Preference Logic

Origin. Sören Halldén, On the Logic of "Better" (Library of Theoria 2, Lund: Gleerup, 1957), gave the first complete calculus of the comparative value concepts "better" and "equal in value", and Georg Henrik von Wright, The Logic of Preference (Edinburgh University Press, 1963), grown from his Edinburgh lectures, fixed the field's name, its prohairetic calculus, and its disputes. The modal reconstruction over a betterness ordering is due to Johan van Benthem, Patrick Girard and Olivier Roy, "Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences" (Journal of Philosophical Logic 38(1): 83–125, 2009); the dynamics began with van Benthem and Fenrong Liu, "Dynamic logic of preference upgrade" (Journal of Applied Non-Classical Logics 17(2): 157–182, 2007), and Liu's Reasoning about Preference Dynamics (Synthese Library 354, Springer, 2011) is the systematic treatment.

Models. An object-language logic of comparison: formulas assert that one proposition is better than or preferred to another, interpreted over a preorder ⪯ on worlds — the betterness relation — from which preferences between propositions are lifted by quantifying over the worlds where each holds. One ordering carries three readings: axiological (better), deontic (Bengt Hansson's dyadic obligation O(A/B) holds when the best B-worlds are A-worlds, so the semantics of Dyadic Deontic Logic is exactly a preference model), and doxastic (the plausibility orders of belief revision and dynamic epistemic logic are the same structure under an epistemic reading). It is not the Preferential Logic of Kraus, Lehmann and Magidor: there the ordering lives in the metalanguage and selects minimal models for a nonmonotonic consequence relation, here betterness is a relation on worlds quantified over by operators of the object language, and the two formalisms answer different questions even where the orderings coincide.

Formalism.

Prohairetic calculus (von Wright 1963): p P q: binary connective, "p is preferred to q", between state descriptions. Asymmetry: (p P q) → ¬(q P p). Transitivity assumed. Contraposition of preference: (p P q) → (¬q P ¬p). Conjunctive expansion: (p P q) ↔ ((p ∧ ¬q) P (¬p ∧ q)). Amplification: expansion of a preference over a fresh atom held fixed on both sides — the ceteris paribus clause made syntactic; incompatible with uniform substitution. Ceteris paribus reading: compare only worlds agreeing on everything other than the matter at issue. Contraposition and conjunctive expansion were contested from the start: Chisholm and Sosa, "On the Logic of 'Intrinsically Better'" (American Philosophical Quarterly 3(3), 1966), answered with a rival axiomatization of intrinsic betterness.

Halldén's calculus (1957): Two primitive connectives: better, and equal in value; betterness asymmetric and transitive, equality an equivalence. Expansion principles through negation; nesting of the connectives permitted in his theory A, barred in theory B. The later reconstruction reads his betterness over the difference worlds: the p∧¬q-worlds against the ¬p∧q-worlds.

Betterness frames: (W, ⪯) with ⪯ reflexive and transitive; w ≺ v iff w ⪯ v and not v ⪯ w. Modalities ⟨⪯⟩φ ("φ holds in some world at least as good"), ⟨≺⟩φ, and the global modality E.

Liftings — preference between propositions from the order on worlds: ∃∃: some ψ-world is better than some φ-world. ∀∃: above every φ-world lies a better ψ-world — axiomatized by Halpern, "Defining Relative Likelihood in Partially-Ordered Preferential Structures" (JAIR 7: 1–24, 1997). ∀∀: every ψ-world is better than every φ-world — von Wright's own reading, on the analysis of van Benthem, Girard and Roy. Halldén's lift: ∀∀ restricted to the difference worlds p∧¬q and ¬p∧q. Every lifting in this spectrum is definable from ⟨⪯⟩, ⟨≺⟩ and E.

Ceteris paribus operator: ⟨Γ⟩φ: φ holds in some world agreeing with the current one on every formula in the set Γ. "Everything else being equal" = Γ collects the facts independent of the comparison. Complete axiomatization for finite Γ (van Benthem, Girard and Roy 2009).

Dynamics: ⇑φ, radical upgrade: ⇑φ(⪯) := (?φ;⪯;?φ) ∪ (?¬φ;⪯;?¬φ) ∪ (?¬φ;⊤;?φ). Every φ-world is promoted above every ¬φ-world; the order inside each zone is kept. Reduction axioms translate upgrade statements into the static language (van Benthem and Liu 2007); the triggers read as suggestions or commands.

Deontic interface: O(A/B) iff the ⪯-best B-worlds are all A-worlds (Hansson, "An Analysis of Some Deontic Logics", Noûs 3(4): 373–398, 1969).

Symbols.

Metatheory. Van Benthem, Girard and Roy (2009) proved completeness for the betterness language over preorders and for the ceteris paribus extension with finite sets Γ, and related the unrestricted operator to infinitary modal logic. The reconstruction turns the old disputes into checkable facts about liftings: each quantifier pattern validates its own principle set, von Wright's axioms come out as properties of the ∀∀ lift read ceteris paribus, and amplification cannot be combined with uniform substitution, which bars it from any axiomatic system with standard rules. Halpern (1997) axiomatized the ∀∃ lift over both partial and total preorders. The deontic reading inherits its metatheory from the same ordering: Hansson's dyadic systems are evaluated by best B-worlds over a betterness preorder, Spohn (1975) proved DSDL3 complete for total preorders satisfying the limit assumption, and the completeness results recorded under Dyadic Deontic Logic — Åqvist's E, F and G, with E and F settled by Parent (2015) — are conditions on this one relation. Van Benthem and Liu (2007) derive reduction axioms for arbitrary definable relation changes, so each such dynamic logic is axiomatized by its static base plus the reductions, and completeness transfers.

Applies to. Dyadic deontic logic — conditional obligation over betterness orderings. Belief revision and plausibility dynamics: soft upgrades in dynamic epistemic and doxastic logic. Ceteris paribus clauses in preference statements, in philosophy and in the preference languages of computer science. Game theory: preference between outcomes in the modal analysis of solution concepts. Deliberation and agenda setting modeled as preference change.

Limitations. There is no single logic of preference: the ∃∃, ∀∃ and ∀∀ lifts validate different principle sets, the choice among them is a modeling decision the formalism cannot make, and different liftings recover different fragments of von Wright's calculus. The ceteris paribus operator is completely axiomatized only for finite Γ; the general operator requires infinitary means. The betterness ordering of the basic systems is primitive — the same externality recorded for Dyadic Deontic Logic — and deriving it from priorities or reasons takes the two-level machinery of Liu (2011), outside the basic language. The framework is qualitative: no degrees, so intensity of preference and trade-offs between criteria are inexpressible. Preferences attach to propositions, not to actions or agents, so strategic and conative structure must be added from outside. The dynamics covers only definable relation changes, and the aggregation of several agents' orderings is outside the basic language. And the KLM boundary bears repeating in the other direction: the representation theorems and rational closure of preferential consequence concern a metalanguage relation over models and say nothing about formulas built with preference operators.

© 2026 Lingenic LLC