Trivalent foundations for a logic of anaphora

Discourse anaphora informs content

Pronouns and their antecedents

Heim 1982, Kamp 1988, a.o.

  1. Andrew has a child.
    She is at school.

  2. Andrew is a parent.
    # She is at school.

  3. Exactly one of the ten balls is not in the bag.
    It is under the sofa.

  4. Exactly nine of the ten balls are in the bag.
    # It is under the sofa.


  • The requirement for an indefinite antecedent is sometimes called the Formal Link Condition.

  • The dynamic view: Content is influenced not just by what is said, but how it is said (Kamp 1981, Heim 1982).

  • In talking, we keep track not just of information concerning how things are, but also referential information concerning what/who the speaker intended to refer to.

    • An utterance of “Andrew has a child” is about a child of Andrew’s in a way that “Andrew is a parent” is not.
    • Another way of saying the same thing: “Andrew has a child” introduces a discourse referent (Karttunen 1969)

The dynamic perspective

  • Dynamic semantics enriches conversational contexts with referential information concerning intended reference (Irene Heim’s notion of a file).

  • Part of the dynamic thesis is that uttering a sentence with an indefinite changes the referential information in the conversational context in a special way.

  • Sentences with pronouns are sensitive to referential information in a way which other expressions aren’t.

  • Dynamic semantics provides an elegant account of the aboutness properties of sentences with indefinites.
    • Dynamic semantics has a problem however: the behaviour of anaphora in complex sentences motivates a theory in which individual lexical items are in charge of referential information flow.

Where we’re going

  • We’ll begin by touring data concerning possible anaphoric dependencies between indefinites and pronouns taken for granted in orthodox dynamic theories — the accessibility generalizations.
  • We’ll lay out some problems and tensions for the accessibility generalizations, concluding that:
    • Orthodox dynamic semantics isn’t classical enough.
    • We need to take into account the role of pragmatics in anaphoric licensing.
  • The goal will be a dynamic semantics in which referential information is passed uniformly from left-to-right.
    • The resulting theory will be rather permissive - we’ll restrict it in the pragmatic component.

Anaphora in complex sentences


“and” is the dynamic connective par excellence: Referential information flows from left-to-right, and flows outwards.

  1. A linguist walked in, and she sat next to a philosopher.
    He yawned.

  2. # She walked in, and a linguist sat down.

In the dynamic parlance, conjunction is internally and externally dynamic. (Groenendijk & Stokhof 1991; henceforth G&S)


Referential information flows from left-to-right, but may flow no further.

  1. If a linguist is here, she’s sitting next to a philosopher.
    # He yawned.

Conditionals are internally dynamic but externally static.


Negation acts like a dam, blocking referential information from flowing further.

  1. It’s not the case that a linguist is here.
    # She yawned.

Negation is externally static.


There are no through-channels between disjuncts; referential information flows neither between disjuncts, nor beyond the disjunction.

  1. # Either a linguist is here, or she’s smoking outside.

  2. Either a linguist is here, or it’s raining.
    # She’s smoking outside.

Disjunction is internally and externally static

Context change potentials

DS puts the lexicon in charge of regulating referential information flow.

If logical connectives are responsible for regulating the flow of information, we make no substantial predictions about how the truth-conditional contribution of the connectives relates to referential information flow.

(Schlenker 2006 a.o. make the same point wrt. presupposition projection.)

An alternative take

To be developed today:

  • Taking the accessibility generalizations to be indicative of the semantic contribution of the logical connectives precludes a principled account of referential information flow.
  • The accessibility generalizations are empirically flawed; accessibility is sensitive to contextual factors.
  • The main point: the logical connectives are exactly what they seem to be - truth-functional operators - referential information flows uniformly from left-to-right.

Accessibility revisited

Double negation

In DS, negation destroys referential information. But consider the following:

  1. Frank doesn’t own no shirt. It’s in the closet.

  2. It’s not the case that Frank doesn’t own any shirt. It’s in the closet.

Referential information can be resurrected by an additional negation (Krahmer & Muskens 1995, Gotham 2019).

