Our object language so far has been a formal language, \(\mathcal{L}\), but we also want to analyse natural languages in similar ways, in the spirit of Montague Grammar.
We’ll develop a compositional semantic theory for a fragment of English that is similar to Heim and Kratzer (1998), except that it will be dynamic. In particular, we will use FCS as our underlying dynamic semantics. See Muskens (1996), Brasoveanu (2007), Charlow (2014), Dotlačil (2013) for related discussion.
In this lecture we will develop a dynamic compositional semantic system without presuppositions. Presuppositions will be added in the next lecture.
It omits certain aspects of natural language semantics, such as plurality or degrees. Also, being extension, it offers no analysis of intensional phenomena like tense/aspect, modality, and attitude. But the system has been extended to cover such phenomena.
For any model \(\mathcal{M} = (\mathcal{D}, \mathcal{I})\),
Denotations in Heim and Kratzer (1998) are all assignment-sensitive (as in PL).
Sentences denote functions of type \((g, t)\).
\(|\!|\textsf{it is raining}|\!|_{\mathcal{M}} = \lambda g_g. \begin{cases} 1 & \textrm{if }\mathcal{I}(\textsf{raining}) = 1\\ 0 & \textrm{otherwise}\end{cases}\)
This is usually written more compactly as follows.
\(|\!|\textsf{it is raining}|\!|_{\mathcal{M}} = \lambda g_g. \textrm{it is raining in }\mathcal{M}\)
The model parameter (the subscript \(\mathcal{M}\)) is normally omitted.
In Heim and Kratzer (1998), the \(g\)-argument is normally saturated with an arbitrary assignment as follows:
\(|\!|\textsf{it is raining}|\!|^g = \textrm{it is raining in }\mathcal{M}\)
To emphasise the functional nature of meaning, we stick to the funcitonal representation.
This example is a constant function, but when a pronoun or trace is present, the assignment argument comes to have an effect.
Phi-features of pronouns introduce presuppositions, but we ignore them until next lecture.
\(|\!|\textsf{she}_3 \textsf{ is happy}|\!| = \lambda g_g. \textrm{g(3) is happy}\)
Intransitive predicates (verbs, nouns, adjectives) denote (partial) functions of type \((g, (e, t))\). Sortal presuppositions, if any, are omitted here.
Transitive predicates denote (partial) functions of type \((g, (e, (e, t)))\).
Certain functional items denote functions from assignments to identity functions.
Quantificational DPs denote (partial) functions of type \((g, ((e,t),t))\). Presuppositions are omitted here.
The compositional rules of Heim and Kratzer (1998) are type-driven. The ‘pendantic versions’ of them also encode presupposition projection, but we will deal with them in the next lecture, so they are omitted.
(Extensional) Functional Application
If \(\alpha\) is a constituent with two daughter constituents, \(\beta\) and \(\gamma\) such that \(|\!|\beta|\!|\in D_{(g, (\sigma, \tau))}\) and \(|\!|\gamma|\!|\in D_{(g, \sigma)}\), then \(|\!|\alpha|\!| = \lambda g_g. |\!|\beta|\!|(g)(|\!|\gamma|\!|(g))\).
Predicate Abstraction
If \(\alpha\) is a constituent with two daughter constituents, \(\beta\) and \(\gamma\), \(\beta\) is a binding index \(i\), and \(|\!|\gamma|\!|\in D_{(g, \tau)}\), then \(|\!|\alpha|\!| = \lambda g_g.\lambda x_e.\ |\!|\gamma|\!|(g[i\mapsto x])\).
There is also a rule for non-branching constituents. We omit it here.
Heim and Kratzer (1998) additionally discuss the possibility that intersective adjectival modification is accounted for by an additional compositional rule, Predicate Modification.
We want sentences to denote functions over information states.
In FCS, information states are sets of world-assignment pairs.
Let us give these things types.
\([\![\textsf{It is raining}]\!] = \lambda c_i. \{\langle w,g\rangle\in c \,|\,\) it is raining in \(w\}\)
Pronouns are interpreted via assignment functions. Again ignoring phi-features:
\([\![\textsf{She}_3\textsf{ smokes}]\!] = \lambda c_i.\ \{\langle w,g\rangle \in c \,|\, g(3)\) smokes in \(w\}\)
We want to derive this meaning by combining \([\![\textsf{she}_3]\!]\) and \([\![\textsf{smokes}]\!]\).
