A proof-theoretic view on individual and collective preference - PowerPoint PPT Presentation

A proof-theoretic view on individual and collective preference Paolo Maffezioli Faculty of Philosophy University of Groningen 1 / 24

◮ Proof-theoretic methods in logic for social choice. ◮ Logic as axiomatic method. ◮ Logic beyond axioms: rule-based calculi. ◮ Method of proof analysis. ◮ Formalize the proofs of impossibility theorems. ◮  Inferentialize ˝ social choice theory. 2 / 24

◮ Proof-theoretic methods in logic for social choice. ◮ Logic as axiomatic method. ◮ Logic beyond axioms: rule-based calculi. ◮ Method of proof analysis. ◮ Formalize the proofs of impossibility theorems. ◮  Inferentialize ˝ social choice theory. 2 / 24

◮ Proof-theoretic methods in logic for social choice. ◮ Logic as axiomatic method. ◮ Logic beyond axioms: rule-based calculi. ◮ Method of proof analysis. ◮ Formalize the proofs of impossibility theorems. ◮  Inferentialize ˝ social choice theory. 2 / 24

◮ Proof-theoretic methods in logic for social choice. ◮ Logic as axiomatic method. ◮ Logic beyond axioms: rule-based calculi. ◮ Method of proof analysis. ◮ Formalize the proofs of impossibility theorems. ◮  Inferentialize ˝ social choice theory. 2 / 24

◮ Proof-theoretic methods in logic for social choice. ◮ Logic as axiomatic method. ◮ Logic beyond axioms: rule-based calculi. ◮ Method of proof analysis. ◮ Formalize the proofs of impossibility theorems. ◮  Inferentialize ˝ social choice theory. 2 / 24

◮ Proof-theoretic methods in logic for social choice. ◮ Logic as axiomatic method. ◮ Logic beyond axioms: rule-based calculi. ◮ Method of proof analysis. ◮ Formalize the proofs of impossibility theorems. ◮  Inferentialize ˝ social choice theory. 2 / 24

Hilbert-style proof theory for individual preference ◮ First-order language where atoms are interpreted as x is at least good as y x � y ◮ First-order axiomatization Axioms for ∀ , ∧ , → , ⊥ Modus Ponens ∀ x ( x � x ) � is reflexive ∀ x ∀ y ∀ z ( x � y ∧ y � z → x � z ) � is transitive ∀ x ∀ y ( x � y ∨ y � x ) � is total 3 / 24

Hilbert-style proof theory for individual preference ◮ First-order language where atoms are interpreted as x is at least good as y x � y ◮ First-order axiomatization Axioms for ∀ , ∧ , → , ⊥ Modus Ponens ∀ x ( x � x ) � is reflexive ∀ x ∀ y ∀ z ( x � y ∧ y � z → x � z ) � is transitive ∀ x ∀ y ( x � y ∨ y � x ) � is total 3 / 24

Hilbert-style proof theory for individual preference ◮ Definitions of > (strict preference) and ∼ (indifference) = d x � y and y � x x > y f x ∼ y = d x � y and y � x f ◮ Theorems ⊢ ∀ x ( x ∼ x ) � is reflexive ⊢ ∀ x ∀ y ∀ z ( x ∼ y ∧ y ∼ z → x ∼ z ) ∼ is transitive ⊢ ∀ x ∀ y ( x ∼ y → y ∼ x ) ∼ is symmetric ∀ x ( x ≯ x ) ⊢ � is irreflexive ⊢ ∀ x ∀ y ∀ z ( x > y ∧ y > z → x > z ) ∼ is transitive ∀ x ∀ y ( x > y → y ≯ x ) ⊢ ∼ is asymmetric 4 / 24

Hilbert-style proof theory for individual preference ◮ Definitions of > (strict preference) and ∼ (indifference) = d x � y and y � x x > y f x ∼ y = d x � y and y � x f ◮ Theorems ⊢ ∀ x ( x ∼ x ) � is reflexive ⊢ ∀ x ∀ y ∀ z ( x ∼ y ∧ y ∼ z → x ∼ z ) ∼ is transitive ⊢ ∀ x ∀ y ( x ∼ y → y ∼ x ) ∼ is symmetric ∀ x ( x ≯ x ) ⊢ � is irreflexive ⊢ ∀ x ∀ y ∀ z ( x > y ∧ y > z → x > z ) ∼ is transitive ∀ x ∀ y ( x > y → y ≯ x ) ⊢ ∼ is asymmetric 4 / 24

