Abstract
In nonradical inquisitive semantics, an intuitionistic Kripke model captures how group knowledge increases throughout a conversation and allows the inquisitive meaning of a sentence to be derived from its classical meaning. In radical inquisitive semantics, as proposed by Groenendijk and Roelofsen, positive and negative ways of reacting to a proposal are captured by the positive and negative inquisitive meanings of a sentence, respectively, which are inductively defined without employing any Kripke-type semantics. This paper demonstrates that, in principle, it is impossible to provide any natural Kripke-type semantics under radical inquisitive semantics. Moreover, an alternative way to establish the semantics is proposed that avoids this negative result.
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Notes
- 1.
To obtain a direct inductive definition of
, the operation “\(\text {MAX}\)” should be placed in front of the right-hand sides of all defining equations of \(\lceil {\varphi }\rceil \) except atomic cases. It is then easily seen that 
= \(\{\,{|p|}\,\}\), 
= \(\{\,{|p|,|q|}\,\}\) and 
= \(\{\,{|p|, |\lnot p|}\,\}\). - 2.
The definition of Groenendijk and Roelofsen [10] is not necessarily the only way to define these notions. However, the present paper accepts these definitions to demonstrate that any Kripke semantics for intuitionistic logic with the strong negation \(\sim \) cannot derive the negative inquisitive meaning of this definition from , as can be accomplished in conservative inquisitive semantics.
- 3.
Reference [10], did not refer to the strong negation in Nelson’s constructive logics.
- 4.
The author would like to thank Daisuke Bekki for sharing Greg Restall’s argument.
- 5.
The author owes this suggestion to Floris Roelofsen.
- 6.
- 7.
The author has presented material related to this paper at several occasions in the past three years and would like to thank the audiences of these events, including the second PhilLog-Math Workshop in Tokyo, the Relating Particles to Evidence and Inference Workshop in Goettingen, the Taiwan Philosophical Logic Colloquium 2012 in Taiwan, and the Second Asian Workshop on Philosophical Logic in Guangzhou. The author especially thanks Jeroen Groenendijk for his discussion during the author’s stay at Amsterdam in 2009 and 2010; Floris Roelofsen for his comments on earlier drafts of the manuscript; and Daisuke Bekki for sharing Greg Restall’s argument on the disjunctive antecedent of conditionals. The author is also grateful to the anonymous referee for their helpful comments. The work of the author was partially supported by a KAKENHI Grant-in-Aid for Young Scientists (B) No. 24700146.
, the operation “
= 
= 
= References
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Sano, K. (2015). Avoiding Impossibility Theorems in Radical Inquisitive Semantics. In: Ju, S., Liu, H., Ono, H. (eds) Modality, Semantics and Interpretations. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47197-5_6
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