Bathroom disjunctions

Again, remember that in DS, negation destroys referential information…

  1. Either there isn’t a bathroom, or it’s upstairs.
  • Referential information destroyed in an initial disjunct can be resurrected in a subsequent disjunct (observation due to Barbara Partee).

  • N.b. this intuitively parallels presupposition projection facts (Beaver 2001):

  1. Either Sam never smoked, or he stopped smoking.

Groenendijk & Stokhof disjunctions

Under certain circumstances, the ban on passing referential information further is lifted (G&S 1991, Kamp & Reyle 1993, Simons 1996):

  1. Either we’re interviewing a linguist, or we’re interviewing a philosopher.
    (Either way) she’s waiting outside.

(G&S chalk this up to a lexical ambiguity, positing an additional, externally dynamic disjunction operator; program disjunction.)

Anaphora and contextual entailment

  • The final issue is, in my opinion, the most telling of all.

f the truth of one of the disjuncts is contextually entailed later in the discourse, anaphora becomes possible (see Rothschild 2017, Mandelkern 2020 for related observations).

  1. A: Either it’s a weekday, or Gabe baked a cake.
    B: It’s Saturday afternoon.
    A: Then, it’s cooling on the windowsill!

Beyond disjunction

Recall that conditionals are also (claimed to be) externally static.

  1. A: If it’s the weekend, then Gabe baked a cake.
    B: It’s Saturday afternoon.
    A: Then, it’s cooling on the windowsill!

Take-away points

  • Double-negation suggests we want a theory that is more classical than, e.g., orthodox DS.
  • Bathroom disjunctions tell us that disjunction isn’t straightforwardly internally static; referential information can be transmitted between disjuncts.
  • G&S disjunctions tell us that disjunction isn’t straightforwardly externally static; referential information can be passed further.
  • Anaphora and contextual entailment tells us that pragmatic factors have a role to play in regulating the availability of subsequent anaphora.

The accessibility generalizations assumed in DS are full of holes. Anaphora licensed by contextual entailment suggests a way of resolving the tension - develop a more classical, more permissive dynamic semantics that can be restricted in the pragmatics.

Anaphora redux

A logical substrate

We’ll develop a theory of anaphora based on the following ideas:

  • The core semantic value of a sentence is a (trivalent) truth-value; referential information is computed in tandem with this logical substrate.
  • The logical connectives operate exclusively on the logical substrate.
    • Referential information is passed from left-to-right uniformly.
    • Incrementality in anaphoric processing is achieved by flipping a single “switch” throughout the grammar.

Enriching the dynamic notion of content

  • Assignments are a store of referential information; they tell us how to fix the value of variables (i.e., pronouns).
  • Sentences express functions from assignments (the input) to sets of assignment-truth-value pairs (the outputs) (Charlow 2014, 2019).
    • Another way of thinking about this: output assignments are polarized.
    • Whence talk of a ‘logical substrate’.
  • Sentences containing neither indefinites nor pronouns polarize the input according to truth-conditional content.

\[ ⟦\text{Aeryn left}⟧ = λ g . \begin{cases} \{(⊤,g)\}&\color{gray}{\text{Aeryn left}}\\ \{(⊥,g)\}&\color{gray}{\text{Aeryn didn't leave}} \end{cases} \]

Pronouns and the third truth-value

  • Sentences with pronouns polarize the input assignment \(g\) depending on truth at \(g\).
  • We’ll assume that assignments are partial; namely, an assignment \(g\) may not deliver a value for a particular index \(n\).

\[ ⟦\text{She}_1\text{ left}⟧^g = \begin{cases} \{(⊤,g)\}&\color{gray}{g(1)\text{ is defined and }g(1)\text{ left}}\\ \{(⊥,g)\}&\color{gray}{g(1)\text{ is defined and }g(1)\text{ didn't leave}}\\ \{(\#,g)\}&\color{gray}{g(1)\text{ is undefined}} \end{cases} \]

  • Our trivalent logical substrate is starting to do some work.

