Franz Huber (California Institute of Technology, USA)

DOI: 10.4018/978-1-59904-849-9.ch198

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TopLet *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 *A*_{1} ∈ **A**, …, *A _{n}* ∈

*4.*ρ(

*A*_{1}∪…∪*A*∪…) = min{ρ(_{n}*A*_{1}), …, ρ(*A*), …}_{n}

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**.

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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