returns the proposition “$\text{Speaker}(c)$ is hungry at $\text{Time}(c)$.”

2. Content $[[e]{]}_c$: Once a context $c$ is fixed, we get an intension (for sentences, a proposition) that can vary across worlds.

   • So $[[\text{I am hungry}]{]}_c$ is a function from $W$ to truth values: in each $w\in W$, it is true iff the speaker in $c$ is hungry at the time in $c$ in world $w$.

3. Extension $[[e]{]}_{c,w}$: Finally, evaluating that content at a world $w$ gives the truth value (for a sentence) or individual (for a referring expression) in that world.

   • Concretely, $[[\text{I am hungry}]{]}_{c,w}=\mathrm{\top }$ if and only if the speaker in context $c$ is hungry at time $\text{Time}(c)$ in the world $w$; otherwise it is $\mathrm{\perp }$.

2.3. Handling Indexicals, Pronouns, and Ambiguity

• Indexicals: “I,” “now,” “yesterday” each have a character telling us how to fetch a referent from the context $c$. For instance: $${\chi }_{\text{I}}(c)=\lambda w.\text{Speaker}(c),{\chi }_{\text{yesterday}}(c)=\lambda w.\text{Time}(c)-1\text{day}.$$ (Here we are eliding many details about how exactly times are modeled, but the idea is that “yesterday” picks out “$\text{Time}(c)$ minus one day.”)

• Pronouns like “he” or “she” can be anaphoric or demonstrative. A simplistic approach is that their character is a function from context to a specific referent, determined by either an antecedent in discourse or a demonstration. Real theories refine how context determines referents.

• Ambiguity / Homonymy: For words like “bank,” the character would also look at the $\text{Disambiguation}(c)$ parameter. Roughly: $${\chi }_{\text{bank}}(c)=\{\begin{array}{ll}\lambda w.\text{river\_bank}& \text{if }\text{Disambiguation}(c)=\text{river}\\ \lambda w.\text{financial\_bank}& \text{if }\text{Disambiguation}(c)=\text{finance}\end{array}$$ This says: if the context indicates the “river” sense, interpret “bank” as $\text{river\_bank}$, etc.

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3. Examples

3.1. Example (1)

“He went to the bank yesterday.”

1. Character step: From the context $c$, we determine:

   • The referent of “he” in $c$ (say $\text{HeRef}(c)$).
   • The relevant sense of “bank” (financial institution vs. river bank).
   • The time referred to by “yesterday,” i.e. $\text{Time}(c)-1$ day.

2. Content (intension): A function $[[\text{He went to the bank yesterday}]{]}_c:W\to \{\mathrm{\top },\mathrm{\perp }\}$, which, for each world $w$, says whether $\text{HeRef}(c)$ went to the contextually chosen “bank” on the day $\text{Time}(c)-1$ in $w$.

3. Extension: For a particular world $w$, the sentence is true ($\mathrm{\top }$) if in $w$, that person performed the relevant “bank‐going” event at the relevant day/time.

3.2. Example (2)

Mary: “I am hungry.”
John: “I am hungry.”

• Both sentences have the same character: the function that, given a context $c$, picks out the speaker of $c$ and asserts that this speaker is hungry at $\text{Time}(c)$.
• But if Mary utters it, the context $c_{\text{Mary}}$ has $\text{Speaker}(c_{\text{Mary}})=\text{Mary}$.
• If John utters it, the context $c_{\text{John}}$ has $\text{Speaker}(c_{\text{John}})=\text{John}$.

Thus:

1. Same character: ${\chi }_{\text{I am hungry}}$.

2. Different contents: $$[[\text{I am hungry}]{]}_{c_{\text{Mary}}}=\lambda w.\text{Mary is hungry at Time}(c_{\text{Mary}})\text{ in }w.$$$$[[\text{I am hungry}]{]}_{c_{\text{John}}}=\lambda w.\text{John is hungry at Time}(c_{\text{John}})\text{ in }w.$$

3. Extension in a specific world $w$: Each of these contents could be true or false depending on whether Mary or John is hungry in $w$ at that time.

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4. Summary of the Formal Steps

1. Define a set of contexts $C$. Each context $c$ carries all relevant parameters for interpreting indexical and ambiguous expressions (speaker, time, location, “disambiguation,” etc.).
2. Define a set of worlds $W$ and for each $w\in W$, a domain $D_w$.
3. Assign a character ${\chi }_e$ to each expression $e$. The character is a function: $${\chi }_e:C\to (W\to \text{ExtensionSpace}).$$
4. Given a context $c$, the content of $e$ in $c$ is $[[e]{]}_c:={\chi }_e(c)$, which is an intensional object (a function from worlds to extensions).
5. Given $c$ and $w$, the extension of $e$ is $[[e]{]}_{c,w}:={\chi }_e(c)(w)$.

This completes a high‐level, formal outline of a semantic model that captures how literal meaning (character) can depend on context, how that yields a context‐fixed content (intension), and how evaluation in a given world yields an extension (truth value or referent).