But \([\![\textsf{she}_3]\!]\) won’t be a single individual in this system.
There are actually several ways of analysing pronouns, but let’s try the following denotation.
\([\![\textsf{she}_3]\!] = \lambda c_i. \{x \,|\,\)for some \(\langle w,g\rangle\in c, g(3) = x\}\)
This function takes an information state \(c\) and returns a set of entities. Let’s say that it’s type is \((i, E)\), where \(E\) is the type of sets of entities.
This might seem like it’s overly complex, and it actually is for a distributive system like the base FCS we are considering. But it becomes convenient when non-distributivity is introduced.
Let’s write \(c(3)\) for \(\{x \,|\,\)for some \(\langle w,g\rangle\in c, g(3) = x\}\). Then we can simply write:
\([\![\textsf{she}_3]\!] = \lambda c_i. c(3)\)
Now we have the denotation of the whole sentence and the denotation of the pronoun, we should be able to figure out the denotation of the verb smokes.
Note that the following doesn’t work.
\([\![\textsf{smokes}]\!] =\lambda c_i.\lambda X_{E}. \{\langle w,g\rangle \in c \,|\,\) each \(x\in X\) smokes in \(w\}\)
With this denotation, we could use the good old (Extensional) Functional Application rule with a minor change:
Functional Application
If \(\alpha\) is a constituent with two daughter constituents, \(\beta\) and \(\gamma\) such that \([\![\beta]\!]\in D_{(i, (\sigma, \tau))}\) and \([\![\gamma]\!]\in D_{(i, \sigma)}\), then \([\![\alpha]\!] = \lambda c_i.\ [\![\beta]\!](c)([\![\gamma]\!](c))\).
The predicted meaning is:
\([\![\textsf{she}_3 \textsf{ smokes}]\!]\)
\(= \lambda c_i.\ [\![\textsf{smokes}]\!](c)([\![\textsf{she}_3]\!](c))\)
\(= \lambda c_i.\ \{\langle w,g\rangle \in c \,|\,\) each \(x\in \{g(3) \,|\, \textrm{for some } w, \langle w,g\rangle\in c\}\) smokes in \(w\})\)
This is incorrect. For example, consider:
There is a\(^3\) linguist. She\(_3\) smokes.
Let zoom in on the following possible worlds.
Let’s suppose that the input context is \(\{\langle w_1, g_1\rangle, \langle w_2, g_2\rangle, \langle w_3, g_3\rangle\}\).
How about the following meaning?
\([\![\textsf{smokes}]\!] =\lambda c_i.\lambda X_{E}. \{\langle w,g\rangle \in c \,|\,\) there is some \(x\in X\) that smokes in \(w\}\)
This also is incorrect. Let’s compute the meaning of the above two-sentence discourse with respect to the input context \(\{\langle w_1, g\rangle, \langle w_2, g'\rangle\}\).
The correct way is to let the denotation of the intranstive verb take a function of type \(i, E\) as its subject, rather than a set of individuals, whose type is \(E\).
\([\![\textsf{smokes}]\!] =\lambda c_i.\lambda d_{(i,E)}. \{\langle w,g\rangle \in c \,|\,\) each \(x\in d(\{\langle w, g\rangle\})\) smokes in \(w\}\)
The compositional rule for combining a pronoun and an intransitive verb is analogous to Intensional Functional Application in Heim and Kratzer (1998), so let’s call it that.
Intensional Functional Application
If \(\alpha\) is a constituent with two daughter constituents, \(\beta\) and \(\gamma\), such that \([\![\beta]\!]\in D_{(i, ((i, \sigma), \tau))}\) and \([\![\gamma]\!]\in D_{(i, \sigma)}\), then \([\![\alpha]\!] = \lambda c_i.\ [\![\beta]\!](c)([\![\gamma]\!])\).
Intransitive nouns and adjectives are analysed analogously.
Transitive predicates take two arguments of type \((i, E)\).
We can analyse an anaphor as a variant of a pronoun.
She\(_8\) likes herself\(_8\)
\([\![\textsf{herself}_8]\!] = \lambda c_i.\ c(8)\)
But of course you might want to pursue a more complicated theory of anaphors.
Indefinites perform random assignment and output a new context, so they cannot be type \((i, E)\). Instead, they are type-lifted to a dynamic generalised quantifier of type \((i, ((i, ((i, E), i)), i))\).
