Natural deduction for a fragment of modal logic of social choice

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1 Natural deduction for a fragment of modal logic of social choice Tin Perkov Polytechnic of Zagreb Av V Holjevca 15, Zagreb, Croatia tinperkov@tvzhr Abstract We define a natural deduction system for a fragment of modal logic of social choice, which suffices to express and prove Arrow s Theorem We prove that the system is sound Introduction A sound and complete modal logic of social choice and, more generally, judgment aggregation, is given in (Ågotnes et al 2011), using a Hilbert-style axiomatization The authors state that it is of additional interest to provide a formal proof of Arrow s Theorem, and make some steps towards it We propose an alternative approach, a Jaśkowski-Fitch-style natural deduction system in which proofs are more intuitive, and we formalize a classical proof of Arrow s Theorem adapted from (Sen 1986), as presented in (Endriss 2011) Language In this section we briefly present the language of Judgment Aggregation Logic (JAL) from (Ågotnes et al 2011), although not in full generality, but adapted to the preference aggregation problem Fix finite sets N of n voters and M of m alternatives The atomic symbols of the language of modal logic of social choice are: a propositional variable p i for each i N, a propositional variable q (x,y) for each pair (x, y) of alternatives, and a special propositional variable σ Formulas are built in the usual way, using Boolean connectives and two modalities and The domain of a model is fixed: it is the set of all pairs (R, (x, y)), where R = (R 1,, R n ) is a voting profile, that is, a tuple of strict linear orders on M (preferences), and (x, y) is a pair of distinct alternatives 1 A model is defined by a social welfare function F, which maps each profile R to a preference F (R) The satisfaction of p i and q (x,y) is also fixed: for any F, we have F, R, (x, y) p i iff (x, y) R i, meaning i prefers x to y, and F, R, (x, y) q (x,y ) iff (x, y) = (x, y ) A social welfare function determines Copyright c 2014, Association for the Advancement of Artificial Intelligence (wwwaaaiorg) All rights reserved 1 In (Endriss 2011), (x, y) is any pair of alternatives, but in this article we demand x and y to be distinct, for the sake of a slight technical convenience the satisfaction of σ: F, R, (x, y) σ iff (x, y) F (R) ( society prefers x to y ) The truth of a formula built using a Boolean connective is defined as usual, and modalities have a standard interpretation, wrt accessibility relations which are apparent from the following: F, R, (x, y) ϕ iff F, R, (x, y) ϕ for all profiles R, and F, R, (x, y) ϕ iff F, R, (x, y ) ϕ for all pairs (x, y ) We use shorthand Aϕ for ϕ Note that A has the semantics of the universal modality We also use diamonds,, and E, respectively The validity (denoted ϕ) and the satisfiability of a formula is defined as usual We work with a fragment without propositional variables q (a,b) and without modalities and, but with the universal modality Since this way we only lose some expressivity concerning pairs, we call this fragment of the language the profiles fragment (PF) Natural deduction system Rules Fix sufficiently large (eg enumerable) sets of symbols P rof = {R 1, R 2, } and V ar = {x 1, x 2, } A proof in our natural deduction system is a sequence of clauses of the form, where R P rof, x, y V ar, x y and ϕ PF, built using the following rules (the last clause in each rule can be appended if the previous clauses are already in the proof, in any order): ψ ψ ( I) ( E) ψ ( E) ψ ( I) ψ ( I) ψ R, (x, y) : ϕ (DN) ( E) R, (x, y) : ϕ R, (x, y) : ( E)