Indefinites and indeterminacy

  • We’ll use sets to model the idea that indefinites extend the input assignment indeterministically (following G&S).
    • Another way of thinking about this — indefinites introduce alternatives.

\[⟦\text{Someone}^1\text{ is here}⟧^g = \begin{cases} \{(\color{red}{⊤},g^{[1 → x]}) \mid x \text{ is here}\}\\ \qquad\color{gray}{\text{someone is here}}\\ \{(\color{red}{⊥},g)\}\\ \qquad\color{gray}{\text{nobody is here}} \end{cases}\]

  • What’s important here is that, due to the logical substrate, we distinguish between:
    • referential information introduced when there is a verifier (the input is extended indeterministically).
    • referential information introduced when there is no verifier (the input remains unchanged).

Illustration i

  1. \(⟦\text{A}^1\text{ triangle is in the circle.}⟧^{[]}\)

\[ = \begin{cases} \{(⊤,[x]) \mid x\text{ is a triangle in the circle}\}\\ \{(⊥,[])\}\qquad\color{gray}{\text{there is no triangle in the circle}} \end{cases} \]

\[ = \{(⊤,[\color{red}{Δ}]),(⊤,[\color{blue}{Δ}])\} \]

Illustration ii

  1. \(⟦\text{A}^1\text{ triangle is in the circle.}⟧^{[]}\)
\[ = \begin{cases} \{(⊤,[x]) \mid x\text{ is a triangle in the circle}\}\\ \{(⊥,[])\}\qquad\color{gray}{\text{there is no triangle in the circle}} \end{cases} \]

\[ = \{(⊥,[])\} \]

Comparison with first-generation Dynamic Semantics

  • If there is a witness for the existential statement, then the true-paired outputs are equivalent to what is delivered by first-generation dynamic theories, such as Groenendijk & Stokhof’s (1991) Dynamic Predicate Logic.
  • If there is no witness for the existential statement, the output is just the input assignment paired with false.
  • In first-generation theories however, if there is no witness then the output set is simply empty; here we crucially keep track of both positive and negative referential information.

Payoff #1: negation

  • Remember, one of the core foundations of the current approach: logical operators are truth-functional.
  • Negation is lifted through the dynamic scaffolding, and applies pointwise to the contained truth-values.

\[ ⟦\text{Nobody is here}⟧ = λ g . \begin{cases} \{(\color{red}{¬\,⊤},g^{[1 → x]}) \mid x\text{ is here}\}\\ \{(\color{red}{¬\,⊥},g)\}\\ \qquad\color{gray}{\text{nobody is here}} \end{cases} \]

\[ = λ g . \begin{cases} \{(\color{red}{⊥},g^{[1 → x]}) \mid x\text{ is here}\}\\ \{(\color{red}{⊤},g)\}\\ \qquad\color{gray}{\text{nobody is here}} \end{cases} \]

  • N.b. referential information survives, but it’s false-tagged.

Double negation

  • Since negation is truth-functional, applying another negation will flip the polarities again.

\[ \begin{aligned}[t] &⟦\text{It's not the case that nobody is here}⟧\\ &= λ g . \begin{cases} \{(\color{red}{¬ ¬\,⊤},g^{[1 → x]} \mid x\text{ is here})\}\\ \{(\color{red}{¬ ¬\,⊥},g)\}\\ \qquad\color{gray}{\text{nobody is here}} \end{cases} \end{aligned} \]

\[ = λ g . \begin{cases} \{(\color{red}{⊤},g^{[1 → x]} \mid x\text{ is here})\}\\ \{(\color{red}{⊥},g)\}\\ \qquad\color{gray}{\text{nobody is here}} \end{cases} \]

  • A doubly negated sentence conveys the same referential information as its positive counterpart!

Interlude: Strong Kleene

  • Before we proceed much further, a remark is due on the semantics we’ll assume for the logical connectives.

    • Recall that a foundational assumption of the current approach is that logical operators are truth-functional; but we have three truth values (\(⊤,⊥,\#\)) to consider, not just two, which opens up a number of possibilities.
  • The interpretation schema we’ll adopt for the logical connectives is Kleene’s strong logic of indeterminacy; (i.e., Strong Kleene).

