Dynamics
Greg Kobele
Semantics II
Winter 2015
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Introduction
Treating pronouns as denoting algorithms for selecting individuals in a context
has enabled us to deal with the context-sensitive nature of lingusitic expressions.
However, from our current vantage point, there are a number of phenomena
which suggest that the context in which a pronoun is interpreted can be modified
by a prior utterance. In other words, we want to develop a system in which
sentences not only depend on contexts for their meaning, but also update the
context passed on to subsequent sentences in a discourse. To motivate this
further, consider the sentences below.
1. A boy smiled.
2. He was happy.
Sentence 2 in isolation (i.e. in an empty context) is deviant; there is no individual that the pronoun can refer to. However, if immediately preceded by sentence
1, sentence 2 is perfectly interpretable. This is so regardless of the context in
which the discourse 1;2 is uttered. Indeed, in an out of the blue context the
discourse 1;2 is interpreted as equivalent to A boy smiled and was happy. Following de Groote [2], we will treat discourses as the fundamental units of speech,
denoting context-sensitive propositions (functions from contexts to truth values). Sentences then are discourse modifiers; they build new discourses from
old. A sentence will denote a function from context-sensitive propositions to
context-sensitive propositions. For the moment, we simplify things by ignoring
(!) context-sensitivity.
We think of T (the proposition true) as a trivial discourse; it is one in which
nothing has been said. This trivial discourse will play an important role later
on. Discourses are built up, somewhat counter-intuitively on this perspective,
from end to start; if [[T ]] is a sentence meaning (a function from discourses to
discourses), then [[T ]] (T) is a discourse as well. This process can be iterated; if
[[S]] is a sentence meaning, then [[S]] ([[T ]] (T)) is also a discourse, representing
the result of first uttering sentence S and then sentence T .
Of course, real discourses proceed forwards, not backwards; we do not know
what the rest of the discourse is when we hear its first sentence. We can see,
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however, that, for any discourse D, the result of uttering first S, then T , followed
by D will be [[S]] ([[T ]] (D)). Thus, uttering first S then T in an out of the blue
context will denote the following discourse modifier.
B [[S]] [[T ]] = λx. [[S]] ([[T ]] x)
We will interpret initial segments of a discourse in terms of function composition;
more poetically, we interpret the period ‘.’ as composing sentences with one
another.
As was the case when we enriched our semantic values to allow for contextsensitivity, we wish here to have a conservative extension of our framework,
which will allow us to ignore dynamic aspects of meaning (as for example when
computing the way the meaning of a sentence is related to the meanings of its
constituent parts) without affecting the resulting denotation.
We do this, as before, by defining operations Dynα and Nydα , for each type
α, which map inherently non-dynamic meanings to dynamic ones.
The relation between non-dynamic types and dynamic ones is a type homomorphism, which can be represented as follows (writing αd for the type which
results from α):
Case
ed
td
(αβ)d
Output
e
tt
(αd β d )
In order to actually define these operations, we need to decide how sentences
influence the meanings of discourses. Although there is a rich literature on just
this topic [1, 3], we will drastically simplify things by assuming that a sentence is
an intersective discourse modifier. Individuals (things of type e) are unaffected
by dynamicization.
Dyne (a) = a
Dynt (φ) = λψ.φ ∧ ψ
Dynαβ (f ) = λA.Dynβ (f (Nydα A))
The above defines how to wrap an inherently static expression in the trappings of dynamism; the result can then be used wherever a truly dynamic expression is expected. The dynamicization of an individual of type e to be itself,
the dynamicization of a static proposition of type t to be the function which
conjoins it with an upcoming discourse, and the dynamicization of a function
of type αβ to be a function from dynamic arguments of type αd to the dynamicization of the result of applying the function to the statification of its
argument.
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Nyde (A) = A
Nydt (Φ) = Φ (T)
Nydαβ (F ) = λa.Nydβ (F (Dynα a))
Given a dynamic term, we can also turn it into a static one. The fundamental
observation is that a dynamic proposition can be turned into a static one simply
by applying it to a static proposition.
References
[1] Nicholas Asher and Alex Lascarides. Logics of Conversation. Cambridge
University Press, 2003.
[2] Philippe de Groote. Towards a montagovian account of dynamics. In
Masayuki Gibson and Jonathan Howell, editors, Proceedings of SALT 16,
pages 1–16, 2006.
[3] William C. Mann and Sandra A. Thompson. Rhetorical structure theory:
Toward a functional theory of text organization. Text, 8(3):243–281, 1988.
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