Reducing Contrastive Knowledge

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 context-basing question. In a slogan: to know is to know the answer to a question. I argue, first, that Schaffer-style 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.

Appendix

1.1 Inquisitive dynamic epistemic logic

Definition 1

Issue

An issue I is a non-empty, 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 $\mathscr {M}, s \models \varphi ^! \;\; and\;\; \mathscr {M}_\varphi , \; s \cap \{w: \mathscr {M}, w \models \varphi \} \models \psi$.²³

For the derived connectives the following support conditions follow:

$\mathscr {M},s \models \lnot \alpha$ iff for any non-empty $t, t\subseteq s$, $\mathscr {M},t \not \models \alpha$

$\mathscr {M},s \models \alpha \vee \beta$ 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 $W_\varphi$

$\Sigma _\varphi (w)= \Sigma (w) \cap \{s : \mathscr {M} ,s \models \varphi \}$

1.2 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. $\square$