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Questions & Indexicality

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The truth conditions of sentences with indexicals like ‘I’ and ‘here’ cannot be given directly, but only relative to a context of utterance. Something similar applies to questions: depending on the semantic framework, they are given truth conditions relative to an actual world, or support conditions instead of truth conditions. Two-dimensional semantics can capture the meaning of indexicals and shed light on notions like apriority, necessity and context-sensitivity. However, its scope is limited to statements, while indexicals also occur in questions. Moreover, notions like apriority, necessity and context-sensitivity can also apply to questions. To capture these facts, the frameworks that have been proposed to account for questions need refinement. Two-dimensionality can be incorporated in question semantics in several ways. This paper argues that the correct way is to introduce support conditions at the level of characters, and develops a two-dimensional variant of both proposition-set approaches and relational approaches to question semantics.

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I am grateful to Floris Roelofsen, Maria Aloni, Tom Schoonen, Gianluca Grilletti, and four anonymous reviewers for their comments on earlier versions of this paper. I would also like to thank the audiences at InqBnB 3, OZSW 2019 and Semantics and Philosophy in Europe 11 for their helpful questions and comments. This paper was written while I was affiliated with the University of Amsterdam and received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 680220). The results in this paper have been presented in a preliminary form in [18].


This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 680220).

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By induction on the complexity of \(\varphi \). The proof relies on the fact that the truth value of a ?-free sentence depends only on the second index, which follows immediately from the truth conditions above. The only non-trivial cases are those for negation and question operator:

(\(\lnot \)) (\(\Rightarrow \)):

Suppose \(s\models \lnot \varphi \). Then for all \(k\in s\), \(\{k\}\not \models \varphi \). Take any \(j,i\in s\). From \(\{i\}\not \models \varphi \) and the induction hypothesis we obtain \(i,i\not \models \varphi \). Since \(\varphi \) is ?-free, it follows that \(j,i\not \models \varphi \) and by the pair semantics for negation that \(j,i\models \lnot \varphi \).

(\(\Leftarrow \)):

Suppose \(s\not \models \lnot \varphi \). Then there is some \(i\in s\) such that \(\{i\}\models \varphi \). By the induction hypothesis we obtain \(i,i\models \varphi \), so \(i,i\not \models \lnot \varphi \). So we found a pair that makes \(\lnot \varphi \) false, as required.

(?) (\(\Rightarrow \)):

Suppose \(s\models {\mathord {?}}\varphi \). Then \(s\models \varphi \) or \(s\models \lnot \varphi \). Take any \(j,i\in s\). In the first case, we have \(j,i\models \varphi \) and \(j,j\models \varphi \) by the induction hypothesis. In the second case, it follows from the semantics of negation that for all \(k\in s\), \(\{k\}\not \models \varphi \). By the induction hypothesis we obtain \(j,j\not \models \varphi \) and \(i,i\not \models \varphi \). Since \(\varphi \) is ?-free, it follows that \(j,i\not \models \varphi \) too. In both cases we can conclude that \(j,i\models {\mathord {?}}\varphi \)

(\(\Leftarrow \)):

Suppose \(s\not \models {\mathord {?}}\varphi \). Then \(s\not \models \varphi \) and \(s\not \models \lnot \varphi \). First, by the induction hypothesis there exists a pair \(k,i\in s\) such that \(k,i\not \models \varphi \). Second, by the semantics of negation we have some \(j\in s\) such that \(\{j\}\models \varphi \). By the induction hypothesis it follows that \(j,j\models \varphi \). Since \(\varphi \) is ?-free, it follows from \(k,i\not \models \varphi \) that \(j,i\not \models \varphi \). So we found a pair ji such that \(j,i\not \models {\mathord {?}}\varphi \).

\(\square \)


By induction on the complexity of \(\varphi \). The semantics of atoms, negation and necessity are already formulated in terms of individual pairs. Conjunction and universal quantification are immediate by unpacking the support conditions. I give only the case for the actuality operator A:

(\(\Rightarrow \)):

Suppose \(s\models A\varphi \). Then \(\{\langle c,w_c \rangle ~|~ \langle c,i\rangle \in s \}\models \varphi \). Take any \(\langle c,i\rangle \in s\). By the induction hypothesis, we obtain \(\{\langle c,w_c\rangle \}\models \varphi \). It follows that \(\{\langle c,i\rangle \}\models A\varphi \).

(\(\Leftarrow \)):

Suppose \(s\not \models A\varphi \). Then \(\{\langle c,w_c \rangle ~|~ \langle c,i\rangle \in s \}\not \models \varphi \). By the induction hypothesis there must be some pair \(\langle c,i\rangle \in s\) such that \(\{\langle c,w_c\rangle \}\not \models \varphi \). It follows that \(\{\langle c,i\rangle \}\not \models A\varphi \).

\(\square \)


By induction on the complexity of \(\varphi \). The atomic case follows from \(\varphi \) being indexical-free. Conjunction, universal quantification and question are immediate by unpacking the support conditions. I give the case for the necessity operator \(\Box \):

(\(\Rightarrow \)):

Suppose \(s\models \varphi \). Then for all \(\langle c,i \rangle \in s\) we have that \(\{c\} \times W \models \varphi \) and by the induction hypothesis that \(C \times W \models \varphi \). So either s is empty or \(C \times W\models \varphi \). In the former case, \(C \times \textsf {indices}(s)\) is also empty and \(\Box \varphi \) is trivially supported. In the latter case, \(\varphi \) is supported in all information states, so \(C \times \textsf {indices}(s)\models \Box \varphi \).

(\(\Leftarrow \)):

Suppose \(s\not \models \Box \varphi \). Since \(s\subseteq C\times \textsf {indices}(s)\), it follows by downward closure that \(C\times \textsf {indices}(s)\not \models \Box \varphi \).

\(\square \)

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van Gessel, T. Questions & Indexicality. J Philos Logic 53, 593–621 (2024).

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