Inquisitive 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
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
. In inquisitive semantics, the semantic content of a sentence captures both the information that the sentence conveys and the issue that it raises. The framework provides a foundation for the linguistic analysis of statements and questions.
It was originally developed by Ivano Ciardelli,
Jeroen Groenendijk
Jeroen Antonius Gerardus Groenendijk (; born 20 July 1949, Amsterdam), is a Dutch logician, linguist and philosopher, working on philosophy of language, formal semantics, pragmatics.
Groenendijk wrote a joint Ph.D. dissertation with Martin St ...
, Salvador Mascarenhas, and Floris Roelofsen.
Basic notions
The essential notion in inquisitive semantics is that of an ''inquisitive
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
''.
* An ''information state'' (alternately a ''classical proposition'') is a set of
possible worlds Possible Worlds may refer to:
* Possible worlds, concept in philosophy
* ''Possible Worlds'' (play), 1990 play by John Mighton
** ''Possible Worlds'' (film), 2000 film by Robert Lepage, based on the play
* Possible Worlds (studio)
* ''Possible Wo ...
.
* An ''inquisitive proposition'' is a nonempty
downward-closed set of information states.
Inquisitive propositions encode informational content via the region of logical space that their information states cover. For instance, the inquisitive proposition
encodes the information that is the actual world. The inquisitive proposition
encodes that the actual world is either
or
.
An inquisitive proposition encodes inquisitive content via its maximal elements, known as ''alternatives''. For instance, the inquisitive proposition
has two alternatives, namely
and
. Thus, it raises the issue of whether the actual world is
or
while conveying the information that it must be one or the other. The inquisitive proposition
encodes the same information but does not raise an issue since it contains only one alternative.
The informational content of an inquisitive proposition can be isolated by pooling its constituent information states as shown below.
* The ''informational content'' of an inquisitive proposition ''P'' is
.
Inquisitive propositions can be used to provide a semantics for the
connectives of
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
since they form a
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
when ordered by the
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
relation. For instance, for every proposition ''P'' there exists a
relative pseudocomplement , which amounts to
. Similarly, any two propositions ''P'' and ''Q'' have a
meet and a
join Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two topo ...
, which amount to
and
respectively. Thus inquisitive propositions can be assigned to formulas of
as shown below.
Given a model
where ''W'' is a set of possible worlds and ''V'' is a valuation function:
#
#_
#
#
#
#
The operators ! and ? are used as abbreviations in the manner shown below.
#
#
Conceptually, the !-operator can be thought of as cancelling the issues raised by whatever it applies to while leaving its informational content untouched. For any formula
, the inquisitive proposition