Ranking Functions

Ranking Functions

Franz Huber (California Institute of Technology, USA)
Copyright: © 2009 |Pages: 5
DOI: 10.4018/978-1-59904-849-9.ch198
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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).
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Ranking 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 AA, then W\AA) and finite intersection (if AA and BA, then ABA). A function ρ from the field A over W into the natural numbers N extended by ∞, ρ: AN∪{∞}, 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(AB) = min{ρ(A), ρ(B)}

If the field of propositions A is closed under countable intersection (if A1A, …, AnA, …, nN, 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 A1A,… AnA, …

  • 4.

    ρ(A1∪…∪An∪…) = min{ρ(A1), …, ρ(An), …}

If the field of propositions A is closed under arbitrary intersection (if BA, then ∩BA) 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 BA:

  • 5.

    ρ(∪B) = min{ρ(A): AB}

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, …

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