  • This is the logic that naturally emerges if we take \(\#\) to stand in for uncertainty whether true or false.

Strong Kleene conjunction

  • In Strong Kleene, conjunctive sentences are:
    • true if both conjuncts are true,
    • false if either conjunct is false.
    • I.e., strong Kleene conjunctions have conjunctive verification conditions and disjunctive falsification conditions.

Truth-table for Strong Kleene conjunction (\(∧^s\)):

\(∧^s\) \(\mathbf{⊤}\) \(\mathbf{⊥}\) \(\mathbf{\#}\)
\(\mathbf{⊤}\) \(⊤\) \(⊥\) \(\#\)
\(\mathbf{⊥}\) \(⊥\) \(⊥\) \(⊥\)
\(\mathbf{\#}\) \(\#\) \(⊥\) \(\#\)

Strong Kleene disjunction

  • In Strong Kleene, disjunctive sentences are:
    • true if either disjunct is true,
    • false if both disjuncts are false.
    • I.e., strong Kleene disjunctions have disjunctive verification conditions and conjunctive falsification conditions.

Truth-table for Strong Kleene disjunction (\(∨^s\))

\(∨^s\) \(\mathbf{⊤}\) \(\mathbf{⊥}\) \(\mathbf{\#}\)
\(\mathbf{⊤}\) \(⊤\) \(⊤\) \(⊤\)
\(\mathbf{⊥}\) \(⊤\) \(⊥\) \(\#\)
\(\mathbf{\#}\) \(⊤\) \(\#\) \(\#\)

On explanatory adequacy

  • The Strong Kleene entries for the connectives don’t need to be stipulated but rather reify how we ordinarily reason about uncertainty.

  • An independent reason to think that Strong Kleene is a reasonable foundation: I agree with Rothschild (2017, p1) that “[…] when the dust has settled, this remains the simplest viable treatment of presupposition projection on the market.

  • Important: the Strong Kleene logic underlying the analysis here is completely symmetric; there is no room for asymmetries or incrementality in the logical substrate.

    • Compare, e.g., George (2007, 2008, 2014) who incrementalizes the logic in order to account for asymmetries in presupposition projection.

Lifting the logical operators

  • In order to lift binary logical operators into a dynamic setting, we’ll adopt much the same strategy as with negation.
    • The truth-functional operator applies pointwise to the values in the logical substrate.
    • This time, we have two sets of outputs to worry about - the referential information ouputed by the first junct as passed in as the input to the second.

  • Some details for the more technically-minded. For any two-place truth-functional operator \(\mathbf{R}\):

\[ ⟦ϕ\,\mathbf{R}\,ψ⟧^g = \{(\color{red}{t\,\mathbf{R}\,u},i)\mid ∃h[(\color{red}{t},h) ∈ ⟦ϕ⟧^g ∧ (\color{red}{u},i) ∈ ⟦ψ⟧^h]\} \]

  • N.b. Charlow (2014, 2019) reifies this algorithm in the compositional semantics via a regime of type-shifters; the current approach can be spelled out in these terms too.

Important: the locus of stipulation on the current approach is in the algorithm for lifting any truth-functional operator into the dynamic domain of referential-information-passing. Nothing is stipulated about the dynamics of the individual logical connectives.

A note on architecture

  • Our algorithm relies on the assumption that the semantics can see the order of the juncts.

  • This has important architectural ramifications regarding the information visible to the semantics.

