Abstract
According to one form of epistemic contrastivism, due to Jonathan Schaffer, knowledge is not a binary relation between an agent and a proposition, but a ternary relation between an agent, a proposition, and a contextbasing question. In a slogan: to know is to know the answer to a question. I argue, first, that Schafferstyle epistemic contrastivism can be semantically represented in inquisitive dynamic epistemic logic, a recent implementation of inquisitive semantics in the framework of dynamic epistemic logic; second, that within inquisitive dynamic epistemic logic, the contrastive ternary knowledge operator is reducible to the standard binary one. The reduction shows, I argue, that Schaffer’s argument in favor of contrastivism is compatible with a binary picture of knowledge. This undercuts the force of the argument in favor of contrastivism.
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Notes
 1.
Schaffer usually presents contrastivism as the view that knowledge is a ternary relation between an agent, a proposition and a contrast proposition q (i.e. (K(a, p, q))), but he takes it for granted that the two are interchangeable:
contrastive knowledge is equivalent to questionrelative knowledge: K(a, p, q) is equivalent to K(a, p, Q), where Q is the question ?\(\{p, q\}\) [i.e. the question whether p or q]. (Schaffer 2007a, p. 241)
 2.
 3.
 4.
 5.
For other formal representations of Schaffer’s contrastivism, see Aloni et al. (2013), Schaffer and Szabó (2014), Hawke (2016) and Groenendijk (2007), the latter being the closest to the one I present here. None of these focus on the question of the reduction of contrastive knowledge. I leave the question of reduction of contrastive knowledge with respect to other formulations out of this paper, due to length constraints. I explain why I choose inquisitive semantics to formulate contrastivism in the end of Sect. 1.2.
 6.
It is fair to doubt the robustness of these intuitions. The issue is tested empirically in (Schaffer and Knobe 2012).
 7.
 8.
 9.
For another logic that combines knowledge and questions, see van Benthem and Minică (2009).
 10.
Using the \(\sigma \) function is equivalent to the (more familiar) use of an epistemic accessibility relation R where \(\sigma (w)=\{v \in W\; \;wRv \}\).
 11.
In some presentations of PAL (e.g. van Ditmarsch et al. 2007), the ‘box’ dual of \(\langle \varphi \rangle \), \([\varphi ]\), is taken as primitive. Apart from duality, the relation between the two is characterized by the PAL validity \(\langle \varphi \rangle \psi \leftrightarrow [\varphi ]\psi \wedge \varphi \).
 12.
For a further discussion about the relation between questions as linguistic objects and the inquisitive disjunction see, e.g. Ciardelli (2016), p. 51.
 13.
If the information state s is sufficient to resolve a question, then any stronger piece of information \(s'\) should be sufficient as well for a resolution of the question. We thus formally require that questions, understood as sets of sets of possible worlds, will be downward closed: if \(s\in Q\) and \(s' \subseteq s\) then \(s' \in Q\) (where s and \(s'\) are information states, and Q a question). The downward closure condition is omitted from Fig. 1 for graphical simplicity.
 14.
In IEL, \(\sigma (w)\) itself is obtained from a more general function, \(\Sigma \), describing the issue entertained by the agent. See the Appendix for the exact details.
 15.
The version of inquisitive epistemic logic presented here (an extension of the inquisitive logic known as InqB) cannot distinguish between the informative content of a sentence and the presuppositional content of a sentence. The two are not the same, however, and a logical analysis of presupposition should be able to distinguish them. For further discussion, including an extension of inquisitive semantics that can distinguish the two notions, see Roelofsen (2015).
 16.
