# Weak Negation in Inquisitive Semantics

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## 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 Fitch-style 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.

## Keywords

Inquisitive semantics Negation Possible worlds Fitch-style natural deduction Denial## Notes

### Acknowledgments

The work on this paper was supported by grant no. 13-21076S of the Czech Science Foundation.

## References

- Adams, E. W. (1975).
*The logic of conditionals. An application of probability to deductive logic*. Dordrecht: D. Reidel Publishing Company.Google Scholar - Aloni, M. (2007). Free choice, modals and imperatives.
*Natural Language Semantics*,*15*, 65–94.CrossRefGoogle Scholar - 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.Google Scholar - Cantwell, J. (2008). The logic of conditional negation.
*Notre Dame Journal of Formal Logic*,*49*, 245–260.CrossRefGoogle Scholar - Ciardelli, I. (2009).
*Inquisitive semantics and intermediate logics*. MSc thesis, Amsterdam.Google Scholar - Ciardelli, I. (2010). A first-order 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.CrossRefGoogle Scholar - 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 [SALT-09]*(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 [TARK-09]*(pp. 71–80).Google Scholar - Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic.
*Journal of Philosophical Logic*,*40*, 55–94.CrossRefGoogle Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - Grice, H. P. (1991). Indicative conditionals. In H. P. Grice (Ed.),
*Studies in the way of words*(pp. 58–85). London: Harvard University Press.Google Scholar - 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.*Google Scholar - Groenendijk, J., Stokhov, M., & Veltman, F. (1996). Coreference and modality. In S. Lappin (Ed.),
*Handbook of contemporary semantic theory*(pp. 179–216). Oxford: Blackwell.Google Scholar - Humberstone, L. (1979). Interval semantics for tense logic: Some remarks.
*Journal of Philosophical Logic*,*8*, 171–196.CrossRefGoogle Scholar - Lewis, C. I., & Langford, C. H. (1932).
*Symbolic logic*. London: Century.Google Scholar - Lewis, L. (1973).
*Counterfactuals*. Oxford: Basil Blackwell.Google Scholar - Lucio, P. (2000). Structured sequent calculi for combining intuitionistic and classical first-order logic. In
*Frontiers of combining systems, lecture notes in computer science*(Vol. 1794, pp. 88–104).Google Scholar - Punčochář, V. (2009).
*Sémantika některých neobvyklých modálních logik*. MSc thesis, Charles University in Prague.Google Scholar - Punčochář, V. (2012). Some modifications of Carnap’s modal logic.
*Studia Logica*,*100*, 517–543.CrossRefGoogle Scholar - Punčochář, V. (2014). Intensionalisation of logical operators. In M. Dančák & V. Punčochář (Eds.),
*The logica yearbook 2013*(pp. 173–185). Milton Keynes: College Publications.Google Scholar - Roelofsen, F. (2013). Algebraic foundations for the semantic treatment of inquisitive content.
*Synthese*,*190*, 79–12.CrossRefGoogle Scholar - Sano, K. (2011). First-order inquisitive pair logic. In
*Logic and its applications, lecture notes in computer science*(Vol. 6521, pp. 147–161).Google Scholar - Simons, M. (2005). Dividing things up: The semantics of or and the modal/or interaction.
*Natural Language Semantics*,*13*, 271–316.CrossRefGoogle Scholar - Stalnaker, R. C. (1968). A theory of conditionals.
*Studies in Logical Theory*,*2*, 98–112.Google Scholar - Stalnaker, R. C. (1999).
*Context and content*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Wansing, H. (2014). Connexive logic. In E. N. Zalta (Ed.),
*The Stanford encyclopedia of philosophy*(Spring 2014 Edition).Google Scholar - Zimmermann, T. E. (2000). Free choice disjunction and epistemic possibility.
*Natural Language Semantics*,*8*, 255–290.CrossRefGoogle Scholar