Abstract
This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’assertions. As a result, a new modal logic emerges. For this logic, a Fitchstyle system of natural deduction is formulated. The main result of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed.
This is a preview of subscription content, access via your institution.
Notes
 1.
At this moment, it would be reasonable to admit also the empty state, but that would lead to some technical complications in the following section and so we avoid these complications by excluding the empty set from the class of states.
 2.
‘Classically true’ means true according to classical logic.
 3.
The fact that the basic framework of inquisitive semantics is not completely suitable for modeling phenomena related to conditionals and modals seems to motivate an extension of inquisitive semantics called suppositional inquisitive semantics (Groenendijk and Roelofsen 2015). The aim of this paper is also to propose an extension of inquisitive semantics as a reaction to this problem. However, it is evident that our extension differs fundamentally from suppositional inquisitive semantics.
 4.
As one of the anonymous reviewers rightly pointed out, Grice’s example could also be interpreted in the sense of (b). So the example is probably not the best illustration of (c). However, it is evident that the reading (c) is also possible here, and we will concentrate on this reading simply because it seems to be more straightforward to extend inquisitive semantics in such a way that the negation of conditionals semantically corresponds to (c). It would be more complicated to adjust the framework so that we would be able to formalize the sentence in the sense of (b) without reducing it to (a).
Grice also admitted (a) as another way of denying conditionals (see Grice 1991, p. 80). Paradoxically, his goal was to defend the material conditional analysis. However, Grice himself admitted that his defense did not lead to a satisfactory explanation of the denial of conditionals (see Grice 1991, p. 83).
 5.
In the original inquisitive semantics without weak negation, it is natural to admit also the empty state. If the first semantic condition (\(s \nvDash \bot \)) is replaced with the condition \(s \vDash \bot \) iff \(s=\emptyset \), we naturally get that every formula is supported by the empty state. As a result, the presence of the empty state does not have any direct impact on the logic determined by the semantics. The situation changes when weak negation is added to the framework. If the empty state was admitted, it would not naturally support every formula and that would complicate the picture in an undesirable way. Of course, it would be possible to introduce a slightly more complex semantic condition for weak negation: \(s \vDash \mathord {\sim }\varphi \) iff \(s\) is empty or \(s \nvDash \varphi \). However, since the empty state would bring no benefit to us, there seems to be no reason for such a complication and we instead decided to exclude the empty state from the class of states.
 6.
It is possible to formulate this claim in a stronger way: If \(\left\ \varphi \right\ \) is closed under binary unions, then \(\square \varphi \equiv \blacksquare \varphi \). It holds that if \(\left\ \varphi \right\ \) is closed under binary unions, it is also closed under arbitrary unions. However, we omit the proof of this fact.
 7.
\(P\) is inquisitive iff \(\bigcup P \notin P\).
 8.
A formalization of the idea that ‘might’sentences draw attention to certain possibilities is proposed in Ciardelli et al. (2011).
 9.
In this and the next section, \(\varphi \), \(\psi \), and \(\chi \) (possibly indexed) will range over the formulas of the language \(L^{\mathord {\sim }}\), unless stated otherwise.
 10.
In the rest of this paper, ‘hyp’ used in the derivations will stand for ‘hypothetical assumption’.