See the appendix for the full formal definition of the support conditions of update clause in IDEL. For further details, see Ciardelli (2016, p. 313).
 17.
In the multiagent case, the update \( \langle \mu \rangle \) will affect the knowledge state of all agents. This can have unintended consequences, for instance making \(\mu ^!\) common knowledge. To avoid this, one can use private announcements that will only affect the epistemic state of a distinguished subset of agents. See van Gessel (2018) for the details.
 18.
 19.
The full reasons for this fact go beyond the technical scope of this paper (see Ciardelli 2016). Note that sentences of the form \(\langle \mu \rangle K \varphi \) are declaratives, and so truth conditional (see Ciardelli 2016, p. 318), meaning, roughly, that they have no inquisitive content. Furthermore, any truth conditional sentence \(\varphi \) of IDEL that does not contain the E operator (see Appendix) is equivalent to a sentence without inquisitive disjunctions (ibid. p. 209).
 20.
Further, note that it is a strength of Schaffer’s argument that it rests on simple examples where \(\varphi \) is nonepistemic. While it is true that sentences with the IDEL E modality are not equivalent to sentences without inquisitive disjunctions in general (see footnote 19), an attempt to build an entire theory of knowledge (according to which all knowledge is contrastive knowledge) from such examples will significantly weaken Schaffer’s argument.
 21.
It might indeed be the case that contrasts can explain away skeptical intuitions; the point is that these contrasts can be expressed in the language of binary knowledge.
 22.
The only difference is that given such a variant of IDEL, the reduction (or translation) happens outside of the object language. The result of the translation still holds.
 23.
Here I am slightly diverging from the formulation of IDEL in Ciardelli (2016). There, the basic dynamic modality appears in the box version as \([\varphi ] \psi \). The semantic clause is
$$\begin{aligned} \mathscr {M}, s \models [ \varphi ] \psi \;\; \Leftrightarrow \;\; \mathscr {M}_\varphi , \; s \cap \{w: \mathscr {M}, w \models \varphi \} \models \psi \end{aligned}$$Ciardelli (2016, p. 313). The truth condition of \([\varphi ]\psi \) relative to a world can then be retrieved to the usual
$$\begin{aligned} \mathscr {M}, w \models [ \varphi ] \psi \Leftrightarrow \mathscr {M}, w \models \varphi \;\; implies\;\; \mathscr {M}_{\varphi }, w \models \psi \end{aligned}$$Moreover, the reduction axioms for the dynamic modality (see later section) are given in terms of \([\varphi ]\psi \). In accordance to the general DEL validity \(\langle \varphi \rangle \psi \leftrightarrow [\varphi ]\psi \wedge \varphi \), I define \(\langle \varphi \rangle \psi \) in IDEL as \([\varphi ]\psi \wedge \varphi ^!\). This explains my choice for the support clause I give for \(\langle \varphi \rangle \psi \). The intuitive difference between \(\langle \varphi \rangle \psi \) and \([\varphi ]\psi \) from DEL persists in IDEL: whenever \(\varphi \) is unsupported, \(\langle \varphi \rangle \psi \) is also unsupported while \([\varphi ]\psi \) is vacuously supported. Thanks to Ivano Ciardelli for his helpful comment on this point.
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Acknowledgements
I would like to thank Johan van Benthem, Ray Briggs, Ivano Ciardelli, Krista Lawlor, Lisa Modifica, Jonathan Schaffer and and two anonymous referees of this journal for many helpful comments, suggestions and corrections on earlier versions of this article.
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Appendix
Appendix
Inquisitive dynamic epistemic logic
Definition 1
Issue
An issue I is a nonempty, downward closed set of information states. If \(t \in I\) then t is said to resolve I.
With the notion of an issue one can define an inquisitive epistemic model.
Definition 2
Inquisitive epistemic model
An inquisitive epistemic model is a triple \(\mathscr {M}=(W,V,\Sigma )\) s.t.

