Group Knowledge in Interrogative Epistemology

Abstract

In this paper we formalize an approach to knowledge that we call Interrogative Epistemology, in the spirit of Hintikka’s “interrogative model” of knowledge. According to our approach, an agent’s knowledge is shaped and limited by her interrogative agenda (as defined by her fundamental questions or “epistemic issues”). The dynamic correlate of this postulate is our Selective Learning principle: the agent’s agenda limits her potential for knowledge-acquisition. Only meaningful information, that is relevant to one’s issues, can really be learnt. We use this approach to propose a new perspective on group knowledge, understood in terms of the epistemic potential of a group of agents: the knowledge that the group may come to possess in common (and thus act upon in a coordinated manner) after all members share their individual information. We argue that the standard notions of group knowledge studied in the literature, ranging from distributed knowledge to common knowledge, do not give us a good measure of a group’s epistemic potential. Common knowledge is too weak and too “static”, focusing on what the agents can coordinate upon only based on their actual, current knowledge (without any intra-group communication), thus disregarding testimonial knowledge. In contrast, the concept of distributed knowledge is too strong, being based on the assumption that agents can completely internalize all the testimonial evidence received from others, irrespective of the limitations posed by their own interrogative agendas. We show that a group’s true epistemic potential typically lies in between common knowledge and distributed knowledge. We propose a logical formalization of these concepts, which comes together with a complete axiomatization, and we use this setting to explain both the triumphs and the failures of collective knowledge, treating examples that range from “collective scientific knowledge” [14, 35, 43] to the so-called “curse of the committee”.

APPENDIX: Completeness and Decidability (Proofs)

The proofs for Theorems 5.1–5.3 follow the standard method used in the completeness proof for epistemic logic with common knowledge and distributed knowledge, as introduced in [15]. The proofs’ plan is as follows:

• 1. Completeness for the Canonical Structure. First, we define a canonical structure. This is a Kripke structure, generated from the syntax in the usual way: possible worlds are maximally consistent theories. The canonical structure is not an interrogative epistemic model, since each modality has its own independent accessibility relation that cannot be reduced to a combination of the other relations. Nevertheless, our axioms are sound for this structure, and moreover they are complete: every maximal consistent theory satisfies itself.

• 2. Decidability via Finite Pseudo-Model Property. We use our version of a Fischer-Ladner closure to filtrate the canonical structure, obtaining a finite “pseudo-model”: this is a structure in which only the individual epistemic and issue modalities and the distributed-knowledge modality have their own independent accessibility relations $\rightarrow _a$, $\approx _a$, $\rightarrow _{Dk}$ (with $\rightarrow _{Dk}\subseteq \rightarrow _a$), while the other modalities are defined exactly as in interrogative epistemic models. This gives us soundness and completeness of our logics with respect to finite pseudo-models, which implies the decidability of these logics.

• 3. Unraveling. In the third step, we unravel the pseudo-model obtained in the previous step: this means that we create all possible histories in the pseudo-model (all paths that can be taken when we follow these one-step relations $\rightarrow _a$, $\approx _a$, $\rightarrow _{Dk}$), thus forming a tree. Then we redefine the relations $\rightarrow _a$ in order to ensure that they include the relation $\rightarrow _{Dk}$; if necessary (for the introspective versions), we also close the relations $\rightarrow _a$ and $\approx _a$ under transitivity (or transitivity and symmetry). Finally, we take the interrogative epistemic model obtained in this way (in which all the other relations are defined as in any epistemic issue model). We define a bounded morphism from this model into the finite pseudo-model obtained in the previous step. Completeness for interrogative epistemic models follows immediately.

• 4. Completeness for the dynamic logics (proof of Theorem 5.3 ). In the last step we show that the dynamic logics can be reduced to their static counterparts, using the reduction axioms and a translation mechanism, which gives us completeness for these logics.

We now give a sketch of the main definitions and results in each step.