 11.
The order of its disjuncts is not important.
 12.
 13.
 14.
The notation is a standard one. \(t_{1},\ldots ,t_{n}\) are terms and \(ve(t_{n})\) is the value of the term in the structure \(v\) relative to the evaluation \(e\). \(e_{x:a}\) is the variant of \(e\) sending \(x\) to \(a\).
 15.
We use the letter nabla since these operators can be understood as quantifier versions of the nabla operator and its dual used in coalgebraic logic.
 16.
The proof of the statement (i) is a ‘meta’proof. It shows that \(\mathord {\sim }p\) is derivable from \(\lozenge \varphi \) on the assumption that \(\lnot p\) is derivable from \(\varphi \). Thus the metaproof shows that in every concrete case, that is in every case in which a concrete formula \(\varphi \) and an atom \(p\) are given such that there is a derivation of \(\lnot p\) from \(\varphi \), there is a concrete derivation of \(\mathord {\sim }p\) from \(\lozenge \varphi \). The derivation can be obtained by substituting the derivation of \(\lnot p\) from \(\varphi \) for the step from 5 to 6 in the metaproof. It is supposed that \(\lnot p\) can be derived from \(\varphi \) alone, without any other assumptions. That guarantees that this derivation does not require any sources from the outer proof even if it is integrated as a subproof of a conditional proof as in the case of the metaproof of the statement (i). It follows that by such step the use of the rule \((\rightarrow I)^{*}\) cannot be made illegitimate.
References
Adams, E. W. (1975). The logic of conditionals. An application of probability to deductive logic. Dordrecht: D. Reidel Publishing Company.
Aloni, M. (2007). Free choice, modals and imperatives. Natural Language Semantics, 15, 65–94.
Bílková, M., Palmigiano, A., & Venema, Y. (2008). Proof systems for the coalgebraic cover modality. In C. Areces & R. Goldblatt (Eds.), Advances in modal logic (Vol. 7, pp. 1–21). London: King’s College Publications.
Cantwell, J. (2008). The logic of conditional negation. Notre Dame Journal of Formal Logic, 49, 245–260.
Ciardelli, I. (2009). Inquisitive semantics and intermediate logics. MSc thesis, Amsterdam.
Ciardelli, I. (2010). A firstorder inquisitive semantics. In M. Aloni, H. Bastiaanse, T. de Jager, & K. Schulz (Eds.), Logic, language, and meaning: Selected papers from the 17th Amsterdam Colloquium (pp. 234–243). Berlin: Springer.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2011). Attention! Might in inquisitive semantics. In E. Cormany, S. Ito, & D. Lutz (Eds.), Proceedings of the 19th conference on semantics and linguistic theory [SALT09] (pp. 91–108). http://elanguage.net/journals/salt/issue/archive.
Ciardelli, I., & Roelofsen, F. (2009). Generalized inquisitive logic: Completeness via intuitionistic Kripke models. In Proceedings of the 12th conference on theoretical aspects of rationality and knowledge [TARK09] (pp. 71–80).
Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40, 55–94.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2012). Inquisitive semantics. In Lecture notes for a course at NASSLLI, held June 18–22, 2012, in Austin, USA. https://sites.google.com/site/inquisitivesemantics/.
Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2013). Inquisitive semantics: A new notion of meaning. Language and Linguistics Compass, 7, 459–476.
Edgington, D. (1995). On conditionals. Mind, 104, 235–329.
Egré, P., & Politzer, G. (2013). On the negation of indicative conditionals. In M. Aloni, M. Franke, F. Roelofsen (Eds.), Proceedings of the nineteenth Amsterdam Colloquium (pp. 10–18).
Fariñas, L., & Herzig, A. (1996). Combining classical and intuitionistic logic, or: Intuitionistic implication as a conditional. In F. Baader & K. Schulz (Eds.), Frontiers in combining systems (pp. 93–102). Dordrecht: Kluwer.
