Inquisitive semantics is a framework inlogic andnatural language semantics. 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.^[1][2] It was originally developed by Ivano Ciardelli,Jeroen Groenendijk, Salvador Mascarenhas, and Floris Roelofsen.^{[3][4][5][6][7]}

The essential notion in inquisitive semantics is that of aninquisitiveproposition.

• Aninformation state (alternately aclassical proposition) is a set ofpossible worlds.
• Aninquisitive proposition is a nonemptydownward-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${\displaystyle \{\{w\},\mathrm{\varnothing }\}}$ encodes the information that{w} is the actual world. The inquisitive proposition${\displaystyle \{\{w\},\{v\},\mathrm{\varnothing }\}}$ encodes that the actual world is either${\displaystyle w}$ or${\displaystyle v}$.

An inquisitive proposition encodes inquisitive content via its maximal elements, known asalternatives. For instance, the inquisitive proposition${\displaystyle \{\{w\},\{v\},\mathrm{\varnothing }\}}$ has two alternatives, namely${\displaystyle \{w\}}$ and${\displaystyle \{v\}}$. Thus, it raises the issue of whether the actual world is${\displaystyle w}$ or${\displaystyle v}$ while conveying the information that it must be one or the other. The inquisitive proposition${\displaystyle \{\{w,v\},\{w\},\{v\},\mathrm{\varnothing }\}}$ 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.

• Theinformational content of an inquisitive propositionP is${\displaystyle \mathrm{info}(P)=\{w\mid w\in t\text{ for some }t\in P\}}$.

Inquisitive propositions can be used to provide a semantics for theconnectives ofpropositional logic since they form aHeyting algebra when ordered by thesubset relation. For instance, for every propositionP there exists arelative pseudocomplement${\displaystyle P^{\ast }}$, which amounts to${\displaystyle \{s\subseteq W\mid s\cap t=\mathrm{\varnothing }\text{ for all }t\in P\}}$. Similarly, any two propositionsP andQ have ameet and ajoin, which amount to${\displaystyle P\cap Q}$ and${\displaystyle P\cup Q}$ respectively. Thus inquisitive propositions can be assigned to formulas of${\displaystyle \mathcal{L}}$ as shown below.

Given a model${\displaystyle \mathfrak{M}=⟨W,V⟩}$ whereW is a set of possible worlds andV is a valuation function:

1. ${\displaystyle [[p]]=\{s\subseteq W\mid \mathrm{\forall }w\in s,V(w,p)=1\}}$
2. ${\displaystyle [[\mathrm{\neg }\phi ]]=\{s\subseteq W\mid s\cap t=\mathrm{\varnothing }\text{ for all }t\in [[\phi ]]\}}$
3. ${\displaystyle [[\phi \wedge \psi ]]=[[\phi ]]\cap [[\psi ]]}$
4. ${\displaystyle [[\phi \vee \psi ]]=[[\phi ]]\cup [[\psi ]]}$

The operators ! and ? are used as abbreviations in the manner shown below.

1. ${\displaystyle !\phi \equiv \mathrm{\neg }\mathrm{\neg }\phi }$
2. ${\displaystyle ?\phi \equiv \phi \vee \mathrm{\neg }\phi }$

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${\displaystyle \phi }$, the inquisitive proposition${\displaystyle [[!\phi ]]}$ expresses the same information as${\displaystyle [[\phi ]]}$, but it may differ in that it raises no nontrivial issues. For example, if${\displaystyle [[\phi ]]}$ is the inquisitive propositionP from a few paragraphs ago, then${\displaystyle [[!\phi ]]}$ is the inquisitive propositionQ.

The ?-operator trivializes the information expressed by whatever it applies to, while converting information states that would establish that its issues are unresolvable into states that resolve it. This is very abstract, so consider another example. Imagine that logical space consists of four possible worlds,w₁,w₂,w₃, andw₄, and consider a formula${\displaystyle \phi }$ such that${\displaystyle [[\phi ]]}$ contains{w₁},{w₂}, and of course${\displaystyle \mathrm{\varnothing }}$. This proposition conveys that the actual world is eitherw₁ orw₂ and raises the issue of which of those worlds it actually is. Therefore, the issue it raises would not be resolved if we learned that the actual world is in the information state{w₃,w₄}. Rather, learning this would show that the issue raised by our toy proposition is unresolvable. As a result, the proposition${\displaystyle [[?\phi ]]}$ contains all the states of${\displaystyle [[\phi ]]}$, along with{w₃,w₄} and all of its subsets.

• Alternative semantics
• Disjunction
• Intermediate logic
• Question
• Responsive predicate
• Rising declarative

• Ciardelli, Ivano; Groenendijk, Jeroen; and Roelofsen, Floris (2019)Inquisitive Semantics. Oxford University Press.ISBN 9780198814788
• https://projects.illc.uva.nl/inquisitivesemantics/

• Semantics
• Non-classical logic
• Systems of formal logic
• Intuitionism
• Philosophical logic

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• Short description matches Wikidata