STEP 1: Soundness and Completeness for the Canonical Structure

The canonical structure for LGK is a Kripke structure

$$M = \left( S, \{\rightarrow _a: a\in G\}, \{\approx _a: a\in G\}, \rightarrow _{Dk}, \rightarrow _{Ck}, \rightarrow _{Gk}, \Vert \bullet \Vert \right) , \text{ where } $$

• S consists of all maximally consistent sets of formulas (“theories”) of LGK;

• $s \rightarrow _a t \text { iff } \forall \varphi (K_a\varphi \in s \Rightarrow \varphi \in t)$;

• $s \approx _a t \text { iff } \forall \varphi (Q_a\varphi \in s \Rightarrow \varphi \in t)$;

• $s \rightarrow _{Dk} t \text { iff } \forall \varphi (Dk\varphi \in s \Rightarrow \varphi \in t)$;

• $s \rightarrow _{Ck} t \text { iff } \forall \varphi (Ck\varphi \in s \Rightarrow \varphi \in t)$;

• $s \rightarrow _{Gk} t \text { iff } \forall \varphi (Gk\varphi \in s \Rightarrow \varphi \in t)$;

• $\Vert p\Vert = \{s \in S; p \in s\}$.

It is easy to see that all the relations are reflexive. When the Positive Introspection axioms of $LGK^+$ are added, one can show that all the relations are transitive; and when the Negative Introspection axioms of $LGK^{\pm }$ are added as well, then all the relations are equivalence relations. For LaGK (and its introspective variants), the canonical structure has to be enriched with another independent accessibility relation $\rightarrow _a^G$, defined in a similar way to the others. The semantics is obvious: each modality is interpreted as the Kripke modality for its associated relation. Soundness and completeness with respect to the canonical structure are proved in the usual way, via the Truth Lemma¹¹: each theory $t\in S$ satisfies a formula $\varphi$ iff $\varphi \in s$.

Step 2: Decidability via Finite Pseudo-Model Property

A pseudo-model for LGK is a Kripke structure $M = ( S, \{\rightarrow _a: a\in G\}, \{\approx _a: a\in G\}, \rightarrow _{Dk})$, satisfying the following conditions: all relations are reflexive; $\rightarrow _a\approx _a\subseteq \rightarrow _a$; and $\rightarrow _{Dk}\subseteq \rightarrow _a$. A pseudo-model for $LGK^+$ is a pseudo-model for LGK in which all the relations are transitive. A pseudo-model for $LGK^{\pm }$ is a pseudo-model for LGK in which all the relations are equivalence relations.

Given a pseudo-model M, the semantics of our basic language is given by interpreting the modalities $K_a, Q_a$ and Dk as Kripke modalities for the corresponding relations, and defining the modalities Ck and Dk in the same way as on interrogative epistemic models.

For the language LaGK, we also define the modality $K_a^G$ in the same way as on interrogative epistemic models (using the appropriate definitions for each of the logics LaGK, $LaGK^+$, $LaGK^{\pm }$).

It is easy to see that each of these logics is sound with respect to the (corresponding) class of pseudo-models.

Let now M be the canonical structure constructed in Step 1, let $\varphi$ be a consistent formula, and let $s_{\varphi }$ be a maximally consistent theory that contains $\varphi$. The Fisher-Ladner closure of $\varphi$ is the smallest set of formulas $\Sigma _{\varphi }$ satisfying the following conditions: $\varphi \in \Sigma _{\varphi }$; $\Sigma _{\varphi }$ is closed under subformulas; $\Sigma _{\varphi }$ is closed under single negations (i.e. if $\psi \in \Sigma _{\varphi }$ is not of the form $\lnot \theta$, then $\lnot \psi \in \Sigma _{\varphi }$); if $K_a\psi \in \Sigma _{\varphi }$ then $Dk\psi \in \Sigma _{\varphi }$; if $K_a\psi \in \Sigma _{\varphi }$ and $\psi$ is not of the form $Q_a\theta$, then $K_a Q_a\psi \in \Sigma _{\varphi }$; if $Ck\psi \in \Sigma _{\varphi }$ then $\bigwedge _{a\in G} K_a Ck \psi \in \Sigma _{\varphi }$; finally, if $Gk\psi \in \Sigma _{\varphi }$ then $\bigwedge _{a\in G} Dk Q_a Gk \psi \in \Sigma _{\varphi }$. For LaGK, we need to add another closure requirement: $K_a^G\psi \in \Sigma _{\varphi }$ implies $Dk Q_a K_a^G \psi \in \Sigma _{\varphi }$.