Grice, H. P. (1991). Indicative conditionals. In H. P. Grice (Ed.), Studies in the way of words (pp. 58–85). London: Harvard University Press.
Groenendijk, J., & Roelofsen, F. (2009). Inquisitive semantics and pragmatics. In J. M. Larrazabal, & L. Zubeldia (Eds.), Meaning, content, and argument: Proceedings of the ILCLI international workshop on semantics, pragmatics, and rhetoric. www.illc.uva.nl/inquisitivesemantics.
Groenendijk, J., & Roelofsen, F. (2015). Towards a suppositional inquisitive semantics. In M. Aher, D. Hole, E. Jerabek, & C. Kupke (Eds.), In Revised selected papers from the 10th international Tbilisi symposium on language, logic, and computation.
Groenendijk, J., Stokhov, M., & Veltman, F. (1996). Coreference and modality. In S. Lappin (Ed.), Handbook of contemporary semantic theory (pp. 179–216). Oxford: Blackwell.
Humberstone, L. (1979). Interval semantics for tense logic: Some remarks. Journal of Philosophical Logic, 8, 171–196.
Lewis, C. I., & Langford, C. H. (1932). Symbolic logic. London: Century.
Lewis, L. (1973). Counterfactuals. Oxford: Basil Blackwell.
Lucio, P. (2000). Structured sequent calculi for combining intuitionistic and classical firstorder logic. In Frontiers of combining systems, lecture notes in computer science (Vol. 1794, pp. 88–104).
Punčochář, V. (2009). Sémantika některých neobvyklých modálních logik. MSc thesis, Charles University in Prague.
Punčochář, V. (2012). Some modifications of Carnap’s modal logic. Studia Logica, 100, 517–543.
Punčochář, V. (2014). Intensionalisation of logical operators. In M. Dančák & V. Punčochář (Eds.), The logica yearbook 2013 (pp. 173–185). Milton Keynes: College Publications.
Roelofsen, F. (2013). Algebraic foundations for the semantic treatment of inquisitive content. Synthese, 190, 79–12.
Sano, K. (2011). Firstorder inquisitive pair logic. In Logic and its applications, lecture notes in computer science (Vol. 6521, pp. 147–161).
Simons, M. (2005). Dividing things up: The semantics of or and the modal/or interaction. Natural Language Semantics, 13, 271–316.
Stalnaker, R. C. (1968). A theory of conditionals. Studies in Logical Theory, 2, 98–112.
Stalnaker, R. C. (1999). Context and content. Oxford: Oxford University Press.
Wansing, H. (2014). Connexive logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2014 Edition).
Zimmermann, T. E. (2000). Free choice disjunction and epistemic possibility. Natural Language Semantics, 8, 255–290.
Acknowledgments
The work on this paper was supported by grant no. 1321076S of the Czech Science Foundation.
Author information
Affiliations
Corresponding author
Appendices
Appendix 1
We will prove all the statements of Lemma 1. (i) If \(\varphi \vdash \lnot p\) then \(\lozenge \varphi \vdash \mathord {\sim }p\).^{Footnote 16}
(ii) \(\mathord {\sim }\blacklozenge \varphi \vdash \lnot \lozenge \varphi \).
(iii) \(\square \varphi \vdash \lozenge \varphi \).
(iv) \(\vdash \lozenge \varphi \vee \lnot \lozenge \varphi \).
(v) \(\lozenge (\varphi \vee \psi ) \vdash \lozenge \varphi \vee \lozenge \psi \).
(vi) \(\square (\varphi \vee \psi ), \lnot \lozenge \psi \vdash \square \varphi \).
(vii) \(\varphi _{1} \oplus \cdots \oplus \varphi _{n}, \varphi _{1} \rightarrow p, \ldots , \varphi _{n} \rightarrow p \vdash p\). Notice that here is the only place in the completeness proof where the rule (R1) is used.
Appendix 2
We will provide a proof that Kreisel–Putnam schema (KP) restricted to the language \(L\) is derivable in our system. This result can be easily obtained by semantic considerations using Theorem 1. Here, we will give a purely syntactic proof of this fact.
In the following text, \(\varphi , \psi , \chi \) (possibly indexed) will range over all formulas from \(L\) and \(\alpha , \beta , \gamma \) (possibly indexed) only over disjunctionfree formulas from \(L\).
Lemma 6
For any \(\alpha , \varphi _{1},\ldots ,\varphi _{n}, \varphi , \psi \):

