1. Basics A (finite) set N of individual voters i, j, k etc. Assume that |N| = n Basic language L: generated by the grammar φ: p | ~φ | ψ ˄ χ from a set A = {p, q, r, …} of basic judgments Individual view: a set Σj  L, where j  N Suppose we have a notion of consistency for L. Usually, individual views are supposed to be consistent. Sometimes, they are supposed to be complete Profile: an n-tuple p = (Σ 1, …Σ n ). For each profile p, an aggregation function F selects a social view: a set F(p)  L.

2. Studying individual and social views Different approaches Study the properties of individual and social views in a metalaguage of L. Extremely powerful, complex mathematical devices Construct different formal logical languages with different expressive powers and formulate in them the axioms Arguments pro: –clearly specification of the concepts needed to formulate the axioms –rigorous comparisons of different languages used to express axioms –non-axiomatizability results –contextualization of axiomatizations to different languages

3. Examples The minimalist approach of M. Pauly (2007a; b) Extensions of the minimal formal language

4. Example: Pauly’s approach Axioms are formulated in a language L C of social judgments defined by: φ  L C iff ψ  L and φ = Cψ; or ψ  L C and φ = ~ψ; or ψ  L C and χ  L C and φ = ψ ˄ χ Semantically, individual and social valuations are defined: –v: L → {1, 0} –V: L C → {1, 0} Individuals evaluate only those sentences which do not appeal to the idea of a collective judgment. Collectively we evaluate sentences which express the fact that the group made a specific judgment on a sentence. Valuations help construct individual and social views. Corollary: v(φ) = 1 and V(Cφ) = 1 are meaningful. But v(Cφ) = 1 and v(Cφ → φ) = 1 are meaningless; V(φ) = 1 and V(Cφ → φ) = 1 are also meaningless.

5. Example: the axiomatization of the majority voting S. Cφ → ~C~φ; T. (Mφ 1 ˄ … Mφ k ˄ ~Cψ 1 … ˄ ~Cψ k ) → (Qφ 1 ˄ Qψ 1 … Qφ k ˄ Qψ k ), where for all v: |{i: v(φ i ) = 1}| = |{i: v(ψ i ) = 1}| E. If φ ≡ ψ is a tautology, then Cφ ≡ Cψ M. C(φ ˄ ψ) → (Cφ ˄ Cψ)

6. A slightly richer language Sometimes combinations of sentences of L and of L C seem to make sense A new language L* Examples: Cφ → φ R. If φ → ψ is a tautology, then (~φ ˄ ψ ˄ Qφ) → Cψ Syntactically, we appeal to a richer language Semantically, only individual valuations are defined: if p  L, then v(p)  {1, 0}; if φ  L, then v(Cφ)  {1,0}; and v(~φ) = 1 - v(φ); v(φ ˄ ψ) = min(v(φ), v(ψ)). with the property that at each voting profile (v 1, … v n ) we have that v i (Cφ) = v j (Cφ) for each i, j.

7. Example: another axiomatization of the majority voting A. Cφ ≡ Cψ whenever |{i: v i (φ) = 1}| = |{i: v i (ψ) = 1}|. R. If φ → ψ is a tautology, then (~φ ˄ ψ ˄ Qφ) → Cψ S. Cφ → ~C~φ; E1. If φ → ψ is a tautology, then Cφ → Cψ Theorem: For each profile p = (Σ 1, …Σ n ), v 1 (Cφ) = 1 iff |{i: v i (φ) = 1}| > n/2.

8. Individual approval operators. I Alternative approach (starting point: Gärdenfors: 2006) A richer language L IC is constructed: A set C 1,... C n of individual approval operators are introduced Sentences of L IC : If φ  L, then φ  L IC ; if φ  L IC, then C j φ  L IC The Theory of individual consistent and complete choice (TICCC). Axioms : C i (φ ˄ ψ) ≡ C i φ ˄ C i ψ for each i  N C i ~φ ≡ ~C i φ for each i  N If φ is a theorem, then C i φ is also a theorem, for each i  N.

9. Individual approval operators. II Corollary: Each individual view is consistent and complete: ~(C i ~φ ˄ C i φ) for each φ and each i  N C i φ ˅ C i ~φ for each φ and each i  N We also have: If φ → ψ is a theorem, then C i φ → C i ψ for each i  N.

10. Iterations C k C i φ is meaningful Theorem: C k C i φ ˅ C k C i ~φ for each i and k  N = the individual k creates within her mind a (complete) image of the individual i’s view, in that for every sentence φ the individual k holds that i either accepts φ, or rejects it. Intuitively, an individual i’ view is broad: it includes not only her attitude toward alternatives of action, but also her view on the attitudes the other individuals have about those alternatives. However, the image needs not be adequate! When for some φ we have both C i φ and C k C i ~φ

11. Broad individual views Let p = (Σ 1, …Σ n ) be a profile. Σ i is consistent and complete. Σ i contains complete images of the individual view of any individual k in N: Lemma: Σ ki = {φ: C k φ  Σ i } is consistent and complete

12. A completeness theorem for Let Ξ be the collection of all consistent and complete sets of sentences of L IC (Ξ is the collection of all potential individual views) Let Θ  Ξ; Θ is comprehensive if if Σ  Θ entails Σ i = {φ: C i φ  Σ}  Θ for each i  N. Voting situation: a pair S = (Θ, N), where Θ is comprehensive. A sentence φ is true at S = (N, Θ) if φ  Σ for each Σ  Θ. Theorem: φ  TICCC iff for each φ is true at each voting situation S. Remember: TICCC has the axioms C i (φ ˄ ψ) ≡ C i φ ˄ C i ψ for each i  N C i ~φ ≡ ~C i φ for each i  N If φ is a theorem, then C i φ is also a theorem, for each i  N.

13. Social approval operators Enrich the language L IC to a new language L SIC by adding a social acceptance operator C: If φ  L IC, then φ  L SIC ; if φ  L SIC, then Cφ  L SIC. Example: a social approval operator C may satisfy Universal Social Voting (USV). For each voting situation S = (Θ, N) and each Σ  Θ, Cφ  Σ iff for all i  N, C i φ  Σ. Theorem. A social approval operator C has the property USV iff the following axioms hold: Cφ → C i φ for each i  N. Pareto optimality (PO): If C i φ is a theorem for all i  N, then Cφ is also a theorem.

14. Social approval operators satisfying USV Let Σ, Σ'  Θ. Define a binary relation R on the set Θ by: R(Σ, Σ') if there is some i  N such that Σ' = Σ i. The idea is that R(Σ, Σ') holds if Σ' is, according to Σ, an individual view which is actually held by some person. Write R Σ for the set of all Σ'  Θ such that R(Σ, Σ'). Then: Cφ  Σ iff φ  Σ', for all Σ'  R Σ. Theorem. Suppose that C satisfies USV. Then: –R Σ ≠ , for all Σ. –If R Σ is a singleton for all Σ, then C satisfies social completeness: Cφ ˅ C~φ. –If R Σ = Θ for all Σ, then C is the consensus voting.