We use superscripts on expressions that perform random assignment to indicate which variables they perform random assignment on (Barwise 1987).
\([\![\textsf{someone}^3]\!] = \lambda c_i. \lambda P_{(i, ((i, E), i))}.\ P(\{\langle w, g'\rangle\ \,|\,\) for some \(\langle w,g\rangle\in c, g[3]g'\) and \(g'(3)\) is a human in \(w\})(\lambda c'_i.\ c'(3))\)
Note that the compositional rule will be Intensional Functional Application here as well.
\([\![\textsf{smokes}]\!] =\lambda c_i.\lambda d_{(i,E)}. \{\langle w,g\rangle \in c \,|\,\) each \(x\in d(\{\langle w, g\rangle\})\) smokes in \(w\}\)
\([\![\textsf{someone}^3 \textsf{ smokes}]\!]\)
\(= \lambda c_i.\ [\![\textsf{someone}^3]\!](c)([\![\textsf{smokes}]\!])\)
\(= \lambda c_i.\ [\![\textsf{smokes}]\!](\{\langle w, g'\rangle\ \,|\,\) for some \(\langle w,g\rangle\in c, g[3]g'\) and \(g'(3)\) is a human in \(w\})(\lambda c'_i.\ c'(3))\)
\(= \lambda c_i.\ \{\langle w,g'\rangle \in \{\langle w, g'\rangle\ \,|\,\) for some \(\langle w,g\rangle\in c, g[3]g'\) and \(g'(3)\) is a human in \(w\}\,|\,\) each \(x\in [\lambda c'_i.\ c'(3)](\{\langle w, g'\rangle\})\) smokes in \(w\}\)
\(= \lambda c_i.\ \{\langle w,g'\rangle \,|\,\) for some \(\langle w,g\rangle\in c, g[3]g'\) and \(g'(3)\) is a human in \(w\) and \(g'(3)\) smokes in \(w\}\)
A(n)-indefinties are analysed in the same way:
\([\![\textsf{a}^8 \textsf{ student}]\!] = \lambda c_i. \lambda P_{(i, ((i, E), i))}.\ P(\{\langle w, g'\rangle\ \,|\,\) for some \(\langle w,g\rangle\in c, g[8]g'\) and \(g'(8)\) is a student in \(w\})(\lambda c'_i.\ c'(8))\)
We can analyse the indefinite determiner a(n) by abstracting over the noun meaning. We will write \(c[\xi]\) for \(\{\langle w, g'\rangle \,|\,\) for some \(\langle w, g\rangle\in c, g[\xi]g'\}\).
\([\![\textsf{a}^5]\!] = \lambda c_i. \lambda Q_{(i, ((i, E), i))}.\lambda P_{(i, ((i, E), i))}. P(Q(c[5])(\lambda c'_i.\ c'(5)))(\lambda c'_i.\ c'(5))\)
These dynamic generalised quantifiers cause type-mismatch in non-subject position, but this issue is a familiar one that is independent from dynamic vs static semantics. One could use one’s favourite way of solving it.
No need for Predicate Abstraction and binding indices to enable variable binding in this system. Dynamicity gives us binding for free.
A\(^9\) cat saw itself\(_9\).
This applies to movement as well. Following Heim and Kratzer (1998), we interpret traces as the same thing as (presupposition-less) pronouns.
Now we’ve got indefinites. The other horn of dynamic semantics is conjunction. Here’s sentential conjunction. Crucially the mode of composition is Intensional Functional Application.
\([\![\textsf{and}]\!] = \lambda c_i.\lambda S_{(i,i)}. \lambda T_{(i, i)}.\ S(T(c))\)
This conjunction is both internally and externally dynamic. So it will give us cross-sentential anaphora.
There is a\(^2\) cat and it\(_2\) is sleeping.
We can type-generalise the above conjunction as follows. First define ‘conjoinable types.’
\(\tau\) is a type that ends in \(i\) iff \((i, \tau)\) is a conjoinable type.
Then type-flexible conjunction for any conjoinable type \(\tau\) will be recursively defined as.