(i)
\(\square \alpha \vdash \alpha \),

(ii)
\(\square \lnot \varphi \vdash \lnot \varphi \),

(iii)
\(\mathord {\sim }(\varphi _{1} \vee \cdots \vee \varphi _{i} \vee \cdots \vee \varphi _{n}) \vdash \mathord {\sim }\varphi _{i}\),

(iv)
\(\mathord {\sim }(\lnot \varphi \rightarrow \alpha ) \vdash \lozenge (\lnot \varphi \wedge \lnot \alpha )\),

(v)
\(\varphi , \lozenge \lnot \varphi \vdash \bot \),

(vi)
\(\square (\varphi \wedge \psi ) \vdash \square \varphi \),

(vii)
\(\lozenge (\varphi \wedge \psi ) \vdash \lozenge \varphi \),

(viii)
if \(\varphi \vdash \psi \) then \(\blacklozenge \varphi \vdash \blacklozenge \psi \).
Proof
(i) Induction on the complexity of \(\alpha \). For atomic formulas, the claim holds due to the rule (R1). For \(\bot \) there is the following derivation:
Suppose that the claim holds for some \(\beta \) and \(\gamma \). We will prove that then it holds also for \(\beta \wedge \gamma \) and \(\beta \rightarrow \gamma \).
(ii) The following derivation can be constructed:
(iii) The following derivation can be constructed:
(iv) The following derivation can be constructed:
(v) The following derivation can be constructed:
(vi) The following derivation can be constructed:
(vii) The following derivation can be constructed:
(viii) Suppose that \(\varphi \vdash \psi \).
The following lemma shows that a restricted version of (KP) schema is derivable in the natural deduction system. Let us remind that \(\alpha _{1},\ldots ,\alpha _{n}\) range over disjunctionfree formulas from \(L\).
Lemma 7
\(\vdash (\lnot \varphi \rightarrow (\alpha _{1} \vee \cdots \vee \alpha _{n})) \rightarrow ((\lnot \varphi \rightarrow \alpha _{1}) \vee \cdots \vee (\lnot \varphi \rightarrow \alpha _{n}))\).
Proof
The following derivation can be constructed:
The following lemma shows that every formula from \(L\) is provably equivalent with some disjunction of disjunctionfree formulas from \(L\). An analogous result (concerning, however, different calculi) was exploited also in Punčochář (2009, (2012, (2014).
Lemma 8
For any \(\varphi \) there are some \(\alpha _{1}, \ldots , \alpha _{n}\) such that \(\varphi \dashv \) \(\vdash \alpha _{1} \vee \cdots \vee \alpha _{n}\).
Proof
This can be proved by induction on the complexity of the formula \(\varphi \). We will show only the inductive step for implication which is most difficult and in which Lemma 7 is used. Suppose that

(a)
\(\psi \dashv \) \(\vdash \beta _{1} \vee \cdots \vee \beta _{k}\) (\(k \ge 1\)),

(b)
\(\chi \dashv \) \(\vdash \gamma _{1} \vee \cdots \vee \gamma _{l}\) (\(l \ge 1\)).
Then the following formulas are provably equivalent:

1.
\(\psi \rightarrow \chi \),

2.
\((\beta _{1} \vee \cdots \vee \beta _{k}) \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l})\),

3.
\(\bigwedge ^{k}_{i=1}(\beta _{i} \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l}))\),

4.
\(\bigwedge ^{k}_{i=1}(\lnot \lnot \beta _{i} \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l}))\),

5.
\(\bigwedge ^{k}_{i=1}((\lnot \lnot \beta _{i} \rightarrow \gamma _{1}) \vee \cdots \vee (\lnot \lnot \beta _{i} \rightarrow \gamma _{l}))\),

6.
\(\bigvee ^{l}_{i_{1}=1} \ldots \bigvee ^{l}_{i_{k}=1}((\lnot \lnot \beta _{1} \rightarrow \gamma _{i_{1}}) \wedge \ldots \wedge (\lnot \lnot \beta _{k} \rightarrow \gamma _{i_{k}}))\).
The equivalence of 1. and 2. follows from the induction hypotheses (a) and (b). 3. can be easily derived from 2. with the help of the rules \((\wedge I)\), \((\vee I)\), \((\rightarrow \) \(E)\), and \((\rightarrow \) \(I)^{*}\). 2. can be derived from 3. using the rules \((\wedge E)\), \((\vee E)\), \((\rightarrow \) \(E)\), and \((\rightarrow \) \(I)^{*}\). The equivalence between 3. and 4. is a consequence of Lemma 6 (i). The derivation of 5. from 4. is an application of Lemma 7 and in the derivation from 5. to 4. it suffices to use the rules \((\vee E)\), \((\vee I)\), \((\rightarrow \) \(E)\), and \((\rightarrow \) \(I)^{*}\). The equivalence of 5. and 6. can be established with the help of the derivable distributive laws.
Theorem 2 is a consequence of Lemmas 7 and 8. Let us stress once again that in this theorem \(\varphi ,\,\psi \) and \(\chi \) range over formulas from \(L\).
Theorem 2
\(\vdash (\lnot \varphi \rightarrow (\psi \vee \chi )) \rightarrow ((\lnot \varphi \rightarrow \psi ) \vee (\lnot \varphi \rightarrow \chi ))\).
Rights and permissions
About this article
Cite this article
Punčochář, V. Weak Negation in Inquisitive Semantics. J of Log Lang and Inf 24, 323–355 (2015). https://doi.org/10.1007/s1084901592192
Published:
Issue Date:
Keywords
 Inquisitive semantics
 Negation
 Possible worlds
 Fitchstyle natural deduction
 Denial