A proof-theoretic view on preference relations and choice functions

The paper develops a Gentzen-style framework for reasoning about basic notions of social choice theory as preference relations and choice functions. First, it aims at providing an inferentialist account of the meaning of such notions in terms of the inference rules governing their use in formal derivations: to this end it is shown how to formulate the axioms of preference relations (reflexivity, completeness, etc.) and choice functions (Sen’s properties α and β, etc.) by means of introduction and elimination rules of natural deduction. The second aim is to provide a logical calculus in which derivations can be automatically generated: a cut and contractionfree sequent system is then introduced and it is shown how the fundamental property of eliminability of the cut rule allows to search systematically for derivations. Since many properties of order relations have an immediate interpretation in terms of preference, order-theoretic notions have played a central role in social choice theory. Traditionally, such properties are formulated in the language of predicate logic (with or without identity) and presented as axioms of Hilbert-style calculi. Since axiomatic theories are difficult to use in practice to derive theorems from axioms and consequently to regiment the proofs in ordinary mathematics, we present the first-order theory of preference relation as calculus based on rules of inference, following the tradition originated with Genzten [2]. In the first part of the paper we heavily rely on [4] where several calculi of sequents for order theories have been presented. Unlike orders in mathematics, preferences in social choice theory are intrinsically multi-agent notions: although it makes sense to think of an option as preferred to another one, it is far more interesting to say that each agent in a group has a preference and to see what happen whens their preferences are combined into a collective one. Secondly, preference are choice-guiding, i.e they must allow agents to choose the best options. Although we will be focusing mostly (Sections 1 and 2) on the second issue by providing a proof-theoretic account of the connection between preferences and choices, in the last section we show how to extend the framework to collective preference. Individual Preference Relations Regardless of whether taken as primitive or defined, there are three most important ordertheoretic notions that provide an account of preference relations. x > y x is strictly preferred to y (or, x is better than y) x ∼ y x and y are indifferent (or, x and y are equally good) x > y x is weakly preferred to y (or, x is at least good as y) If weak preference is taken to be primitive then strict preference and indifference will be defined as follows: x > y =df x > y and y x x ∼ y =df x > y and y > x Alternatively, we could take both > and ∼ as primitive and define > as their union, i.e. 1Discussions with N. Tennant have been an invaluable source of inspiration. x ∼ y =df x > y or x > y Usually, formal simplicity is preferred to conceptual clarity and therefore > is taken as the only primitive relation. In the standard theory of orders, > is assumed to be a partial order, i.e. a reflexive (x > x), transitive (x > y and y > z implies x > z), and antisymmetric (x > y and y > x implies x = y) binary relation. However, the latter does not correspond to any intuitively acceptable property of preference relation because it makes perfectly sense to think of two equally good distinct alternatives, i.e. x ∼ but x 6= y which contradicts antisymmetry. Therefore, we shall assume that > is only reflexive and transitive, a preorder. These properties will be given as rules of natural deduction [2]. x > x x > y y > z x > z In most applications it turns out to be useful to assume that > is complete, i.e. either x > y or y > x holds. This corresponds to the following rule (with φ an arbitrary formula) x > y (i) .... φ y > x (i) .... φ φ (i) The label (i) denotes that the formulas below the inference line are discharged, i.e. the conclusion does not depend on them. Also the meanings of the defined relations > and ∼ are given inferentially. For each notion, an introduction rule makes explicit the conditions under which x > y or x ∼ y can be concluded. x > y y x x > y x > y y > x x ∼ y Conversely, elimination rules specify what can be concluded from x > y or x ∼ y. Elimination rules presented here are in a general form (also called “parallel”in [7]) . The standard (or “serial”in [7]) rules are special cases when φ and the discharged formula coincide. x > y x > y (i) .... φ φ (i) x > y y x (i) .... φ φ (i) x ∼ y x > y (i) .... φ φ (i) x ∼ y y > x (i) .... φ φ (i) From the fact that > is a preorder (together with the definition of > and ∼ in terms of >) it follows that (i) > is an irreflexive, asymmetric and transitive; (ii) ∼ is an equivalence relation; (iii) > and ∼ are incompatible. One of the properties we shall use in the next and that can be easily derived is the equivalence of x > y and y ≯ x which essentially depends on the the completeness of >. x > y y > x (2) x y (1)