\([\![\textsf{and}_{\tau}]\!] = \begin{cases} \lambda c_i. \lambda S_{(i,i)}. \lambda T_{(i, i)}.\ S(T(c)) & \textrm{if } \tau = (i,i)\\ \lambda c_i. \lambda P_{(i,(\sigma_1, \sigma_2))}. \lambda Q_{(i,(\sigma_1, \sigma_2))}.\lambda x\in D_{\sigma_1}.\\ \hspace{1em} [\![\textsf{and}_{(i,\sigma_2)}]\!](c)(\lambda c'.\ P(c')(x))(\lambda c'.Q(c')(x)) & \textrm{if } \tau = (i, (\sigma_1, \sigma_2)) \end{cases}\)
This allows us to account for cases like the following, without scoping out the indefinite (cf. Muskens 1996).
We can analyse discourse conjunction with an invisible conjunction.
A\(^5\) cat is sleeping. It\(_5\) is happy.
Or it can be analyzed as a pragmatic phenomenon along the following lines.
If someone utters \(\phi\) in context \(c\), then:
The most natural disjunction in this non-distributive system is the externally dynamic one:
\([\![\textsf{or}]\!] = \lambda c_i.\lambda S_{(i,i)}. \lambda T_{(i, i)}.\ S(c)\cup T(c)\)
Exercise: Define a type-flexible version of this.
We can have a DPL-style disjunction, which is both externally and internally static, as well.
\([\![\textsf{or}]\!] = \lambda c_i.\lambda S_{(i,i)}. \lambda T_{(i, i)}.\ \{\langle w,g\rangle \in c \,|\,\) for some \(g'\), \(\langle w,g'\rangle\in S(c)\) or \(\langle w,g'\rangle\in T(c)\}\)
Which entry is better is an empirical question. Stone disjunction suggests that the former is necessary.
She\(_3\) has a\(^1\) cat or she\(_3\) has a\(^1\) dog. She\(_3\) loves it\(_1\).
Negation can be defined as in FCS.
\([\![\textsf{not}]\!] = \lambda c_i.\lambda S_{(i,i)}.\ \{\langle w,g\rangle\in c \,|\,\)for no \(g'\), \(\langle w,g'\rangle\in S(c)\}\)
In English, the canonical position of negation is between the subject and VP, so we want to use the following version.
\([\![\textsf{not}]\!] = \lambda c_i.\lambda P_{(i, ((i, E), i))}.\lambda d_{(i, E)}.\ \{\langle w,g\rangle\in c \,|\,\)for no \(g'\), \(\langle w,g'\rangle\in P(c)(d(c))\}\)
Here we could use Extensional Functional Application for \(P\).
\([\![\textsf{not}]\!] = \lambda c_i.\lambda P_{((i, E), i)}.\lambda d_{(i, E)}.\ \{\langle w,g\rangle\in c \,|\,\)for no \(g'\), ,\(\langle w,g'\rangle\in P(d(c))\}\)
This is one special case of in the flexible-type series. For a negated quantifier like not everyone, we might need another type.
Exercise: Define flexible-type negation.
The denotations of proper names could be analysed as type-\((i, E)\) functions:
\([\![\textsf{Mary}]\!] = \lambda c_i.\ \{m\}\)
\([\![\textsf{she}_3]\!] = \lambda c_i.\ c(3)\)
But this is not good enough. Besides all the issues associated with this simple view of proper names, such proper names won’t be able to bind pronouns. (Recall that we don’t have Predicate Abstraction!)
Mary saw herself\(_3\).
We could of course introduce a rule like Predicate Abstract to enable binding, but instead, it’s more straightforward to turn proper names into dynamic quantifiers (cf. Montagovian individuals).
\([\![\textsf{Mary}^4]\!]\)
\(= \lambda c_i.\lambda P_{(i, ((i, E), i))}.\ P(\{\langle w, g'\rangle\ \,|\,\) for some \(\langle w,g\rangle\in c, g[4]g'\) and \(g'(4)=m\})(\lambda c'_i.\ c'(4))\)
\(= \lambda c_i. \lambda P_{(i, ((i, E), i))}.\ P(\{\langle w, g[4\mapsto m]\rangle\ \,|\,\langle w,g\rangle\in c\})(\lambda c'_i.\ c'(4))\)
You might want to allow pronouns to bind other pronouns as well, given Binding Theoretic considerations (see Büring 2005, for example).
\([\![\textsf{she}^4_2]\!] = \lambda c_i. \lambda P_{(i, ((i, E), i))}.\ P(\{\langle w, g[4\mapsto g(2)]\rangle\ \,|\,\langle w,g\rangle\in c\})(\lambda c'_i.\ c'(4))\)