Basing Belief on Quasi-Factive Evidence

Abstract

Topological semantics have proved to be a very fruitful approach in formal epistemology, two noticeable representatives being the interior semantics and topological evidence models. In this paper, we introduce the concept of quasi-factive evidence as a way to account for untruthful evidence in the interior semantics. This allows us to import concepts from topological evidence models, thereby connecting the two frameworks in spite of their apparent disparities. This approach sheds light on the interpretation of belief in the interior semantics, and gives meaning to concepts that used to be essentially technical: the closure-interior semantics can be interpreted as the condition of existence of a quasi-factive justification, while the extremally disconnected spaces are now characterized as those where the available information is always consistent. But our most important result is the equivalence between the interior-closure-interior semantics and what we call the strengthening condition, along with a sound and complete axiomatization. Finally, we build on this strengthening condition to introduce a notion of relative plausibility.

Appendix: General Topology

Definition A.1

Let $X$ be a set. A topology on a $X$ is a collection of sets $\tau \subseteq \mathcal {P}({X})$ such that:

• $\tau$ contains $\varnothing$ and $X$,

• $\tau$ is closed under arbitrary unions, i.e. if $(U_i)_{i \in I} \in \tau ^I$, then $\bigcup _{i \in I}U_i \in \tau$,

• $\tau$ is closed under finite intersections, i.e. for all $U\in \tau$ and $V\in \tau$, we have $U\cap V\in \tau$.

The pair $(X,\tau )$ is then said to be a topological space, and elements of $\tau$ are said to be open in $X$.

Alternatively, a topology can be presented by a generating family, also called a subbase:

Definition A.2

Let $B\subseteq \mathcal {P}({X})$ be a collection of subsets of $X$. The topology generated by $B$ is

$$\begin{aligned} \tau \,{{:}} {=}\,\left\{ \bigcup _{i \in I} \bigcap _{j=1}^{n_i} U^i_j \ \biggr |\ I \text { is a set and for all } i \in I, U^i_1, \dots , U^i_{n_i} \in B\right\} . \end{aligned}$$

This is the smallest topology containing the elements of $B$.

A subset is called closed if its complement is open. A subset $A \subseteq X$ is called a neighbourhood of a point $x\in X$ if there exists an open set $U$ such that $x\in U$ and $U\subseteq A$.

Definition A.3

Let $Y\subseteq X$. The subspace topology on $Y$ is defined by $\tau _{Y} \,{{:}} {=}\,\{U\cap Y\mid U\in \tau \}$. The pair $(Y,\tau _{Y})$ is a topological space and is called a subspace of $(X,\tau )$.

Two crucial topological notions are the interior and closure operators.

Definition A.4

Let $A \subseteq X$. The interior $\text {Int} _{}(A)$ of A is the set of all points of which A is a neighbourhood:

$$\begin{aligned} \text {Int} _{}(A) \,{{:}} {=}\,\{x\in X\mid \exists U\in \tau ,\ x\in U\subseteq A\}. \end{aligned}$$

Definition A.5

Let $A \subseteq X$. The closure $\text {Cl} _{}(A)$ of A is the set of all points whose every open neighbourhood intersects A:

$$\begin{aligned} \text {Cl} _{}(A) \,{{:}} {=}\,\{x\in X\mid \forall U\in \tau ,\ x\in U\implies U\cap A \ne \varnothing \}. \end{aligned}$$

We summarize in Table 7 a collection of standard properties of these operators, that we extensively use in this article.

Table 7 Classical properties of the interior and closure operators