W is a set of possible worlds,

V is a valuation function, \(V: \mathscr {P} \rightarrow P(W)\), and

\(\Sigma \) is a state map, \(\Sigma : W \rightarrow P(P(W))\), taking worlds and returning issues.
The information state of the agent at a world, \(\sigma (w)\), is defined as \(\sigma (w) = \bigcup \Sigma (w)\).
Definition 3
The language of IDEL
The entire language of IDEL is defined inductively as:
We abbreviate \(\varphi \rightarrow \bot \) as \(\lnot \varphi \), \(\lnot (\lnot \varphi \wedge \lnot \psi )\) as \(\varphi \vee \psi \), as \(?\alpha \), and \(\varphi ^! \wedge [\varphi ]\psi \) as \(\langle \varphi \rangle \psi \).
Definition 4
Support
Let \(\mathscr {M}\) be an inquisitive epistemic model and s an information state in \(\mathscr {M}\). Then:

1.
\(\mathscr {M},s \models p\) iff \(w \in V(p)\) for all \(w \in s\)

2.
\(\mathscr {M},s \models \bot \) iff \(s= \emptyset \)

3.
iff \(\mathscr {M},s \models \varphi \) or \(\mathscr {M},s \models \varphi \)

4.
\(\mathscr {M},s \models \varphi \wedge \psi \) iff \(\mathscr {M},s \models \varphi \) and \(\mathscr {M},s \models \psi \)

5.
\(\mathscr {M},s \models \alpha \rightarrow \varphi \) iff for any \( t \subseteq s \), if \(\mathscr {M}, t \models \alpha \), then \(\mathscr {M}, t \models \varphi \)

6.
\(\mathscr {M},s \models K \varphi \) iff for any \(w \in s\), \(\mathscr {M},\sigma (w) \models \varphi \)

7.
\(\mathscr {M},s \models E \varphi \) iff for any \(w \in s\) and for any \(t \in \Sigma (w)\), \(\mathscr {M},t \models \varphi \)

8.
\(\mathscr {M}, s \models \langle \varphi \rangle \psi \) iff \(\mathscr {M}, s \models \varphi ^! \;\; and\;\; \mathscr {M}_\varphi , \; s \cap \{w: \mathscr {M}, w \models \varphi \} \models \psi \).^{Footnote 23}
For the derived connectives the following support conditions follow:
\(\mathscr {M},s \models \lnot \alpha \) iff for any nonempty \(t, t\subseteq s\), \(\mathscr {M},t \not \models \alpha \)
\(\mathscr {M},s \models \alpha \vee \beta \) iff there are \(t_1, t_2\) s.t. \(s=t_1 \cup t_2\), \(\mathscr {M},t_1 \models \alpha \) and \(\mathscr {M},t_2 \models \beta \).
To understand clause 8., we need a definition of an updated model:
Definition 5
Updated model
An inquisitive epistemic model \(\mathscr {M}\) after the announcement that \(\varphi \), \(\mathscr {M}_\varphi =(W_\varphi , V_\varphi , \Sigma _\varphi )\) is defined as:
\(W_\varphi = W \cap \{w: \mathscr {M}, w \models \varphi \}\)
\(V_\varphi = V\) restricted to \(W_\varphi \)
\(\Sigma _\varphi (w)= \Sigma (w) \cap \{s : \mathscr {M} ,s \models \varphi \}\)
Proof of Fact 1
Proof
PAL will suffice for the proof, since IDEL is a conservative extension of PAL (Ciardelli 2016).
Assume for reductio that Fact 1 is false, and consider the sentence \(\langle p \rangle K p\). Thus, there is a sentence \(K \varphi ^*\) s.t. \(\vdash \langle p \rangle K p \leftrightarrow K \varphi ^*\). By the reduction axioms of PAL, we have that \(\langle p \rangle K p\) is provably equivalent to p, so \(\vdash p \leftrightarrow K \varphi ^*\). Consider a T model containing only w and v with the universal relation, and let p be true in w and false in v. Since \(w \models p\), \(w \models K\varphi ^*\). Since the model is S5, \(v \models K\varphi ^*\). On the other hand, since \(v \models \lnot p\), we have that \(v \models \lnot K \varphi ^*\), contradiction. \(\square \)
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Cohen, M. Reducing Contrastive Knowledge. Erkenn (2019). https://doi.org/10.1007/s10670019001694
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