To make the matter more complicated, there are other uses of pronouns besides the above three (see below).
Ideally, we want to have a uniform and coherent picture for all uses of pronouns, but this turns out to be a quite tricky task, as we will see.
The building has a flag in front of
Received wisdom: pronouns are variables (Montague 1973; Heim and Kratzer 1998; Büring 2005).
E.g., in the Heim & Kratzer system, variables are distinguished from each other via indices (ignoring gender and number):
The discourse anaphoric use is often left undiscussed in textbooks like Heim and Kratzer (1998). One might think that the above pragmatics of assignment functions could be extended to it, but in order to do so, we will be explicit about pronouns’ discourse antecedent. Dynamic semantics is a formally explicit theory of discourse anaphora.
Sidenote: Variable-free semantics (Jacobson 1999, 2000, 2002) treats pronouns as functions of type \(\langle e,e\rangle\), but its core ideas are not so different from the variable-ful theory.
There are some remaining issues for the pronouns-as-variables analysis, as has been discussed extensively throughout the history of formal semantics (Geach 1962; Partee 1970; Evans 1977a, 1977b, 1980; Heim 1990; Neale 1990; Krifka 1996; Heim and Kratzer 1998; Jacobson 1999, 2000, 2002; Büring 2005; Elbourne 2005; Brasoveanu 2007, among many others; See Nouwen 2020 for an overview).
We’ll come back to these issues later (hopefully).
The man who gave his paycheck to his wife was wiser than the man who gave
Every wise man gave his paycheck to his wife. Every stupid man gave
Most of the students that attended my seminar wrote a paper for it.
It’s a truism that truth-conditional meaning does not exhaust everything there is to meaning in natural language.
For one, we have pragmatic inferences:
In addition, ample evidence has been raised that anaphoric meaning constitutes a separate dimension of meaning from truth-conditional meaning.
Expressions with contextually equivalent truth-conditional meaning can have different anaphoric properties (Kamp 1981; Heim 1982).
To formalise this idea, Karttunen (1976) introduced the idea of discourse referents.
Discourse referents are often formalised in dynamic semantics (if there’s time we’ll talk about other ways of thinking about them).
A related issue is donkey anaphora.
In static semantics, the following example can be accounted for by ‘inverse linking’:
A farmer who owned a donkey vaccinated it.
[a donkey] \(7\) [[a farmer who owned \(t_7\)] [ vaccinated it\(_7\) ]]_{}) iff \(g = g'\) and \([\![\tau_1 ]\!]^g_{\mathcal{M}} = [\![\tau_2 ]\!]^g_{\mathcal{M}}\) c. \((g, g')\in [\![\Pi\tau_1\cdots \tau_n]\!]_{\mathcal{M}}\) iff \(g = g'\) and \(([\![\tau_1]\!]^g_{\mathcal{M}}, \dots, [\![\tau_n]\!]^g_{\mathcal{M}})\in \mathcal{I}(\Pi)\) d. \((g, g')\in [\![\neg\phi]\!]_{\mathcal{M}}\) iff \(g = g'\) and there is no \(f\) such that \((g, f)\in [\![\phi]\!]_{\mathcal{M}}\) e. \((g, g')\in [\![(\phi\land\psi)]\!]_{\mathcal{M}}\) iff for some \(f\), \((g, f)\in [\![\phi]\!]_{\mathcal{M}}\) and \((f, g')\in [\![\psi]\!]_{\mathcal{M}}\) f. \((g, g')\in [\![(\phi\lor\psi)]\!]_{\mathcal{M}}\) iff \(g = g'\) and for some \(f\), \((g, f)\in [\![\phi]\!]_{\mathcal{M}}\) and/or \((g, f)\in [\![\psi]\!]_{\mathcal{M}}\) g. \((g, g')\in [\![(\phi\to\psi)]\!]_{\mathcal{M}}\) iff \(g = g'\) and for each \(f\) such that \((g, f)\in [\![\phi]\!]_{\mathcal{M}}\), there is \(f'\) such that \((f, f')\in [\![\psi]\!]_{\mathcal{M}}\) h. \((g, g')\in [\![\exists \xi[\phi] ]\!]_{\mathcal{M}}\) iff for some \(f\), \(g[\xi]f\) and \((f, g')\in [\![\phi]\!]_{\mathcal{M}}\) i. \((g, g')\in [\![\forall \xi[\phi] ]\!]_{\mathcal{M}}\) iff \(g = g'\) and for each \(f\) such that \(g[\xi]f\), there is \(f'\) such that \((f, f')\in [\![ \phi ]\!]_{\mathcal{M}}\)
The model parameter is often omitted.
Groenendijk and Stokhof (1991) discuss in detail why they chose the definitions they chose.
