Anaphora and negation

Abstract

One of the central questions of discourse dynamics is when an anaphoric pronoun is licensed. This paper addresses this question as it pertains to the complex data involving anaphora and negation. It is commonly held that negation blocks anaphoric potential, for example, we cannot say “Bill doesn’t have a car. It is black”. However, there are many exceptions to this generalization. This paper examines a variety of types of discourses in which anaphora on indefinites under the scope of negation is felicitous. These cases are not just of intrinsic interest, but I argue present serious problems for the dynamic semantic framework, which builds the licensing facts into the semantics. I argue in favor of adopting a dynamic pragmatics, a theory that explains context change through general Gricean principles, and combining it with a static, d-type theory of anaphora, in which pronouns go proxy for definite descriptions.

A formal implementation

In this appendix, I briefly outline the formal implementation of both context and the semantics of pronouns, showing how these work together to account for basic cases of discourse anaphora.

• 1. Context: A context C contains the following elements:

• DR, the set of discourse referents, modeled as indices. These indices are the domain of the assignment functions.

• WG, a set of world/assignment function pairs $\langle$w, g$\rangle$ such that w $\in$ W, where W is the set of all possible worlds, and g is one of the possible assignments of indices in DR to entities in w, for each possible such assignment g.⁴²

• CG, the common ground, modeled a context set, i.e.{w $\in$ W $\mid$ w is possible given conversational presuppositions}

• QUD, the set of questions under discussion.

• 2. Syntax: The syntax of the language includes variables x, y, z, with or without subscripts, numerical indices, logical constants (proper names), predicates (from English), the logical connectives $\wedge$, $\vee$, $\lnot$, the indefinite article a, the definite article the, the generalized quantifiers every, some, most, few, no.

• 3. Semantics:

• A model M = $\langle$W, D$_e$, D$_t$, I$\rangle$ where W = set of worlds w, D$_e$ = domain of entities e, D$_t$ = {0,1}, and I = interpretation function

• For each constant c, I(c) = some e $\in$ D$_e$

• For n-place predicates p, I(p) = {$\langle$w,{$\langle$e$_1$,...e$_n\rangle$, $\langle$e$_1$,...e$_n \rangle$...}$\rangle$, $\langle$w,{$\langle$e$_1$,...e$_n \rangle$, $\langle$e$_1$,...e$_n \rangle$...}$\rangle$...}, such that w $\in$ W and e$_1$,...e$_n$ $\in$ D$_e$, and the second member of the n-tuple is a set of entities if p is a 1-place predicate and a set of n-tuples of entities otherwise.

• The logical connectives have their ordinary static denotations.⁴³

• The quantifiers have standard denotations from generalized quantifier theory, e.g.:

• a. $\llbracket$[every x: $\phi$]($\psi$)$\rrbracket ^{M, w, c,h}$ = 1 iff $\forall$e in D$_e$ s.t. $\llbracket \phi \rrbracket ^{h[x\rightarrow e]} = 1$ at w, $\llbracket \psi \rrbracket ^{h[x\rightarrow e]} = 1$ at w

• b. $\llbracket$[some x: $\phi$]($\psi$)$\rrbracket ^{M, w, c,h} = 1$ iff $\exists$e in D$_e$ s.t. $\llbracket \phi \rrbracket ^{h[x\rightarrow e]} = 1$ at w & $\llbracket \psi \rrbracket ^{h[x\rightarrow e]} = 1$ at w

• The assignment functions relative to which denotations are calculated are distinct from the ones in WG. The assignment functions in WG are functions from indices to entities, the other assignment functions are functions from variables to entities or indices. I reserve ‘g’ for the former and ‘h’ for the latter.

• An has the same denotation as the existential quantifier some.

• Pronouns are equivalent to definite descriptions with null overt material, and the definite article is treated as a generalized quantifier that presupposes discourse uniqueness:

  $$ \begin{aligned}&\llbracket \text {[The x:} \phi ] (\psi )\rrbracket ^{M, w, c, h} \\&\quad = {\left\{ \begin{array}{ll} \text {defined if } \exists !\text {n} \in \text {DR s.t. for all} \ \langle \text {w,g} \rangle \in \text {WG}, \llbracket \phi \rrbracket ^{h[x\rightarrow g(n)]} = 1 \ \text {at w}\\ \text {1 iff } \exists \text {e in D}_e \ \text {s.t.} \ \llbracket \phi \rrbracket ^{h[x\rightarrow e]} = 1 \ \text {at w} \ \& \llbracket \psi \rrbracket ^{h[x\rightarrow e]} = 1 \ \text {at w}\\ \text {0 otherwise} \end{array}\right. } \end{aligned}$$

The clause defines the discourse uniqueness presupposition as the requirement that there is exactly one index n in DR such that the (possibly complex, massively conjoined) property associated with that index is exactly the same as the property denoted by the descriptive material in the definite description. The presupposition is a condition on definedness. The relevant descriptive material on the definite description that must satisfy discourse uniqueness is the completed description. The completed description occurs via the mechanism of quantifier domain restriction. If defined, The F is G is true iff there is at least one F that is G.

