Social Choice and Defeasible Reasoning

I’m the author of the article on “Defeasible Reasoning” in the Stanford Encyclopedia of Philosophy. In revising my entry this month, I came across a fascinating idea proposed in 1991 by Sten Lindström (in an article first published in Theoria in 2022). As I argue in my article, the best approach to formalizing a defeasible or non-monotonic logic is one in which we say that a set of formulas Gamma defeasibly entails a formula phi just in case phi is true in all the preferred models of Gamma. This can be implemented in a version of modal logic, where the preferred models of the non-modal language are represented by preferred worlds in the modal theory.

But when is one world preferred to another? Lindström’ s idea was to import the principles of rational choice theory, as developed in the mid-20th century by Kenneth Arrow, Amartya Sen, and others. Each principle of social choice theory gives us a corresponding constraint on our preference orderings of worlds. Lindström uses a set selection function: a function that inputs a set of worlds (the worlds verifying the premises) and outputs another set of worlds (the preferred worlds).

Let C be the nonmonotonic consequence function. Cn = classical (monotonic) consequence. S is a selection function, defined over the powerset of the set of worlds. S(X) are the preferred worlds in set X. C and Cn are defined over sets of formulas.

Chernoff

S(X) ∩ Y ⊆ S(X ∩ Y)

C(Γ ∪ Δ ) ⊆ Cn(C(Γ) ∪ Δ),

The Chernoff condition, which is equivalent to Sen’s condition alpha, states that if one adds the information Δ to that in Γ, the defeasible consequences of the combined set cannot include more information than the result of simply adding Δ to the defeasible consequences of Γ alone. In terms of preferred worlds, this means that if x is a preferred model in A, then it is also a preferred model in any subset of A that includes x.

Path Independence

S(S(X) ∪ S(Y)) = S(X ∪ Y)

C(C(Γ) ∩ C(Δ)) = C(Cn(Γ) ∩ Cn(Δ))

Gamma

Let F be a nonempty set of sets of worlds. Then ⋂ _{(X ∈ F)}S(X) ⊆ S(⋃_{(X ∈ F)} X). I.e., if x is a best choice in every set X in a family of sets, then x is also a best choice in their union.

Gamma has a simple finitary consequence: S(X) ∩ S(Y) ⊆ S(X ∪ Y)

This semantic condition corresponds to the following syntactic rule:

∩ _{(Γ ∈ F)} Cn(Γ) ⊆ Cn(∪ _{Γ ∈ F} C(Γ)) where F is any non-empty family of sets of sentences.

Sen

If S(X) ∩ S(Y) ≠ ∅, then S(X ∩ Y) ⊆ S(X) ∩ S(Y)

This has a finitary consequence: S(X) ∩ S(Y) ⊆ S(X ∪ Y).

This is Sen’s property β. If C(Γ) ∪ C(Δ) is L-consistent, then C(Γ) ∪ C(Δ) ⊆ C(Γ ∪ Δ).

Arrow (independence of irrelevant alternatives)

If S(X) ∩ Y ≠ ∅, then S(X ∩ Y) = S(X) ∩ Y

If C(Γ) union Δ is L-consistent, then C(Γ ∪ Δ) = Cn(C(Γ ∪ Δ))

Consistency preservation

If X ≠ ∅, then S(X) ≠ ∅

If ⊥ ∉ Cn(Γ), then ⊥ ∉ C(Γ).

Arrow is equivalent to Chernoff plus Sen. Consistency preservation plus Arrow imply Chernoff, Cautious monontonicity and Gamma.