Gentzen-style proof theory for individual preference ◮ Systematic proof-search procedure. ◮ Sequent calculi Γ , ∆ multisets (lists without order) of formulas interpreted as � Γ → � ∆ Γ ⇒ ∆ ◮ One axiom. ◮ Logical rules. ◮ Structural rules. 5 / 24

Gentzen-style proof theory for individual preference ◮ Systematic proof-search procedure. ◮ Sequent calculi Γ , ∆ multisets (lists without order) of formulas interpreted as � Γ → � ∆ Γ ⇒ ∆ ◮ One axiom. ◮ Logical rules. ◮ Structural rules. 5 / 24

Gentzen-style proof theory for individual preference ◮ Systematic proof-search procedure. ◮ Sequent calculi Γ , ∆ multisets (lists without order) of formulas interpreted as � Γ → � ∆ Γ ⇒ ∆ ◮ One axiom. ◮ Logical rules. ◮ Structural rules. 5 / 24

Gentzen-style proof theory for individual preference ◮ Systematic proof-search procedure. ◮ Sequent calculi Γ , ∆ multisets (lists without order) of formulas interpreted as � Γ → � ∆ Γ ⇒ ∆ ◮ One axiom. ◮ Logical rules. ◮ Structural rules. 5 / 24

Gentzen-style proof theory for individual preference ◮ Systematic proof-search procedure. ◮ Sequent calculi Γ , ∆ multisets (lists without order) of formulas interpreted as � Γ → � ∆ Γ ⇒ ∆ ◮ One axiom. ◮ Logical rules. ◮ Structural rules. 5 / 24

Gentzen-style proof theory for individual preference ◮ Weakening, Contraction and Cut Γ ⇒ ∆ Γ ⇒ ∆ W W Γ ⇒ ∆ , ϕ ϕ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ, ϕ ϕ, ϕ, Γ ⇒ ∆ C C Γ ⇒ ∆ , ϕ ϕ, Γ ⇒ ∆ ϕ, Γ ′ ⇒ ∆ ′ Γ ⇒ ∆ , ϕ CUT Γ , Γ ′ ⇒ ∆ ′ , ∆ ◮ Admissibility of the structural rules. 6 / 24

Gentzen-style proof theory for individual preference ◮ Weakening, Contraction and Cut Γ ⇒ ∆ Γ ⇒ ∆ W W Γ ⇒ ∆ , ϕ ϕ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ, ϕ ϕ, ϕ, Γ ⇒ ∆ C C Γ ⇒ ∆ , ϕ ϕ, Γ ⇒ ∆ ϕ, Γ ′ ⇒ ∆ ′ Γ ⇒ ∆ , ϕ CUT Γ , Γ ′ ⇒ ∆ ′ , ∆ ◮ Admissibility of the structural rules. 6 / 24

Gentzen-style proof theory for individual preference P, Γ ⇒ ∆ , P ⊥ , Γ ⇒ ∆ ϕ, ψ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ Γ ⇒ ∆ , ψ ϕ ∧ ψ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ ∧ ψ Γ ⇒ ∆ , ϕ ψ, Γ ⇒ ∆ ϕ, Γ ⇒ ∆ , ψ ϕ → ψ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ → ψ ϕ ( x ) , ∀ xϕ ( x ) , Γ ⇒ ∆ Γ ⇒ ∆ , ϕ ( y ) y / ∈ Γ , ∆ ∀ xϕ ( x ) , Γ ⇒ ∆ Γ ⇒ ∆ , ∀ xϕ ( x ) G3c where P is either x � y or x > y or else x ∼ y . 7 / 24

Cut admissibility in presence of axioms ◮ Rules for � , ∼ and > s.t. admissibility results preserved ◮ G3c + ⇒ x ∼ x ( ∼ is reflexive) x ∼ y ⇒ y ∼ x ( ∼ is symmetric) x ∼ y, y ∼ z ⇒ x ∼ z ( ∼ is transitive) ◮ Counter-example to cut admissibility. x ∼ y ⇒ y ∼ x y ∼ x, x ∼ z ⇒ y ∼ z CUT x ∼ y, x ∼ z ⇒ y ∼ z 8 / 24

