Dynamic semantics is a framework in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
and
natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dynamic semantics, knowing the meaning of a sentence means knowing "the change it brings about in the information state of anyone who accepts the news conveyed by it."
In dynamic semantics, sentences are mapped to functions called ''context change potentials'', which take an input context and return an output context. Dynamic semantics was originally developed by
Irene Heim and
Hans Kamp in 1981 to model
anaphora, but has since been applied widely to phenomena including
presupposition
In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include ...
,
plurals,
questions,
discourse relations, and
modality
Modality may refer to:
Humanities
* Modality (theology), the organization and structure of the church, as distinct from sodality or parachurch organizations
* Modality (music), in music, the subject concerning certain diatonic scales
* Modaliti ...
.
Dynamics of anaphora
The first systems of dynamic semantics were the closely related ''File Change Semantics'' and ''
discourse representation theory'', developed simultaneously and independently by
Irene Heim and
Hans Kamp. These systems were intended to capture
donkey anaphora
Donkey sentences are sentences that contain a pronoun with clear meaning (it is bound semantically) but whose syntactical role in the sentence poses challenges to grammarians. Such sentences defy straightforward attempts to generate their formal ...
, which resists an elegant compositional treatment in classic approaches to semantics such as
Montague grammar __notoc__
Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on mathematical logic, especially higher-order predicate logic and lambda calculus, and makes ...
.
Donkey anaphora is exemplified by the infamous donkey sentences, first noticed by the medieval logician
Walter Burley
Walter Burley (or Burleigh; 1275 – 1344/45) was an English scholastic philosopher and logician with at least 50 works attributed to him. He studied under Thomas WiltonHarjeet Singh Gill, ''Signification in language and culture'', Indian In ...
and brought to modern attention by
Peter Geach
Peter Thomas Geach (29 March 1916 – 21 December 2013) was a British philosopher who was Professor of Logic at the University of Leeds. His areas of interest were philosophical logic, ethics, history of philosophy, philosophy of religion and ...
.
::Donkey sentence (relative clause): Every farmer who owns a donkey beats it.
::Donkey sentence (conditional): If a farmer owns a donkey, he beats it.
To capture the empirically observed truth conditions of such sentences in
first order logic, one would need to translate the
indefinite noun phrase "a donkey" as a
universal quantifier
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
scoping over the variable corresponding to the pronoun "it".
:: FOL translation of donkey sentence: :
While this translation captures (or approximates) the truth conditions of the natural language sentences, its relationship to the syntactic form of the sentence is puzzling in two ways. First, indefinites in non-donkey contexts normally express
existential
Existentialism ( ) is a form of philosophical inquiry that explores the problem of human existence and centers on human thinking, feeling, and acting. Existentialist thinkers frequently explore issues related to the meaning, purpose, and valu ...
rather than universal quantification. Second, the syntactic position of the donkey pronoun would not normally allow it to be
bound by the indefinite.
To explain these peculiarities, Heim and Kamp proposed that natural language indefinites are special in that they introduce a new ''discourse referent'' that remains available outside the syntactic scope of the operator that introduced it. To cash this idea out, they proposed their respective formal systems that capture donkey anaphora because they validate ''Egli's theorem'' and its corollary.
::Egli's theorem:
::Egli's corollary:
Update semantics
''Update semantics'' is a framework within dynamic semantics that was developed by
Frank Veltman
Frank or Franks may refer to:
People
* Frank (given name)
* Frank (surname)
* Franks (surname)
* Franks, a medieval Germanic people
* Frank, a term in the Muslim world for all western Europeans, particularly during the Crusades - see Farang
Cur ...
.
In update semantics, each formula
is mapped to a function
Intersective update was proposed by Robert Stalnaker in 1978 as a way of formalizing the speech act of assertion.
In Stalnaker's original system, a context (or ''context set'') is defined as a set of possible worlds representing the information in the common ground of a conversation. For instance, if
C = \ this represents a scenario where the information agreed upon by all participants in the conversation indicates that the actual world must be either
w,
v, or
u. If
![\varphi!.html" ;"title="varphi.html" ;"title="![\varphi">![\varphi!">varphi.html" ;"title="![\varphi">![\varphi!= \, then updating
C with
\varphi would return a new context
C varphi= \. Thus, an assertion of
\varphi would be understood as an attempt to rule out the possibility that the actual world is
u .
