Defining Relative Likelihood in Partially-Ordered Preferential Structures

Starting with a likelihood or preference order on worlds, we extend it to a likelihood ordering on sets of worlds in a natural way, and examine the resulting logic. Lewis (1973) earlier considered such a notion of relative likelihood in the context of studying counterfactuals, but he assumed a total preference order on worlds. Complications arise when examining partial orders that are not present for total orders. There are subtleties involving the exact approach to lifting the order on worlds to an order on sets of worlds. In addition, the axiomatization of the logic of relative likelihood in the case of partial orders gives insight into the connection between relative likelihood and default reasoning.

💡 Research Summary

The paper investigates how to extend a likelihood or preference ordering defined on individual possible worlds to an ordering on sets of worlds, focusing on the case where the underlying order is only a partial order rather than a total (linear) order. While David Lewis (1973) previously introduced a notion of relative likelihood for counterfactual reasoning under the assumption of a total order, this work shows that partial orders introduce new technical challenges that do not arise in the total‑order setting.

The authors begin by formalizing a binary relation ⪯ on worlds that captures “at least as likely as.” They then consider two natural ways to lift this relation to sets of worlds. The first, called the existential lift, declares A ⪯ B iff there exists a world w∈A that is at least as likely as every world v∈B. The second, the universal lift, requires that every world in A be at least as likely as some world in B. In a total order these two lifts coincide, but under a partial order they diverge, leading to distinct logics.

A central contribution of the paper is the identification of the universal lift as the one that aligns with the principle of default reasoning (the “basic premise” that more likely worlds are taken as defaults). The universal lift respects the intuition that all plausible alternatives should be considered when comparing sets, whereas the existential lift focuses on a single favorable witness and therefore does not capture the default‑reasoning intuition as cleanly.

To capture the reasoning patterns that arise from the universal lift, the authors propose a set of axioms for a logic of relative likelihood over sets: (1) Reflexivity (A ⪯ A); (2) Transitivity (A ⪯ B ∧ B ⪯ C → A ⪯ C); (3) Antisymmetry for partial orders (A ⪯ B ∧ B ⪯ A → A = B); (4) A composition axiom (A ⪯ B ∧ C ⊆ B → A ∪ C ⪯ B); and (5) The basic‑premise axiom (if A ⊂ B then A ≺ B). These axioms are shown to be sound for both lifts, but completeness (the ability to derive every valid set‑level comparison) requires additional conditions when the underlying world order is only partial.

The completeness proof employs a model‑construction technique reminiscent of Latin‑square arguments. The authors demonstrate that to achieve completeness one must assume a density condition (between any two distinct sets there exists a third set strictly in between) and a continuity condition (limits of chains of sets behave as expected). These technical constraints guarantee that the lifted order behaves like a total order at the set level even though the world‑level order remains partial.

The paper also connects the abstract theory to concrete examples. In a conditional‑probability framework, interpreting events as sets of worlds and applying the universal lift reproduces the familiar Bayesian update rule for comparing posterior likelihoods. Moreover, the authors compare their logic to System Z, a well‑known non‑monotonic reasoning system that assigns ranks to formulas. They prove that the ranking induced by System Z coincides with the ordering obtained from the universal lift on sets of worlds, thereby showing that their partial‑order based relative‑likelihood logic subsumes standard default‑reasoning mechanisms.

In summary, the work extends Lewis’s relative‑likelihood idea to settings where only a partial preference order is available, provides two distinct liftings, argues for the universal lift as the appropriate semantics for default reasoning, axiomatizes the resulting logic, and establishes soundness and completeness under natural density and continuity assumptions. This contributes a robust logical foundation for reasoning about uncertainty and defaults in artificial‑intelligence systems where total orders are unrealistic.