Truth-tracking by belief revision
Call plausibility space a triple of the form \((S,\mathcal{O},\preceq)\) where \(\mathcal{O}\), the set of observables, is a set of subsets of \(S\), the set of possible worlds, with both \(S\) and \(\mathcal{O}\) being countable and \(\bigcap\{o\in\mathcal{O}\mid s\in o\}=\{s\}\) for all \(s\in S\), and where \(\preceq\) is a total pre-order over \(S\), with \(s\preceq t\) meant to express that \(s\) is at least as plausible as \(t\). An iterative revision method \(R\) maps a pair consisting of a plausibility space and an observable to a plausibility space; given a plausibility space \(\mathbb S\) and a sequence \(\sigma=(o_1,\dots,o_k)\) of members of \(\mathcal{O}\), \(R(\mathbb S,\sigma)\) is then inductively defined as \(\mathbb S\) if \(k=0\), and as \(R\Bigl(R\bigl(\mathbb S,(o_1,\dots,o_{k-1})\bigr),o_k\Bigr)\) otherwise; any set of possible worlds that are all minimal in \(R(\mathbb S,\sigma)\) is then identified as a proposition that is believed after observing \(o_1\), \dots, \(o_k\). A number of properties of \(R\) are considered, that have to hold for any \(\mathbb S\), \(k\geq 1\) and \(o_1\), \dots, \(o_k\), in particular: \begin{itemize} \item conservative if \(o_k\) is believed in case it is believed after seeing \(o_1\), \dots, \(o_{k-1}\); \item strongly conservative if the plausibility spaces after seeing \(o_1\), \dots, \(o_{k-1}\) and after seeing \(o_1\), \dots, \(o_k\) are the same in case \(o_k\) is believed after seeing \(o_1\), \dots, \(o_{k-1}\). \end{itemize} Also, 3 specific one-step revision methods are defined (the conditions bring no change to the set of observations): \begin{itemize} \item [COND] (for conditioning): the total preorder of \(R(\mathbb S,o)\) is the restriction of the total preorder of \(\mathbb S\) to the set of possible worlds of \(R(\mathbb S,o)\), which are the possible worlds of \(\mathbb S\) that belong to \(o\). \item [LEX] (for lexicographic revision): the possible worlds of \(R(\mathbb S,o)\) are those of \(\mathbb S\), and \(s\) is at least as plausible as \(t\) in \(R(\mathbb S,o)\) if \(s\in o\) while \(t\notin o\), or \(s\) is at least as plausible as \(t\) in \(\mathbb S\) with either \(\{s,t\}\subseteq o\), or \(\{s,t\}\cap o=\varnothing\). \item [MINI] (for minimal revision): the possible worlds of \(R(\mathbb S,o)\) are those of \(\mathbb S\); if \(s\in o\) and is at least as plausible in \(\mathbb S\) as any possible world which \(o\) belongs to, while this is not the case for \(t\), then \(s\) is at least as plausible as \(t\) in \(R(\mathbb S,o)\); otherwise \(s\) is at least as plausible as \(t\) in \(R(\mathbb S,o)\) if it is so in \(\mathbb S\). \end{itemize} It is shown in particular that neither COND nor LEX are strongly conservative, whereas MINI is. The authors then consider whether, by properly choosing the initial plausibility space \(\mathbb S\), an iterative revision method \(R\) can be (a) universal (learning method), in that given \(s\in S\) and an infinite enumeration \(e\) of all members of \(\mathcal{O}\) that contain \(s\) (a data stream for \(s\)), \(R(\mathbb S,\sigma)\) has, for cofinitely many initial segments \(\sigma\) of \(e\), \(s\) as unique most plausible possible world (so \(\{w\}\) is the only nonempty believed property); in case \(\mathbb S\) can be chosen to be well-founded, then \(R\) is said to be standardly universal. The main results of the paper are: \begin{itemize} \item COND and LEX are universal, whereas MINI isn't. \item No conservative belief revision method is standardly universal. \end{itemize} Finally, the paper investigates two restrictions to the paradigm: one assumes that for all \(o\in\mathcal{O}\), \(S\setminus o\in\mathcal{O}\), making sure that data streams are closed under negation; the other one allows for finitely many errors in data streams, that are eventually corrected. It is shown in particular that whereas COND and MINI are not universal on data streams that are subjected to those conditions, LEX is.
scientific article; zbMATH DE number 7138883
scientific article