From a formal perspective, intersective update can be taken as a recipe for lifting one's preferred static semantics to dynamic semantics. For instance, if we take classical propositional semantics as our starting point, this recipe delivers the following intersective update semantics.
* Intersective update semantics based on classical propositional logic:
#
C = \
#
C neg \varphi= C - C varphi
#
C varphi \land \psi= C varphi\cap C psi
#
C varphi \lor \psi= C varphi\cup C psi
The notion of intersectivity can be decomposed into the two properties known as ''eliminativity'' and ''distributivity''. Eliminativity says that an update can only ever remove worlds from the context—it can't add them. Distributivity says that updating
C with
\varphi is equivalent to updating each singleton subset of
C with
\varphi and then pooling the results.
* Eliminativity:
\varphi is ''eliminative'' iff
C varphi\subseteq C for all contexts
C
* Distributivity:
\varphi is ''distributive'' iff
C varphi= \bigcup\
Intersectivity amounts to the conjunction of these two properties, as proven by
Johan van Benthem.
The test semantics for modals
The framework of update semantics is more general than static semantics because it is not limited to intersective meanings. Nonintersective meanings are theoretically useful because they contribute different information depending on what information is already present in the context. For instance, if
\varphi is intersective, then it will update any input context with the exact same information, namely the information encoded by the proposition
![\varphi!.html"_;"title="varphi.html"_;"title="![\varphi">![\varphi!">varphi.html"_;"title="![\varphi">![\varphi!/math>._On_the_other_hand,_if__\varphi_is_nonintersective,_it_could_contribute___![\varphi!.html"_;"title="varphi.html"_;"title="![\varphi">![\varphi!">varphi.html"_;"title="![\varphi">![\varphi!/math>_when_it_updates_some_contexts,_but_some_completely_different_information_when_it_updates_other_contexts.
Many_natural_language_expressions_have_been_argued_to_have_nonintersective_meanings._The_nonintersectivity_of_epistemic_modals_can_be_seen_in_the_ ![\varphi!.html"_;"title="varphi.html"_;"title="![\varphi">![\varphi!">varphi.html"_;"title="![\varphi">![\varphi!/math>._On_the_other_hand,_if__\varphi_is_nonintersective,_it_could_contribute___![\varphi!.html"_;"title="varphi.html"_;"title="![\varphi">![\varphi!">varphi.html"_;"title="![\varphi">![\varphi!/math>_when_it_updates_some_contexts,_but_some_completely_different_information_when_it_updates_other_contexts.
Many_natural_language_expressions_have_been_argued_to_have_nonintersective_meanings._The_nonintersectivity_of_epistemic_modals_can_be_seen_in_the_felicity_(linguistics)">infelicity_of_''epistemic_contradictions''.
:Epistemic_contradiction:_#It's_raining_and_it_might_not_be_raining.
These_sentences_have_been_argued_to_be_bona_fide_logical_contradictions,_unlike_superficially_similar_examples_such_as_![\varphi!.html"_;"title="varphi.html"_;"title="![\varphi">![\varphi!">varphi.html"_;"title="![\varphi">![\varphi!/math>._On_the_other_hand,_if_
_\varphi_is_nonintersective,_it_could_contribute_
__![\varphi!.html"_;"title="varphi.html"_;"title="![\varphi">![\varphi!">varphi.html"_;"title="![\varphi">![\varphi!/math>_when_it_updates_some_contexts,_but_some_completely_different_information_when_it_updates_other_contexts.
Many_natural_language_expressions_have_been_argued_to_have_nonintersective_meanings._The_nonintersectivity_of_epistemic_modals_can_be_seen_in_the_felicity_(linguistics)">infelicity_of_''epistemic_contradictions''.
:Epistemic_contradiction:_#It's_raining_and_it_might_not_be_raining.