The behavior of externally static connectives is motivated by the following observations in English.
The internal dynamicity and stativity of the binary connectives are motivated analogously.
It is important to notice that some of these connectives could be defined in other ways, while keeping the classical aspect of meaning, e.g.:
More on this point later.
Notation: \(\phi\equiv \psi\) iff for each \(\mathcal{M}\), \([\![\phi]\!]_{\mathcal{M}} = [\![\psi]\!]_{\mathcal{M}}\)
(See Groenendijk and Stokhof (1991) for more discussion.)
Unlike in PL, double negation cancellation is not valid in DPL, i.e., for some \(\phi\), \(\phi\not\equiv\neg\neg\phi\).
Because \(\neg\) makes a formula externally static, when \(\phi\) is not a test, i.e. it introduces a new discourse referent, then \(\phi\not\equiv\neg\neg\phi\). E.g. \(\exists x[Px] \not\equiv \neg\neg\exists x[Px]\).
In fact, whenever \(\phi\) is a test, \(\phi\equiv\neg\neg\phi\).
\(\neg\neg\) is sometimes abbreviated as \(!\) (Groenendijk and Stokhof 1991 call it \(\diamond\)), and called a closure operator (or assertion operator). \(!\phi\) is always a test.
Certain classical equivalences hold in DPL.
For any formulae \(\phi\) and \(\psi\):
But certain others don’t.
For instance, \(\exists x[Px] \not\equiv \exists y[Py]\), because they give rise to different anaphoric possibilities.
Similarly, the following pairs are not equivalent for some \(\phi\) and \(\psi\). This is due to the difference in external dynamicity.
But DPL validates some new equivalences that do not hold in PL.
\(g[\![(\exists\xi[\phi] \land \psi)]\!]g'\)
iff for some \(f\), \(g[\![\exists\xi[\phi]]\!]f\) and \(f[\![\psi]\!]g'\)
iff for some \(f\), for some \(f'\), \(g[\xi]f'\) and \(f'[\![\phi]\!]f\) and \(f[\![\psi]\!]g'\)
iff for some \(f'\), \(g[\xi]f'\) and for some \(f\), \(f'[\![\phi]\!]f\) and \(f[\![\psi]\!]g'\)
iff for some \(f'\), \(g[\xi]f'\) and \(f'[\![\phi\land\psi]\!]g'\)
iff \(f'[\![\exists \xi[\phi\land\psi] ]\!]g'\)
\(g[\![(\exists\xi[\phi] \to \psi)]\!]g'\)
iff \(g = g'\) and for each \(f\), if \(g[\![\exists\xi[\phi]]\!]f\), then there is \(f'\) such that \(f[\![\psi]\!]f'\)
iff \(g = g'\) and for each \(f\), if there is \(f''\) such that \(g[\xi]f''\) and \(f''[\![\phi]\!]f\), then there is \(f'\) such that \(f[\![\psi]\!]f'\)
iff \(g = g'\) and for each \(f\), for each \(f''\), if \(g[\xi]f''\) and \(f''[\![\phi]\!]f\), then there is \(f'\) such that \(f[\![\psi]\!]f'\)
iff \(g = g'\) and for each \(f\), for each \(f''\), if \(g[\xi]f''\), then if \(f''[\![\phi]\!]f\), then there is \(f'\) such that \(f[\![\psi]\!]f'\)
iff \(g = g'\) and for each \(f''\), if \(g[\xi]f''\), then for each \(f\), if \(f''[\![\phi]\!]f\), there is \(f'\) such that \(f[\![\psi]\!]f'\)
iff \(g = g'\) and for each \(f''\), if \(g[\xi]f''\), then \(f''[\![\phi \to \psi]\!]f\)
iff \(g[\![ \forall \xi [\phi \to \psi] ]\!]g'\)
Nonetheless, for every DPL formula, there is a truth-conditionally equivalent PL formula (Groenendijk and Stokhof 1991 for a proof).
If a farmer owns a donkey, he beats it.
Every farmer who owns a donkey beats it.
Externally dynamic disjunction (Groenendijk and Stokhof 1991; Stone 1992; Rothschild 2017).
Giorgos has a cat or he has a dog, and he will bring
This could be accounted for by \(\dot{\lor}\).
\(g[\![(\phi\dot{\lor}\psi)]\!]g'\) iff \(g[\![\psi]\!]g'\) and/or \(g[\![\phi]\!]g'\)
Bathroom anaphora, originally due to Barbara Partee.
Either this house doesn’t have a bathroom or
This is harder to account for. One would need externally dynamic negation, so we have double negation cancellation. In fact, such a version of negation might be necessary anyway.
It’s not true that John doesn’t have a car. It’s in the garage.