Disbelief as the Dual of Belief

Abstract

The duality of truth and falsity in a Boolean algebra of propositions is used to generate a duality of belief and disbelief. To each additive probability measure that represents belief there corresponds a dual additive measure that represents disbelief. The dual measure has its own peculiar calculus, in which, for example, measures are added when propositions are combined under conjunction. A Venn diagram of the measure has the contradiction as its total space. While additive measures are not self‐dual, the epistemic state of complete ignorance is represented by the unique, monotonic, non‐additive measure that is self‐dual in its contingent propositions. Convex sets of additive measures fail to represent complete ignorance since they are not self‐dual.

Notes

[1] If degrees of belief are the same as degrees of confirmation, then we cannot always associate these degrees with probabilities; or so I have argued in Norton (2003, 2005, 2007a).

[2] For a more precise characterization, including an axiom system, see Marciszewski (1981), ‘Algebraic Structures’, section 6.7, 89, and section 6.9, 910, for a discussion of the dualities of Boolean Algebra. For a lengthier treatment of an axiom system and its self‐duality, see Goodstein (1963, ch. 2).

[3] The paired proofs are too lengthy for a footnote. See Goodstein (1963, ch. 2). In a simpler example, we start with the axioms A∨∼A = Ω and Ø∨A = A, substitute Ø for A in the first and ∼Ø for A in the second to recover Ø ∨ ∼Ø = Ω and Ø ∨ ∼Ø = ∼Ø, so that Ω = ∼Ø. In the dual proof, we start with dual axioms A & ∼A = Ø and Ω & A = A, substitute Ω for A in the first and ∼Ω for A in the second to recover Ω & ∼Ω = Ø and Ω &∼Ω = ∼Ω, to recover the dual theorem, Ø = ∼Ω.

[4] Massey (1992) has used this duality to argue for the indeterminacy of translation.

[5] Although Hajek (2003) urges that it does not handle all cases well, I introduce conditional measures with this ratio definition. It gives us the simplest, first approach to the calculus of dual additive measures and avoids the greater complication of an axiom system for conditional measures.

[6] We replace (A ⇒ B) = Ω by its dual ∼(B ⇒ A) = Ø, using Equations 1f and 1a. The latter is equivalent to (B ⇒ A) = Ω, or more simply, B ⇒ A.

[7] To see this, note that H = (H∨∼E) & (H∨E) and, when H ⇒ E, (H∨E) = E. Therefore, for a dual additive measure, M(H∨∼E) = M(H) − M(H∨E) = M(H) − M(E) = M(H) +M(∼E) − 1; so that M(∼E|H) = M(∼E∨H)/M(H) = 1+ (M(∼E)−1)/M(H).

[8] If we presume that the contradiction Ø corresponds to an area or point in the diagram, that area or point must satisfy contradictory requirements. Since Ø ⇒ A and Ø ⇒ B for any pair of mutually exclusive propositions A and B, the area or point corresponding to Ø must be contained within both of the disjoint areas corresponding to A and B.

[9] The condition ‘generically’ rules out the special case in which m(A) = m(B) and M(A) = M(B).

[10] The trivial exception is the outcome space of one proposition, A. Then the additive measure m(A) = m(∼A) = 1/2 is self dual in the one contingent proposition A.

[11] A set of measures is convex if, whenever m₁and m₂ are in the set, then so is λm₁ + (1−λ)m₂, for all 0 < λ < 1. See Kyburg and Pittarelli (1996) for discussion and an inventory of the problems raised by the convexity of the sets.

[12] This would correspond to replacing the set {m,M} = {M, m} by the ordered pair <m, M>, where the first member is reserved for the additive measure and the second for the dual additive measure. The ordered pair is not self‐dual, for, under the negation map (Equation 7) <m, M> → <M, m> ≠ <m, M>. In addition, in this simple case, the information in the ordered pair <m, M> is equivalent to the additive measure m and thus unable to represent ignorance for reasons given earlier.

Additional information

Notes on contributors

John D. Norton

John D. Norton is at the Department of History and Philosophy of Science and Center for Philosophy of Science, University of Pittsburgh.