It is easy to see that the Fisher-Ladner closure of $\varphi$ is always finite.

We define an equivalence relation on the set S of all maximally consistent theories in the canonical structure M above, by putting:

$$s\cong _{\varphi } t \text{ iff } s\cap \Sigma _{\varphi }=t\cap \Sigma _{\varphi }.$$

We can now define a filtration $M^f = \left( S^f, \{\rightarrow _a^f: a\in G\}, \{\approx _a^f: a\in G\}, \rightarrow _{Dk}^f,\right. \left. \rightarrow _{Ck}^f, \rightarrow _{Gk}^f, \Vert \bullet \Vert ^f\right)$ of the canonical structure M above, by putting:

• $S^f = \{[s]: s\in S\}$, where [s] is the equivalence class of s modulo $\cong _{\varphi }$;

• $[s]\rightarrow _a^f [t]$ if and only if $\forall K_a\psi \in \Sigma _{\varphi } (K_a\psi \in s \Rightarrow \psi \in t)$; for $LGK^+$, we use instead a transitive version: $[s]\rightarrow _a^f [t]$ if and only if $\forall K_a\psi \in \Sigma _{\varphi } (K_a\psi \in s \Rightarrow K_a \psi \in t)$; while for $LGK^{\pm }$, we use a transitive and symmetric version: $[s]\rightarrow _a^f [t]$ if and only if $\forall K_a\psi \in \Sigma _{\varphi } (K_a\psi \in s \Leftrightarrow K_a \psi \in t)$;

• $[s]\rightarrow _{Dk}^f [t]$ if and only if $\forall Dk\psi \in \Sigma _{\varphi } (Dk\psi \in s \Rightarrow \psi \in t)$; again, we use the corresponding transitive version for $LGK^+$, and the transitive and symmetric version for $LGK^{\pm }$;

• $[s]\approx _a^f [t]$ if and only if $\forall Q_a\psi \in \Sigma _{\varphi } (Q_a\psi \in s \Leftrightarrow Q_a \psi \in t)$;

• $\rightarrow _{Ck}^f= (\bigcup _{a\in G} \rightarrow _a^f)^+$;

• $\rightarrow _{Gk}^f = (\bigcup _{a\in G} \rightarrow _{Dk}^f\approx _a^f)^+$;

• for LaGK, $LaGK^+$, $LaGK^{\pm }$, we add the appropriately filtrated version for $\rightarrow _a^{G}$;

• $\Vert p\Vert =\{[s]: s\in \Vert p\Vert _M\}=[\{[s]: p\in s\}$, for $p\in \Phi \cap \Sigma$, and $\Vert p\Vert =\emptyset$ for $p\in \Phi \setminus \Sigma$.

It is now easy to check that $M^f$ is a finite pseudo-model, and that it is indeed a filtration ¹² of the canonical Kripke structure M.

By the usual properties of filtration, it follows that, for all formulas $\psi \in \Sigma _{\varphi }$, we have:

$$[s]\models _{M^f} \psi \,\, \text{ iff } \,\, s \models _M \psi \,\, \text{ iff } \,\, \psi \in s.$$

Hence, $\varphi$ is satisfied at state $[s_{\varphi }]$ in $M^f$.

This gives us completeness with respect to finite pseudo-models, and thus decidability.

Step 3: Unraveling

Let $\varphi , s_{\varphi }, \Sigma _{\varphi }$ be as above, and $M^f$ be the finite pseudo-model constructed in Step 2. A history in $M^f$ (with origin $[s_{\varphi }]$) is any finite sequence $h := ([s_0], R_0, [s_1],\dots ,R_{n-1}, [s_n])$ such that

• for all $k\le n: [s_k] \in S^f$;

• $[s_0] = [s_{\varphi }]$;

• for all $k< n: R_k\in \{\rightarrow _a^f : a\in G\}\cup \{\approx _a^f : a\in G\}\cup \{\rightarrow _{Dk}^f\}$;

• for all $k< n: [s_k] R_k [s_{k+1}]$.