Payoff 2: Dynamic conjunction

  • Applying this algorithm to \(∧^s\) gives us something equivalent to DPL conjunction wrt positive referential information.
    • We can see this most clearly if we just compute the positive extension (\(⟦.⟧_+\); the referential information paired with \(\color{red}{⊤}\)).

\[ ⟦\text{Someone}^1\text{ walked in}⟧_+^g = \{g^{[1 → x]} \mid x\text{ walked in}\} \]

\[ ⟦\text{She}_1\text{ sat down}⟧_+^g = \{g \mid \text{sat down }g(1)\text{ and }g(1)\text{ is defined}\} \]

\[ \begin{aligned}[t] &⟦\text{Someone}^1\text{ walked in and she}_1\text{ sat down}⟧_+^g\\ & = \{g^{[1 → x]}\mid x\text{ walked in and sat down}\} \end{aligned} \]

  • Conjunction is asymmetric despite a symmetric logical substrate.
  • N.b., when we take into account the possibility of either conjunct being false the predictions diverge from orthodox DS. We’ll come back to this when we discuss the pragmatics.

Payoff 3: Bathroom disjunctions

Let’s now return to bathroom disjunctions.

  1. Either there isn’t a\(^1\) bathroom, or it\(_1\)’s upstairs.

Applying our algorithm to \(∨^s\) gives us a disjunctive recipe for dynamically verifying disjunctive sentences:

\[ ⟦ϕ ∨ ψ⟧_+^g = \begin{aligned}[t] \{i | ∃h[h ∈ ⟦ϕ⟧^g_+ ∧ (*,i) ∈ ⟦ψ⟧^h]\}\\ ∪ \{i | ∃h[(*,h) ∈ ⟦ϕ⟧^g ∧ i ∈ ⟦ψ⟧^h_+]\} \end{aligned} \]

  • In plain English, we can compute the positive referential information conveyed by the disjunctive sentence by:
    • passing the positive referential information of the first disjunct into the second, and gathering all of the outputs, positive or otherwise.
    • passing the outputs of the first disjunct (positive or otherwise) into the second, and retaining just the positive outputs.
  • For our purposes, the second clause is important: we can pass the negative ouputs of the first disjunct into the second, and if there are positive outputs we dynamically verify the disjunctive sentence.

  • Because of how negation works, if there is a bathroom (\(b\)), this will be stored in the negative information conveyed by the first disjunct “there isn’t a bathroom”:

\[ \{(⊥,[b])\} \]

  • The false-tagged assignment licenses a pronoun in the subsequent sentence, which will be paired with true if there is indeed a bathroom upstairs:

\[ ⟦\text{it}_1\text{'s upstairs}⟧^{[b]} = \{(⊤,[b])\mid b\text{ is upstairs}\} \]

N.b. an apparently bad prediction that you may notice at this point is that we predict a bathroom disjunction can license a subsequent pronoun, at least if there is indeed a bathroom.

  1. Either there isn’t a bathroom, or it’s upstairs.
    # It is very clean.

The dynamic connective par excellence

  • One way of thinking about what we’re doing: we take DPL conjunction to be the dynamic connective par excellence, and generalize it.
    • Referential information is always piped between juncts (from left-to-right), and can in principle be passed further.

This leads to a problem with external staticity.

  1. Either a\(^1\) linguist is here or it’s raining.
    # She\(_1\)’s smoking outside.

We predict that if a linguist is indeed here, anaphora should be licensed in (22).

  • Just how bad is this result? In the next section i’ll argue that, in stating accessibiliy generalizations, we haven’t been paying enough attention to the pragmatic component.
  • Once we understand some of the independently motivated pragmatic constraints on asserting complex, sentences, we’ll be in a position to understand apparent external staticity.
    • A move foreshadowed by the contextual entailment facts.

The pragmatics of anaphora

Pragmatics via Stalnaker and Heim

  • In order to keep things simple, we’ve been operating in a purely extensional setting.

  • To return to one of the themes explored at the beginning of this talk, we’d like a theory which addresses both worldly, and referential information.

  • Following Heim (1982), we’ll treat the context set (which we’ll write as \(C\)), as a set of world-assignment pairs.

  • Before anything has been uttered, every world in \(C\) is simply paired with the initial assignment \([]\).


