Truth approximation, belief merging, and peer disagreement

Abstract

In this paper, we investigate the problem of truth approximation via belief merging, i.e., we ask whether, and under what conditions, a group of inquirers merging together their beliefs makes progress toward the truth about the underlying domain. We answer this question by proving some formal results on how belief merging operators perform with respect to the task of truth approximation, construed as increasing verisimilitude or truthlikeness. Our results shed new light on the issue of how rational (dis)agreement affects the inquirers’ quest for truth. In particular, they vindicate the intuition that scientific inquiry, and rational discussion in general, benefits from some heterogeneity in opinion and interaction among different viewpoints. The links between our approach and related analyses of truth tracking, judgment aggregation, and opinion dynamics, are also highlighted.

Appendix: Proofs

We first introduce some notation and terminology. Given a c-theory $T$, a completion of $T$ is any constituent $c_i$ of $\fancyscript{L}_{n}$ which agrees with all the claims of $T$. One can check that $c_i\in \fancyscript{R}_{T}$ iff $c_i$ is a completion of $T$. The reversal of a constituent is another constituent, which is the conjunction of all the negations of the conjuncts of the former constituent. By extension, the reversal of a sentence $X$ is the sentence which contains, in its range, the reversal of each constituent appearing in $\fancyscript{R}_{X}$ (Oddie 1986, p. 50). One can check that, given a c-theory $T$, the reversal of $T$ is just the conjunction of the negations of the claims of $T$; this was called the “specular” of $T$ by Cevolani et al. (2011, p. 186).

Proof of result(8) Let be $E=[T_1,\dots ,T_k]$ a profile of $k$ c-theories. Recall from (3) that $E^{\circ } = \bigvee \{c_i: \varDelta _{ min }({E},{c_i})\text { is minimal}\}$, where $\varDelta _{}(E,c_i)= \sum _{T_j\in E} \varDelta _{ min }({T_j},{c_i})$. Note that, if $T_j$ is a c-theory, the closest constituent in $\fancyscript{R}_{T_j}$ to $c_i$ is the completion of $T_j$ which agree with $c_i$ on all the basic sentences which are indeterminate in $T_j$; thus, $\varDelta _{ min }({T_j},{c_i})$ is just the number of mismatches between $T_j$ and $c_i$. In turn, $\varDelta _{}(E,c_i)$ is the sum of the number of mismatches between $c_i$ and each $T_j$ in $E$. Moreover, one can check that this sum can be equivalently expressed as the the sum, for each basic sentence $b$ appearing in $c_i$, of the number of theories in $E$ rejecting $b$. This means that $c_i$ is the closer to $E$, the smaller the number of theories in $E$ rejecting each of its claims.

Let be $c_i$ and $c_h$ two constituents differing only because $b$ is accepted in $c_i$ and rejected in $c_h$. Then $c_i$ is closer than $c_h$ to $E$ iff: 1) either some $T_j$ in $E$ accepts $b$ and no $T_j$ in $E$ rejects $b$; 2) or there are in $E$ both theories accepting $b$ and theories rejecting $b$, but the former are more than the latter. This means that all the constituents closest to $E$ will agree on each basic sentence $b$ such that the number of theories in $E$ accepting $b$ is greater than the number of those rejecting it. The conjunction of such $b$ is exactly the c-theory $E^{\circ }$.

Proof of result(9) Let be $T_1$ and $T_2$ two c-theories. As Fig. 2 shows, they can be expressed, respectively, as $T_1\equiv O_{T_1T_2}\wedge C_{T_1T_2}\wedge X_{T_1T_2}$ and $T_1\equiv O_{T_2T_1}\wedge C_{T_2T_1}\wedge X_{T_2T_1}$. Note also that, on the one hand, the claims of $O_{T_1T_2}=O_{T_2T_1}$, of $X_{T_1T_2}$, and of $X_{T_2T_1}$ are rejected neither by $T_1$ nor by $T_2$. On the other hand, each claim of $C_{T_1T_2}$ and of $C_{T_2T_1}$ is rejected by one theory and accepted by the other. From result (8), it follows that $T_1\circ T_2 = O_{T_1T_2}\wedge X_{T_1T_2} \wedge X_{T_2T_1}$.

Proof of result(10) Let be $T_1$ and $T_2$ two c-theories. Note first that, as one can easily check, measure $Vs _{\phi }$ as defined in (7) is additive in the sense that, given a c-theory $T$, $Vs _{\phi }(T)=\sum _{b} Vs _{\phi }(b)$ where $b$ is a claim of $T$. Thus, e.g., $Vs _{\phi }(T_1)= Vs _{\phi }(O_{T_1T_2})+ Vs _{\phi }(C_{T_1T_2})+ Vs _{\phi }(X_{T_1T_2})$. From result (9), it follows that $T_1\circ T_2 = O_{T_1T_2}\wedge X_{T_1T_2} \wedge X_{T_2T_1}$. Then $Vs _{\phi }(T_1\circ T_2)> Vs _{\phi }(T_1)$ iff $Vs _{\phi }(O_{T_1T_2})+ Vs _{\phi }(X_{T_1T_2})+ Vs _{\phi }(X_{T_2T_1})> Vs _{\phi }(O_{T_1T_2})+ Vs _{\phi }(C_{T_1T_2})+ Vs _{\phi }(X_{T_1T_2})$ iff $Vs _{\phi }(X_{T_2T_1})> Vs _{\phi }(C_{T_1T_2})$. Similarly, $Vs _{\phi }(T_1\circ T_2)> Vs _{\phi }(T_2)$ iff $Vs _{\phi }(X_{T_1T_2})> Vs _{\phi }(C_{T_2T_1})$.

Proof of result(11) Let be $T_1$ and $T_2$ two c-theories and suppose that there is no weak disagreement between $T_1$ and $T_2$, i.e., that their extra parts are “empty”. From result (9) it follows that $T_1\circ T_2 = O_{T_1T_2}$. Moreover, from result (10) it follows that $Vs _{\phi }(T_1\circ T_2) > Vs _{\phi }(T_1)$ iff $Vs _{\phi }(C_{T_1T_2})<0$ and $Vs _{\phi }(T_1\circ T_2)> Vs _{\phi }(T_2)$ iff $Vs _{\phi }(C_{T_2T_1})<0$. Note that, by definition, the conflicting part of $T_1$ is the reversal of the conflicting part of $T_2$, and vice versa. One can check that, if $\phi \ge 1$, then $Vs _{\phi }(C_{T_1T_2})>0$ iff $Vs _{\phi }(C_{T_2T_1})<0$. In fact, $Vs _{\phi }(C_{T_1T_2})>0$ iff, by definition (7), $cont _{t}(C_{T_1T_2},c_\star )>\phi cont _{f}(C_{T_1T_2},c_\star )$, i.e., if $T$ makes more matches than mistakes. But since, by definition, $cont _{t}(C_{T_1T_2},c_\star )= cont _{f}(C_{T_2T_1},c_\star )$ and $cont _{f}(C_{T_1T_2},c_\star )= cont _{t}(C_{T_2T_1},c_\star )$, this means that $C_{T_2T_1}$ makes more mistakes than matches and hence, given $\phi \ge 1$, that $Vs _{\phi }(C_{T_2T_1})<0$. It then follows that $Vs _{\phi }(T_1\circ T_2) > Vs _{\phi }(T_1)$ iff $Vs _{\phi }(T_1\circ T_2)< Vs _{\phi }(T_2)$, and vice versa.