For any history $h := ([s_0], R_0, [s_1],\dots ,R_{n-1}, [s_n])$, we write $last(h)=[s_n]$.

The unraveling of $M^f$ around $[s_{\varphi }]$ is a tree-structure $\vec {M} = (\vec {S}, R_{\rightarrow _a}, R_{\approx _a}, R_{\rightarrow _{Dk}})$ such that

• $\vec {S} = \{h; h \text { history in } M^f \}$;

• $h R_{\rightarrow _a} h' \text { iff } h'= (h, \rightarrow _a^f, [s'])$ for some $s'$;

• $h R_{\approx _a} h' \text { iff } h'= (h, \approx _a^f, [s'])$ for some $s'$;

• $h R_{\rightarrow _{Dk}} h' \text { iff } h'= (h, \rightarrow _{Dk}^f, [s'])$ for some $s'$.

We now convert $\vec {M}$ into an interrogative epistemic model $\vec {M} = (\vec {S}, \rightarrow , \approx , \Vert \bullet \Vert )$ as follows:

• we put $\approx _a:= (R_{\approx _a}\cup R_{\approx _a}^{-1})^*$, i.e. the least equivalence relation that includes $R_{\approx _a}$;

• for LGK, we take $\rightarrow _a := \approx _a\cup (R_{\rightarrow _a}\approx _a) \cup (R_{\rightarrow _{Dk}}\approx _a)$; for $LGK^+$, we take $\rightarrow _a := (\approx _a\cup R_{\rightarrow _a}\cup R_{\rightarrow _{Dk}})^*$; and for $LGK^{\pm }$, we put $\rightarrow _a := (\approx _a\cup R_{\rightarrow _a}\cup R_{\rightarrow _a}^{-1}\cup R_{\rightarrow _{Dk}}\cup R_{\rightarrow _{Dk}}^{-1})^*$, i.e. the least equivalence relation that includes $R_{\approx _a}\cup R_{\rightarrow _a}\cup R_{\rightarrow _{Dk}}$.

• $\Vert p\Vert =\{h\in \vec {S}: last(h)\in \Vert p\Vert _{M^f}\}$.

It is easy now to check that the map $last: \vec {S}\rightarrow S^f$, which maps each history $h\in \vec {S}$ into its last element $last(h)\in S^f$, is a bounded morphism ¹³ between pseudo-models $M^f$ and $\vec {M}$. Since bounded morphisms preserve the truth of modal formulas, it follows that our consistent formula $\psi$ is satisfied at history $([s_{\varphi }])$ in the interrogative epistemic model $\vec {M}$. This finishes our completeness proof for the static logics (Theorems 5.1 and 5.2).

Step 4: Completeness for the dynamic logics

We now prove Theorem 5.3. It is easy to see that the Reduction Axioms are sound. For completeness, we show that there exists a translation function t from formulas of our dynamic languages to formulas in the corresponding static languages, defined recursively as follows: $t(p)=t([!]p):=p$; $t(\lnot \varphi ):= \lnot t(\varphi )$; $t(\varphi \wedge \psi ):= t(\varphi ) \wedge t(\psi )$; $t(\Delta \varphi ):=\Delta t(\varphi )$ for all operators $\Delta \in \{K_a, Q_a, Dk, Ck, Gk, K_a^G\}$; $t([!]\lnot \varphi ):= t(\lnot [!]\varphi )$; $t([!] (\varphi \wedge \psi )):= t([!]\varphi ) \wedge t([!]\psi )$; $t([!] K_a\varphi )= t([!] K_a^G \varphi ):= t(K_a^G [!] \varphi )$; $t([!] Dk \varphi ):= t(Dk [!]\varphi )$; $t([!] Ck \varphi )=t([!] Gk \varphi ):=Gk t([!]\varphi )$.

Now we can show that, for every formula $\varphi$ of our dynamic languages LGK! and LaGK!, we have:

• $t(\varphi )$ is a formula in the corresponding static language (LGK or LaGK);

• the formula $\varphi \Leftrightarrow t(\varphi )$ is a theorem in the corresponding dynamic logic LGK! or LaGK!.

The proof is by induction, using the Reduction Axioms. This result immediately implies Theorem 5.3.