Quantified preference logic

The logic of reason-based preference advanced in Osherson and Weinstein (2012) is extended to quantifiers. Basic properties of the new system are discussed.

💡 Research Summary

The paper “Quantified Preference Logic” builds on the reason‑based preference framework introduced by Osherson and Weinstein (2012) and extends it to the full expressive power of first‑order logic. The authors begin by motivating the need for quantification in preference reasoning: many natural statements about choice, such as “Every voter prefers policy A to policy B” or “There exists a candidate who is preferred by all voters,” cannot be captured by a propositional‑only system. To address this, they define a new formal language that adds the usual first‑order symbols (variables, predicates, ∀, ∃) to the binary preference operator “≻”. A formula ϕ ≻ ψ reads “ϕ is preferred to ψ,” and the operator can be nested inside the scope of quantifiers.

Semantically, the paper introduces a two‑layer model. The first layer is a standard first‑order structure M = (D, I) where D is a non‑empty domain and I interprets predicate symbols. On top of this, two functions are added: a reason function ρ that maps each formula to a set of justifying reasons, and a utility function u : D → ℝ that assigns a numeric value (or, in restricted fragments, a finite rank) to each individual. The truth condition for a preference formula is defined in terms of these functions: ϕ ≻ ψ holds in a world w if, for every reason set R that is applicable to both ϕ and ψ, the average (or weighted) utility of the individuals satisfying ϕ under R exceeds that of the individuals satisfying ψ under the same R. This definition preserves the original intuition that preferences are justified by reasons and are compared only when the same reasons are in play.

The authors then present a set of axioms and inference rules that capture the logical behavior of the extended language. Classical first‑order axioms are retained, while new axioms govern the preference operator: (i) non‑reflexivity (¬(ϕ ≻ ϕ)), (ii) transitivity ((ϕ ≻ ψ ∧ ψ ≻ χ) → ϕ ≻ χ), and (iii) quantifier preservation (e.g., ∀x ϕ ≻ ψ → ϕ ≻ ∀x ψ). The quantifier preservation axioms rely on continuity‑type assumptions about ρ and u with respect to variable substitution, ensuring that quantification does not arbitrarily disrupt established preferences.

A natural‑deduction proof system is introduced. In addition to the usual introduction and elimination rules for ∧, ∨, →, ¬, ∀, and ∃, the system contains a “preference transitivity rule” and a “quantifier‑preference rule” that allow one to move a preference statement across a quantifier when the appropriate side conditions on variables are satisfied. The main metatheoretic results are: (1) soundness – any formula derivable in the system is true in all quantified preference models; (2) completeness – any formula that is valid in all models can be derived using the proof system. The completeness proof adapts Henkin’s construction for first‑order logic, extending it with canonical reason and utility assignments that respect the preference axioms.

Complexity and decidability are examined in depth. In the unrestricted language, where the domain may be infinite and utilities are real numbers, the satisfiability problem is shown to be highly undecidable (essentially as hard as first‑order logic with arithmetic). However, for several natural fragments the authors obtain positive results. When the domain is finite, the utility function is restricted to a finite set of ranks, and the number of nested preference operators is bounded, the satisfiability problem becomes NP‑complete. Moreover, if utilities are binary (0/1) and the reason function is deterministic, model checking can be performed in polynomial time. These results delineate a clear trade‑off between expressive power and computational tractability.

The paper concludes with a comparative discussion of quantified preference logic versus modal preference logics. Modal approaches model preferences across possible worlds, whereas the quantified system works directly at the level of individuals, allowing statements about “all” or “some” agents to be expressed without resorting to higher‑order modalities. In terms of expressive power, quantified preference logic strictly subsumes the propositional fragment of modal preference logics, while sharing similar worst‑case complexity for certain fragments.

Finally, the authors acknowledge limitations and outline future work. The current framework treats reasons and utilities as independent; a more realistic model would allow reasons to influence utilities dynamically, perhaps via a probabilistic or game‑theoretic extension. Incorporating uncertainty about reasons, learning mechanisms, and richer preference aggregation operators are identified as promising directions for further research.