These_sentences_have_been_argued_to_be_bona_fide_logical_contradictions,_unlike_superficially_similar_examples_such_as_Moore's_paradox">Moore_sentences,_which_can_be_given_a_pragmatics_(linguistics).html" "title="Moore's_paradox.html" ;"title="felicity_(linguistics).html" ;"title="varphi">![\varphi!.html" ;"title="varphi.html" ;"title="![\varphi">![\varphi!">varphi.html" ;"title="![\varphi">![\varphi!/math>. On the other hand, if \varphi is nonintersective, it could contribute ![\varphi!.html" ;"title="varphi.html" ;"title="![\varphi">![\varphi!">varphi.html" ;"title="![\varphi">![\varphi!/math> when it updates some contexts, but some completely different information when it updates other contexts.
Many natural language expressions have been argued to have nonintersective meanings. The nonintersectivity of epistemic modals can be seen in the felicity (linguistics)">infelicity of ''epistemic contradictions''.
:Epistemic contradiction: #It's raining and it might not be raining.
These sentences have been argued to be bona fide logical contradictions, unlike superficially similar examples such as Moore's paradox">Moore sentences, which can be given a pragmatics (linguistics)">pragmatic
Pragmatism is a philosophical movement.
Pragmatism or pragmatic may also refer to:
*Pragmaticism, Charles Sanders Peirce's post-1905 branch of philosophy
* Pragmatics, a subfield of linguistics and semiotics
*'' Pragmatics'', an academic journal i ...
explanation.
:Epistemic contradiction principle: \varphi \land \Diamond \neg \varphi \models \bot
These sentences cannot be analysed as logical contradictions within purely intersective frameworks such as the relational semantics for modal logic. The Epistemic Contradiction Principle only holds on the class of Kripke frame, relational frames such that Rwv \Rightarrow (w=v) . However, such frames also validate an entailment from \Diamond \varphi to \varphi. Thus, accounting for the infelicity of epistemic contradictions within a classical semantics for modals would bring along the unwelcome prediction that "It might be raining" entails "It is raining". Update Semantics skirts this problem by providing a nonintersective denotation for modals. When given such a denotation, the formula \Diamond \neg \varphi can update input contexts differently depending on whether they already contain the information that \varphi provides. The most widely adopted semantic entry for modals in update semantics is the ''test semantics'' proposed by Frank Veltman
Frank or Franks may refer to:
People
* Frank (given name)
* Frank (surname)
* Franks (surname)
* Franks, a medieval Germanic people
* Frank, a term in the Muslim world for all western Europeans, particularly during the Crusades - see Farang
Cur ...
.
*The test semantics for modals: C Diamond \varphi = \begin
C & \text C varphi\neq \varnothing \\
\varnothing & \text
\end
On this semantics, \Diamond \varphi tests whether the input context could be updated with \varphi without getting trivialized, i.e. without returning the empty set. If the input context passes the test, it remains unchanged. If it fails the test, the update trivializes the context by returning the empty set. This semantics can handle epistemic contradictions because no matter the input context, updating with \varphi will always output a context that fails the test imposed by \Diamond \neg \varphi .[For a complete derivation of the Epistemic Contradiction Principle within Update Semantics, see for instance Goldstein (2016), p. 13. This derivation crucially depends on a particular definition of entailment, as well as an intersective semantic entry for \neg and a treatment of \land as updating consecutively with the conjuncts in their linear order.]
See also
* Conversational scoreboard
* Donkey anaphora
Donkey sentences are sentences that contain a pronoun with clear meaning (it is bound semantically) but whose syntactical role in the sentence poses challenges to grammarians. Such sentences defy straightforward attempts to generate their formal ...
* Discourse representation theory
* Formal semantics of programming languages
* Hans Kamp
* Import-Export
* Irene Heim
* Modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
* Scope (formal semantics)
Notes
External links
Dynamic Semantics, Stanford Encyclopedia of Philosophy
Dynamic Semantics Notes, Daniel Rothschild
Dynamic Semantics and Pragmatic Alternatives, ESSLLI 2017 Course Notes
{{Non-classical logic
Semantics
Logic
Philosophy of language
Non-classical logic
Systems of formal logic
Linguistic modality