  • We can mechanically intensionalize our theory by adding a world parameter to our interpretation function.
    • outputs are now truth-value/world/assignment triples

\[⟦\text{Someone}^1\text{ is here}⟧^{\color{#20A5BA}{w},\color{#10A778}{g}} = \begin{cases} \{(\color{#C30771}{⊤},\color{#20A5BA}{w},\color{#10A778}{g^{[1 → x]}}) \mid x \text{ is here in }w\}\\ \qquad\color{gray}{\text{someone is here}}\\ \{(\color{#C30771}{⊥},\color{#20A5BA}{w},\color{#10A778}{g})\}\\ \qquad\color{gray}{\text{nobody is here in }w} \end{cases}\]

\[ \begin{aligned}[t] &⟦\text{She}_1\text{ left}⟧^{\color{#20A5BA}{w},\color{#10A778}{g}}\\ &= \begin{cases} \{(\color{#C30771}{⊤},\color{#20A5BA}{w},\color{#10A778}{g})\}&\color{gray}{g(1)\text{ is defined and }g(1)\text{ left in }w}\\ \{(\color{#C30771}{⊥},\color{#20A5BA}{w},\color{#10A778}{g})\}&\color{gray}{g(1)\text{ is defined and }g(1)\text{ didn't leave in }w}\\ \{(\color{#C30771}{\#},\color{#20A5BA}{w},\color{#10A778}{g})\}&\color{gray}{g(1)\text{ is undefined}} \end{cases} \end{aligned} \]

Stalnaker’s bridge

  • We update \(C\) with a sentence \(ϕ\) (\(C[ϕ]\)) by evaluating \(ϕ\) wrt each world-assignment pair in \(C\), and retaining the true-tagged outputs.

  • Stalnaker’s bridge (von Fintel 2008) demands that for assertion of \(ϕ\) to be successful, \(ϕ\) cannot be undefined at any point in \(C\).

  • A direct consequence: a sentence with a pronoun indexed \(n\) carries a referential presupposition that \(n\) is defined.

\[\{(\color{#20A5BA}{w},\color{#10A778}{[]}),(\color{#20A5BA}{w},\color{#10A778}{[a]})\}[\text{she}_1\text{ left}] \text{ is undefined}\]

\[\begin{aligned}[t] &\{(\color{#20A5BA}{w},\color{#10A778}{[a]}),(\color{#20A5BA}{w},\color{#10A778}{[b]})\}[\text{she}_1\text{ left}]\\ &= \{(\color{#20A5BA}{w},\color{#10A778}{[x]})\mid x\text{ left in }w\text{ & }x ∈ \{a,b\}\} \end{aligned}\]

Indefinites and updates

Some things that fall out automatically:

  • Updating \(C\) with an existential statement results in a new context-set that licenses the referential presupposition of a subsequent coindexed pronoun.

\[ \begin{aligned}[t] &C[\text{Andrew has a}^1\text{ child}]\\ &= \{(\color{#20A5BA}{w},\color{#10A778}{g^{[1 → x]}})\mid (w,g) ∈ C\,\&\,x\text{ a child of A's in }w\} \end{aligned} \]

\[ \begin{aligned}[t] &C[\text{Andrew is a parent}]\\ &= \{(\color{#20A5BA}{w},\color{#10A778}{g})\mid (w,g) ∈ C\,\&\,\text{A has a child in }w\} \end{aligned} \]

  • A witness to the existential statement is a necessary but not sufficient condition for licensing subsequent anaphora; using an indefinite is crucial.

A pragmatic constraint on disjunction

  • We now have everything we need to understand apparent external staticity with disjunction

  • We can independently observe that an utterance “P or Q” is only felicitous if P and Q are both real possibilies wrt the context set.

    • \(C\) shouldn’t entail the truth or falsity of either disjunct.

Context: It’s common ground that Alice was in the audience.

  1. # Either someone was in the audience or the event was a disaster.

We use this to account for apparent external staticity.

Accounting for external staticity

Consider the following space of logical possibilities, which we’ll use to represent a context set in which “Either someone\(^1\) was in the audience, or the event was a disaster” might be felicitously uttered:

  • \(\color{#20A5BA}{w_{ad}}\): \(a\) was in the audience; the event was a disaster.
  • \(\color{#20A5BA}{w_{a¬d}}\): \(a\) was in the audience; the event was successful.
  • \(\color{#20A5BA}{w_{∅d}}\): Nobody was in the audience; the event was a disaster.
  • \(\color{#20A5BA}{w_{∅¬d}}\): Nobody was in the audience; the event wasn’t a disaster.