I treat quantifier domain restriction following Stanley and Szabo (2000). All quantifiers are restricted by a variable that shares a node with the noun at the level of syntax. The variable is of the form f(i), where i is an individual variable that can either be bound or get a value from the context and f is a contextually determined function from individuals to quantifier domains, which in the intensional version is a property, a function from worlds to sets of individuals. The domain of the quantifier is determined by combining the denotation of the overt predicate (if any) and the result of f(i). (In Stanley and Szabo’s extensional version, they combine by set intersection. In the intensional version, we can think of them combining by something like looks exactly like predicate modification, except for the fact that the noun and variables share the same node.) For example, take the sentence ‘every student was happy’ in a particular context of use, formally written as [Every x: $\langle$student, f(i)$\rangle$ x] (happy x). In a particular context, f might be the function from events to a function from worlds to their participants at that world, and the value for i might be the 2019 Met Gala. Hence, f(i) yields a function from worlds to the set of all the participants of the 2019 Met Gala at that world, and combining that with the denotation of students (i.e. a function from worlds to students at that world), we get students at the 2019 Met Gala as the restrictor property. Thus we get the restricted universal claim that every student at the 2019 Met Gala was happy.

As I said, pronouns are treated as definite descriptions with null overt material, and so the descriptive material comes entirely from the quantifier domain restriction.⁴⁴ Let’s see how this applies to our central example:

(57)	a.	A woman walked in.
	a’.	[An x: woman x] (walked in x)45
	b.	She sat down.
	b’.	[The $x: \langle \emptyset$,f(i)$\rangle x$] (sat down x)

Introduction and update of discourse referents are modeled as change to DR and WG. (57a) adds a novel index, say 1, to DR, and changes WG such that the new set contains all the 1-variants of the input assignment functions, such that 1 is now assigned to an individual in I(woman) and I(walked.in), relative to its paired world. An 1-variant of an assignment function g is here defined as all the possible extensions of g that assign 1 to something in the relevant interpretation.

Consider the following toy model:

• 1. W = {$w_1$, $w_2$, $w_3$}

• 2. D$_e$ = {Alice, Bob, Carol, David, Emily, Francine}

• 3. I(woman) = {$\langle w_1$, {Alice, Carol, Emily, Francine}$\rangle$, $\langle w_2$, {Alice, Carol, Emily, Francine}$\rangle$, $\langle w_3$, {Alice, Carol, Emily, Francine}$\rangle$}

• 4. I(walked in) = {$\langle w_1$, {Alice, Carol, Bob}$\rangle$, $\langle w_2$, {Bob, Carol, David, Emily}$\rangle$, $\langle w_3$, {Alice, Emily, Francine}$\rangle$}

• 5. I(sat down) = {$\langle w_1$, {Alice, Bob, Francine}$\rangle$, $\langle w_2$, {Carol, Emily}$\rangle$, $\langle w_3$, {Francine}$\rangle$}

For simplicity assume a initial context in which CG = W and DR = {}. I suppress the QUD in what follows. The pragmatic effect on the context of asserting (57a) is to add a novel discourse referent for a woman who walked in, and to eliminate all worlds in CG that are incompatible with a woman walking in:

Context after (57a) is asserted:

• 1. CG = W $\cap$ $\llbracket$a woman walked in$\rrbracket ^{M,c,h}$ = W$_1$

• 2. DR = {1}

• 3. WG = {$\langle w_1$, g$_1$: 1$\rightarrow$ Alice$\rangle$, $\langle w_1$, g$_2$: 1$\rightarrow$ Carol$\rangle$, $\langle w_2$, g$_3$: 1$\rightarrow$ Carol$\rangle$, $\langle w_2$, g$_4$: 1$\rightarrow$ Emily$\rangle$, $\langle w_3$, g$_5$: 1$\rightarrow$ Alice$\rangle$, $\langle w_3$, g$_6$: 1$\rightarrow$ Emily$\rangle$, $\langle w_3$, g$_7$: 1$\rightarrow$ Francine$\rangle$}

The discourse referent serves as the value for the salient individual i in (57b), and the QDR function that goes in for the variable f gathers all the information associated with that discourse referent, yielding the set of all possible individuals who could be a witness for the discourse referent relative to each possible world:

Contextual QDR function for anaphoric pronouns:

For some index n, f(n) = {$\langle$w, {e$_1$...e$_n$}$\rangle$ $\mid$ w is the first member of a tuple in WG and {e$_1$...e$_n$} is the set of all entities assigned to n by some g which is the second member of a tuple that has w as its first member}

This will yield a function from worlds w to the set of all women who walked in at w. (57b) is then true at a world iff there is at least one object in that domain of possible witnesses that sat down, i.e. it is true iff there is at least one woman who walked in and sat down.

Context after (57b) is asserted:

• 1. CG = W$_1$ $\cap$ $\llbracket$some woman walked in and sat down$\rrbracket ^{M,c,h}$

• 2. DR = {1}

• 3. WG = {$\langle w_1$, g$_1$: 1$\rightarrow$ Alice$\rangle$, $\langle w_2$, g$_3$: 1$\rightarrow$ Carol$\rangle$, $\langle w_2$, g$_4$: 1$\rightarrow$ Emily$\rangle$, $\langle w_3$, g$_7$: 1$\rightarrow$ Francine$\rangle$}

The effect here is again purely pragmatically motivated. Eliminating incompatible assignment functions captures the fact that conversational participants use discourse referents to track what properties hang together as satisfied by a single witness according to the discourse.

For the cases that involve accommodation, such as the external felicity cases central to this paper, the relevant discourse referent is accommodated in the context before the semantic machinery does its work. For example, take (39): when we get to the pronoun ‘it’, there is no unique discourse referent to which it can be resolved. So a discourse referent for Bryan’s Paris apartment is added to the context, and then semantics works in the same way as described.