The Representation of Belief

Abstract

I derive a sufficient condition for a belief set to be representable by a probability function: if at least one comparative confidence ordering of a certain type satisfies Scott’s axiom, then the belief set used to induce that ordering is representable. This provides support for Kenny Easwaran’s project of analyzing doxastic states in terms of belief sets rather than credences.

Appendix

The results of Section 3 generalize to the case of a belief threshold of 1. First, a new definition of belief representability is required.

Definition 7 (B₁-Representable)

Let $\mathcal {X}$be a finite Boolean algebra,and let $B\subseteq \mathcal {X}$be a belief set. Say that B is b ₁ -representable just in casethere is a probability function P r such that

• (i) if P r(p) = 1then p ∈ B,and

• (ii) if P r(p) < 1then p∉B.

Second, the $\mathcal {C}$ construction needs to be adjusted for the 1 threshold. The new construction consists of two parts. For the first, define the following two sets.

• ${D_{1}^{1}}$ = {〈p,⊤〉∣p ∈ B}.

• ${D_{2}^{1}}$ $=\{\langle p,\bot \rangle \mid p\in \mathcal {X}\}$.

Then let $\succeq _{B}^{\star _{1}}={D_{1}^{1}}\cup {D_{2}^{1}}$.

${D_{1}^{1}}$ says that if A believes p, then A is at least as confident in p as in the tautological proposition. Given that the notion of b₁-representability under scrutiny here takes the threshold for belief to be 1, ${D_{1}^{1}}$ is extremely plausible: it implies absolute certainty in all propositions believed. ${D_{2}^{1}}$ is plausible for same reasons D ₃ is plausible (see Section 3).

For the second part of the construction, let $\mathcal {K}_{1}$ be the set of total comparative confidence orderings ≽ that contain $\succeq _{B}^{\star _{1}}$ as a subset, and that also satisfy the following conditions.

• $(\mathcal {C}_{1})$ 〈⊥,⊤〉∉ ≽.

• $(\mathcal {C}_{2}^{1})$ For all p∉B, 〈p,⊤〉∉ ≽.

As before, $(\mathcal {C}_{1})$ ensures that according to every ordering in $\mathcal {K}_{1}$, A is strictly more confident in ⊤ than in ⊥. $(\mathcal {C}_{2}^{1})$ ensures that if p is not believed, then A is not at least as confident in p as in ⊤. Since the orderings in $\mathcal {K}_{1}$ are total, this implies that if p is not believed, then A is strictly more confident in ⊤ than in p.

The following definition provides a succinct way to refer to this construction.

Definition 8 (Constructed from B in the manner of $\mathcal {C}$-1)

Let $\mathcal {X}$be a finite Boolean algebra, let $B\subseteq \mathcal {X}$be a belief set, and let $\mathcal {K}_{1}$be the set of total comparative confidence orderings that contain$\succeq _{B}^{\star _{1}}$and that satisfy $(\mathcal {C}_{1})$and $(\mathcal {C}_{2}^{1})$.Then $\mathcal {K}_{1}$is the set of comparative confidence orderings constructed from B in the manner of $\boldsymbol {\mathcal {C}-1}$.

As before, we need an account of how comparative confidence orderings can induce belief sets.

Definition 9 (Induced₁ Belief Set)

Let $\mathcal {X}$be a finite Boolean algebra, and let $\succeq \;\subseteq \mathcal {X}\times \mathcal {X}$be a comparative confidence ordering. Let B _≽be the belief set which consists of all and only the propositions p such that p ≽⊤.Call B _≽the belief set induced ₁ by ≽.

Now to derive a sufficient condition for b₁-representability that is analogous to Theorem 3.4. Proofs are included only when they are significantly different from the proofs of the corresponding lemmas and theorems in Section 3.

Lemma 2

Let $\mathcal {X}$ be a finite Boolean algebra, let $B\subseteq \mathcal {X}$ be a belief set, and let $\mathcal {K}_{1}$ be the set of comparative confidence orderings constructed from B in the manner of $\mathcal {C}$ -1. Each $\succeq \;\in \mathcal {K}_{1}$ satisfies (A ₁), (A ₂), and (A ₃).

Theorem A.1

Let $\mathcal {X}$ be a finite Boolean algebra, let $B\subseteq \mathcal {X}$ be a belief set, and let $\mathcal {K}_{1}$ be the set of comparative confidence orderings constructed from B in the manner of $\mathcal {C}$ -1. Then for each $\succeq \;\in \mathcal {K}_{1}$ , ≽satisfies (SA) if and only if ≽is c-representable.

Theorem A.2

Let $\mathcal {X}$ be a finite Boolean algebra, and let $\succeq \;\subseteq \mathcal {X}\times \mathcal {X}$ be a comparative confidence ordering that induces ₁ the belief set B _≽.Let P r be a probability function that c-represents ≽.Then P r b ₁ -represents B _≽.

Proof

If P r(p) = 1then since P r(⊤) = 1,it follows that $Pr(p)\geqslant Pr(\top )$.So by Definition 4, p ≽⊤.Therefore, by Definition 9, p ∈ B _≽.

If P r(p) < 1then since P r(⊤) = 1,it follows that P r(p) < P r(⊤).So by Definition 4, p ≺⊤.Definition 9 implies that p∉B _≽.

Therefore, by Definition 7, P r b₁-represents B _≽. □

Theorem A.3

Let $\mathcal {X}$ be a finite Boolean algebra, let $B\subseteq \mathcal {X}$ be a belief set, and let $\mathcal {K}_{1}$ be the set of comparative confidence orderings constructed from B in the manner of $\mathcal {C}$ -1. Suppose that some $\succeq \;\in \mathcal {K}_{1}$ induces ₁ the belief set B _≽.Then B _≽ = B.

Proof

First, take p ∈ B _≽. By Definition 9, p ≽⊤. By condition $(\mathcal {C}_{2}^{1})$, if p∉B, then p ≽ ⊤. Thus, p ∈ B.So B _≽⊆ B.

Second, take p ∈ B. Then $\langle p,\top \rangle \in {D_{1}^{1}}$,from which it follows that 〈p,⊤〉∈ ≽.Thus, by Definition 9, p ∈ B _≽.So B ⊆ B _≽.

Therefore, B _≽ = B. □

Finally, here is the sufficient condition for b ₁-representability.

Theorem A.4 (Sufficient Condition for B₁-Representability)

Let $\mathcal {X}$ be a finite Boolean algebra, let $B\subseteq \mathcal {X}$ be a belief set, and let $\mathcal {K}_{1}$ be the set of comparative confidence orderings constructed from B in the manner of $\mathcal {C}$ -1. Suppose that some $\succeq \;\in \mathcal {K}_{1}$ satisfies (SA). Then B is b ₁ -representable.

Proof

By Theorem A.1, ≽is c-representable by some probability function P r.Let B _≽be the belief set induced₁by ≽.By Theorem A.2, P r b₁-represents B _≽.Since B _≽ = B by Theorem A.3, P r b₁-represents B. □