Updating with the disjunctive sentence will:

  • Knock out the final world (\(\color{#20A5BA}{w_{∅¬d}}\)).
  • Extend assignments paired with \(a\)-worlds with \(a\) (\(\color{#20A5BA}{w_{ad}}\),\(\color{#20A5BA}{w_{a¬d}}\)).
  • Fail to extend the assignment paired with \(\color{#20A5BA}{w_{∅d}}\).

\[ \left\{\begin{aligned}[c] &(\color{#20A5BA}{w_{ad}},\color{#10A778}{[]})\\ &(\color{#20A5BA}{w_{a¬d}},\color{#10A778}{[]})\\ &(\color{#20A5BA}{w_{∅d}},\color{#10A778}{[]})\\ &(\color{#20A5BA}{w_{∅¬d}},\color{#10A778}{[]})\\ \end{aligned}\right\}\left[\begin{aligned}[c] &\text{someone}^1\text{ was in the audience}\\ &\text{or the event was a disaster} \end{aligned}\right] \]

\[ = \{(\color{#20A5BA}{w_{ad}},\color{#10A778}{[a]}),(\color{#20A5BA}{w_{a¬d}},\color{#10A778}{[a]}),(\color{#20A5BA}{w_{∅d}},\color{#10A778}{[]})\} \]

  • Note that the resulting context set is not one in which the presupposition of a subsequent pronoun will be licensed.
    • This reflects the fact that a witness for the existential statement is not contextually entailed post-assertion!
    • We therefore get the appearance of external staticity - this does not however motivate saying anything special about the dynamics of disjunction. Moreover, our system doesn’t allow us to say anything special.

Addressing the role of contextual entailment

  • The explanation for anaphora and contextual entailment falls out straightforwardly.
    • After using an indefinite, if a witness for the existential statement is subsequently contextually entailed, we predict anaphora to be possible.
  • For example, a subsequent utterance might knock out the problematic world assignment pair:

\[ \left\{\begin{aligned}[c] &(\color{#20A5BA}{w_{ad}},\color{#10A778}{[a]})\\ &(\color{#20A5BA}{w_{a¬d}},\color{#10A778}{[a]})\\ &(\color{#20A5BA}{w_{∅d}},\color{#10A778}{[]})\\ \end{aligned}\right\}\left[\text{the event wasn't a disaster}\right] \]

\[ = \{(\color{#20A5BA}{w_{ad}},\color{#10A778}{[a]}),(\color{#20A5BA}{w_{a¬d}},\color{#10A778}{[a]})\} \]

This correctly predicts the subsequent availability a pronoun.

  1. A: Either someone was in the audience or the event was a disaster.
    B: The event wasn’t a disaster.
    A: In that case, I hope she enjoyed it!

Disjunctive subordination

An exception that proves the rule: Anaphora appears to be possible when not disrupted by uncertainty:

  1. Each linguist [met with a philosopher or read a philosophy paper]

  2. They each [found her interesting or couldn’t understand it]

(N.b. these cases involve quantificational subordination, and require an extension of the system to a plural setting ala van den Berg 1996)

Negating conjunctions

I’ll note here that we can extend this general explanation to negations of conjunctive sentences, which we otherwise predict can license subsequent anaphora if the first conjunct is true second conjunct is false.

  1. It’s not the case that [anyone walked in and they sat down].
    # They remained standing.

The explanation takes advantage of the fact that an utterance of the form “not (P and Q)” typically requires “not P” and “not Q” to be real possibilities in \(C\). See Elliott (2020) for details.

G&S disjunctions

We now straightforwardly account for G&S disjunctions (discussed earlier).

  1. Either we’re interviewing a\(^1\) linguist
    or we’re interviewing a\(^1\) philosopher.

Updating a context set in which both disjuncts are real possibilities will extend every assignment with a linguist/philosopher at index \(1\).