Cut admissibility in presence of axioms ◮ Rules for � , ∼ and > s.t. admissibility results preserved ◮ G3c + ⇒ x ∼ x ( ∼ is reflexive) x ∼ y ⇒ y ∼ x ( ∼ is symmetric) x ∼ y, y ∼ z ⇒ x ∼ z ( ∼ is transitive) ◮ Counter-example to cut admissibility. x ∼ y ⇒ y ∼ x y ∼ x, x ∼ z ⇒ y ∼ z CUT x ∼ y, x ∼ z ⇒ y ∼ z 8 / 24

Cut admissibility in presence of axioms ◮ Rules for � , ∼ and > s.t. admissibility results preserved ◮ G3c + ⇒ x ∼ x ( ∼ is reflexive) x ∼ y ⇒ y ∼ x ( ∼ is symmetric) x ∼ y, y ∼ z ⇒ x ∼ z ( ∼ is transitive) ◮ Counter-example to cut admissibility. x ∼ y ⇒ y ∼ x y ∼ x, x ∼ z ⇒ y ∼ z CUT x ∼ y, x ∼ z ⇒ y ∼ z 8 / 24

Cut admissibility in presence of axioms ◮ How can we restore cut admissibility? ◮ Systematic approaches: cut admissibility once and for all. ◮ Criteria for a new rule to be  good ˝ w.r.t. cut admissibility. 9 / 24

Cut admissibility in presence of axioms ◮ How can we restore cut admissibility? ◮ Systematic approaches: cut admissibility once and for all. ◮ Criteria for a new rule to be  good ˝ w.r.t. cut admissibility. 9 / 24

Cut admissibility in presence of axioms ◮ How can we restore cut admissibility? ◮ Systematic approaches: cut admissibility once and for all. ◮ Criteria for a new rule to be  good ˝ w.r.t. cut admissibility. 9 / 24

Axioms as inference rules ◮ Extension by inference rules ◮ G3c + x ∼ x, Γ ⇒ ∆ Ref Γ ⇒ ∆ y ∼ x, x ∼ y, Γ ⇒ ∆ Sym x ∼ y, Γ ⇒ ∆ x ∼ z, x ∼ y, y ∼ z, Γ ⇒ ∆ Trans x ∼ y, y ∼ z, Γ ⇒ ∆ 10 / 24

Axioms as inference rules ◮ Extension by inference rules ◮ G3c + x ∼ x, Γ ⇒ ∆ Ref Γ ⇒ ∆ y ∼ x, x ∼ y, Γ ⇒ ∆ Sym x ∼ y, Γ ⇒ ∆ x ∼ z, x ∼ y, y ∼ z, Γ ⇒ ∆ Trans x ∼ y, y ∼ z, Γ ⇒ ∆ 10 / 24

Axioms as inference rules ◮ Extension by inference rules ◮ G3c + x ∼ x, Γ ⇒ ∆ Ref Γ ⇒ ∆ y ∼ x, x ∼ y, Γ ⇒ ∆ Sym x ∼ y, Γ ⇒ ∆ x ∼ z, x ∼ y, y ∼ z, Γ ⇒ ∆ Trans x ∼ y, y ∼ z, Γ ⇒ ∆ 10 / 24

Axioms as inference rules ◮ Extension by inference rules ◮ G3c + x ∼ x, Γ ⇒ ∆ Ref Γ ⇒ ∆ y ∼ x, x ∼ y, Γ ⇒ ∆ Sym x ∼ y, Γ ⇒ ∆ x ∼ z, x ∼ y, y ∼ z, Γ ⇒ ∆ Trans x ∼ y, y ∼ z, Γ ⇒ ∆ 10 / 24

Axioms as inference rules ◮ Extension by inference rules ◮ G3c + x ∼ x, Γ ⇒ ∆ Ref Γ ⇒ ∆ y ∼ x, x ∼ y, Γ ⇒ ∆ Sym x ∼ y, Γ ⇒ ∆ x ∼ z, x ∼ y, y ∼ z, Γ ⇒ ∆ Trans x ∼ y, y ∼ z, Γ ⇒ ∆ 10 / 24

Axioms as inference rules ◮ x ∼ y, x ∼ z ⇒ y ∼ z has a cut-free derivation y ∼ z, y ∼ x, x ∼ y, x ∼ z ⇒ y ∼ z Trans y ∼ x, x ∼ y, x ∼ z ⇒ y ∼ z Sym x ∼ y, x ∼ z ⇒ y ∼ z ◮ The new rules are ◮ applied bottom-up ◮ logic-free ◮ left-hand side only ◮ cumulative 11 / 24