Abstract
Ranking functions have been introduced under the name of ordinal conditional functions in Spohn (1988; 1990). They are representations of epistemic states and their dynamics. The most comprehensive and up to date presentation is Spohn (manuscript).
TopRanking Functions
Let W be a non-empty set of possibilities or worlds, and let A be a field of propositions over W. That is, A is a set of subsets of W that includes the empty set ∅ (∅ ∈ A) and is closed under complementation with respect to W (if A ∈ A, then W\A ∈ A) and finite intersection (if A ∈ A and B ∈ A, then A∩B ∈ A). A function ρ from the field A over W into the natural numbers N extended by ∞, ρ: A → N∪{∞}, is a (finitely minimitive) ranking function on A if and only if for all propositions A, B in A:
- 1.
ρ(W) = 0
- 2.
ρ(∅) = ∞
- 3.
ρ(A∪B) = min{ρ(A), ρ(B)}
If the field of propositions A is closed under countable intersection (if A1 ∈ A, …, An ∈ A, …, n ∈ N, then A1∩…∩An∩… ∈ A) so that A is a σ-field, a ranking function ρ on A is countably minimitive if and only if it holds for all propositions A1 ∈ A,… An ∈ A, …
If the field of propositions A is closed under arbitrary intersection (if B ⊆ A, then ∩B ∈ A) so that A is a γ-field, a ranking function ρ on A is completely minimitive if and only if it holds for all sets of propositions B ⊆ A:
- 5.
ρ(∪B) = min{ρ(A): A ∈ B}
A ranking function ρ on A is regular just in case ρ(A) < ∞ for each non-empty or consistent proposition A in A.
Key Terms in this Chapter
Pointwise Ranking Function: A function ? from the set of worlds W into the natural numbers N, ?: W ? N, is a pointwise ranking function on W if and only if ?(w) = 0 for at least one world w in W.
Degree of Disbelief: An agent’s degree of disbelief in the proposition A is the number of information sources providing the information A that it would take for the agent to give up her disbelief that A if those information sources were independent and minimally positively reliable.
Belief: An agent with ranking function ?: A ? N?{8} believes A if and only if ?(W\A) > 0 – equivalently, if and only if ?(W\A) > ?(A).
Ranking Function: A function ? on a field of propositions A over a set of worlds W into the natural numbers extended by 8, ?: A ? N?{8}, is a (finitely minimitive) ranking function on A if and only if for all propositions A, B in A: ?(W) = 0, ?(Ø) = 8, ?(A?B) = min{?(A), ?(B)}.
Belief Set: The belief set of an agent with ranking function ?: A ? N?{8} is the set of propositions the agent believes, Bel? = {A ? A: ?(W\A) > 0}.
Conditional Ranking Function: The conditional ranking function ?(·|·): A×A ? N?{8} based on the ranking function ? on A is defined such that for all propositions A, B in A: ?(A|B) = ?(AnB) – ?(B) if A ? Ø, and ?(Ø|B) = 8.
Completely Minimitive Ranking Function: A ranking function ? on a ?-field of propositions A is completely minimitive if and only if ?(?B) = min{?(A): A ? B} for each set of propositions B ? A.
Degree of Entrenchment: An agent’s degree of entrenchment for the proposition A is the number of information sources providing the information A that it takes for the agent to give up her disbelief in A.
Countably Minimitive Ranking Function: A ranking function ? on a s-field of propositions A is countably minimitive if and only if ?(A1?…?An?) = min{?(A1), …, ?(An), …} for all propositions A1 ? A,… An ? A, …