  • The presupposition of a subsequent co-indexed pronoun will be satisfied, exactly because G&S disjunctions contextually entail a witness.
  • It’s crucial that the indefinites in each disjunct are co-indexed (this has interesting consequences for Heim’s novelty condition - ask me about this in the Q&A).

Internal staticity and Hurford disjunctions

Recall that disjunctions appear to be internally static

  1. # Either a linguist is here, or she’s smoking outside.

Since our algorithm for passing referential is modeled on dynamic conjunction, there’s no obvious reason why (29) should be unacceptable on the intended reading.

Simons’ observation

In order to explain what’s going on here, we’ll begin with Simons’ (1996) observation that the following sentence is deviant:

  1. # Either someone is in the audience,
    or the person in the audience is sitting down.

What Simons observes is that, when a disjunct Strawson entails the other, the disjunction is odd.

  • In other words, if the presupposition of the second disjunct is satisfied, in entails the first.
  • This is a special case of Hurford’s constraint (HC): a disjunction is odd if one disjunct entails the other.

Hurford’s constraint in a dynamic setting

Since we want to accommodate the possibility of anaphora between disjuncts, we need to state this constraint in a way that accommodates referential information flow.

\(\text{P or Q}\)” is odd if
\(⟦\text{(not P) and Q}⟧_+^g = ∅ ∨ ⟦\text{P and not Q}⟧_+^g = ∅, ∀ g\)

In plain English: a disjunction is odd if there is no way of verifying the disjunction other than by verifying both disjuncts.

  1. # Someone\(^1\) is in the audience, or she\(_1\) is sitting down.

Here, if the first disjunct is false, the second is always undefined

Wrapping up

The bottom line

  • We’ve achieved a dynamic semantics which is up-front about what exactly it stipulates.
    • Concretely, the locus of stipulation is in the statement of the algorithm for passing referential information, which we stipulate passes information from left-to-right.
  • The idea is that there is a single switch which gives rise to incrementality in anaphoric processing; this isn’t localized to the lexical entries of individual connectives.
    • This makes a clear predictions with respect to acquisition, and cross-linguistic uniformity with respect to referential information flow.

  • In developing a more principled theory of anaphora, what we’ve learned is that the literature has essentially been mistaken in taking the accessibility generalizations at face value.
    • In order to maintain a parsimonious semantic theory, due care needs to be taken to address the role of pragmatic factors.
    • Developing an understanding of the pragmatics of referential information is essential in order to improve on our understanding of the semantic component.
  • As we’ve seen, it’s possible to retain some of the appealing aspects of dynamic semantics - such as the dynamic notion of content - while improving upon the stipulative nature of extant dynamic theories.

Thank you!


I’m deeply grateful to Simon Charlow, Keny Chatain, Enrico Flor, Danny Fox, Matthew Gotham, Julian Grove, Nathan Klinedinst, Matt Mandelkern, and Yasu Sudo for feedback which greatly improved this work.

I’d also like to thank audiences at Rutgers, NYU, and LENLS 17 for their feedback.


Ext 1: Donkey anaphora

  • Although material implication doesn’t give a realistic semantics for conditionals, it’s interesting to consider what this system predicts when we apply our algorithm for logical connectives.
    • Unlike orthodox dynamic semantics, we systematically predict so-called “weak” or “existential” truth-conditions for donkey sentences (Kanazawa 1994, Chierchia 1995); ask me more about this in the Q&A.
    • Such readings are attested, but strong readings must be derived via some kind of strengthening mechanism.

Ext 2: Generalized quantifiers and modal subordination

The algorithm we’ve outlined for the logical connectives needs to be generalized to quantifiers such as every linguist and most philosophers.

  • This is necessary in order to develop a principled account of donkey anaphora which doesn’t face the proportion problem.

Another interesting extension is modal subordination. In Elliott (2020b) I develop an extend this system with epistemic modality, in order to account for what I call conjunctive bathroom sentences.

  1. Maybe there is no bathroom and maybe it is upstairs.