2 R, (x, y) : ϕ R, (x, y) : ϕ ( E) R, (x, y) : ϕ ( I) R, (x, y) : Aϕ R, (x, y ) : ϕ (AE) R, (x, y ) : Eϕ (EI) where R and (x, y ) are any (including R and (x, y)) We call these rules -introduction ( I), -elimination ( E) and so on, as usual The following (hypothetical) rules ( E, I, I, E and EE) induce subproofs, depicted in the box The first clause in a subproof is the assumption Boxes can be nested, and clauses outside of a box can be used as premises of rules inside the box ψ ( I) where R and (x, y ) are any R, (x, y) : ϕ R, (x, y) : ϕ ( E) ϕ ( E) R, (x, y) : ϕ ( I) R, (x, y) : Eϕ R, (x, y ) : ϕ (EE) where R and (x, y ) are new, ie did not appear in the proof before Next we have rules with subproofs, but no assumptions R, (x, y) : ϕ R, (x, y) : ϕ ( I) R, (a, b) : ϕ R, (x, y) : Aϕ (AI) where R is new in the proof, and (a, b) is new in case no symbols from V ar that were already in the proof are used in this subproof Otherwise R, (a, b) : ϕ stands for several clauses we need to infer before we are allowed to apply the rule, that is, all R, (a, b) : ϕ such that (a, b) is equal to (x, y ), where (x, y ) is new, or a is old and used in the subproof and b = y, or b is old and used in the subproof and a = x, or a and b are both old and used in the subproof Next we have rules which reflect properties of the frame (transitivity and antisymmetry) We call these structural rules R, (x, y) : p R, (y, z) : p (T r) R, (x, z) : p R, (x, y) : p R, (y, x) : p (As) R, (x, y) : p R, (y, x) : p (As) where p denotes any p i or σ, throughout the article And finally, to address the universal domain assumption, we have the following rule (UD): R, (x 1, y 1 ) : i C1 p i R, (x 2, y 2 ) : i C2 p i R, (x k, y k ) : i Ck p i (apply structural rules and E, whenever applicable) (if the box contains, strike the box out) where R is new, pairs are any, C j N for all j, and conjunctions are just shorthands for sequences of atomic clauses (or alternatively, first apply E whenever applicable, then structural rules) The rule reflects a type of reasoning in an informal proof, which begins like this: let R be a profile such that voters from C 1 rank x 1 before y 1, and so on The rule says that we can assume any number of these (limited only by M ), thus specify a profile partially or completely, and then inside the dashed box check if there actually exists such profile, ie if a specified profile is well-defined If we conclude at any point, we cannot use the clauses from the box later in the proof, otherwise we can use them as if they where not in a box A proof can end at any point, provided all boxes are completed We say that a formula ϕ is a theorem of this system, and we write ϕ, if there is a proof (with all boxes completed) which ends with a clause R, (x, y) : Aϕ Soundness Throughout this section, for a fixed proof, v denotes a function which maps each R P rof to a profile R v and each x V ar to an alternative x v, such that x v y v for each pair (x, y) which appears in the proof We call such v a valuation Lemma 1 (Soundness of UD) Let Σ be the set of clauses from all dashed boxes (appended using UD rule) that appear in a proof and do not contain Let F be any social welfare function Then there is a valuation v such that for any clause in Σ we have F, R v, (x v, y v ) ϕ Proof Define v arbitrarily on V ar Put x v < R i y v if and only if the clause R, (x, y) : p i is in Σ Then < R i is a partial order for each R Irreflexivity is implied by the definition

3 of valuation, and to show transitivity, let x v < R i y v < R i z v, thus R, (x, y) : p i and R, (y, z) : p i are in Σ If x z, since all applicable structural rules are applied in dashed boxes, we also have R, (x, z) : p i Σ, so x v < R i z v If x = z, we infer R, (x, y) : p i, thus R, (x, y) : Now, for all R and i, define Ri v as an arbitrarily chosen strict linear order on M which extends < R i It is easy to see that thus defined profiles R v satisfy the claim We say that a clause is in the scope of an assumption due to a hypothetical rule if it is contained in the box starting from that assumption We say that a clause is in the scope of an assumption due to UD rule simply if it appears in the proof after that assumption We consider each assumption to be in the scope of itself Theorem 1 (Soundness) If ϕ, then ϕ Proof We prove the following claim, which clearly implies the theorem: let F be arbitrary Let be a clause in a proof, which is in the scope of a (possibly empty) set Σ of assumptions R 1, (x 1, y 1 ) : ϕ 1,, R k, (x k, y k ) : ϕ k Let v be any valuation such that F, R v 1, (x v 1, y v 1) ϕ 1,, F, R v k, (xv k, yv k ) ϕ k Then F, R v, (x v, y v ) ϕ Let be the n-th clause in a proof We prove the claim by induction on n The base case is easy Assume the claim holds for all k < n If the n-th clause is an assumption, the claim obviously holds, so assume the n-th clause is a conclusion of a rule Suppose this rule is -introduction So, let ψ be the n-th clause, obtained by I from premises and Let v be any valuation which satisfies all assumptions ψ is in the scope of But these include all assumptions the premises are in the scope of Hence, by induction hypothesis we have F, R v, (x v, y v ) : ϕ and F, R v, (x v, y v ) : ψ, thus F, R v, (x v, y v ) : ϕ ψ, as desired The cases of E, I, double negation rule, E, E, E, AE, I, EI and structural rules are similar For the case of I, let v be any valuation which satisfies all assumptions of the n-th clause ψ If F, R v, (x v, y v ) ϕ, then the inductive hypothesis implies F, R v, (x v, y v ) ψ, thus F, R v, (x v, y v ) ϕ ψ The cases of I and E are similar For I, let v be any valuation which satisfies all assumptions of the n-th clause R, (x, y) : ϕ Then v also satisfies all assumptions of the previous clause R, (x, y) : ϕ, so the induction hypothesis implies F, R v, (x v, y v ) ϕ Since R is new, R v is arbitrary, so F, R v, (x v, y v ) ϕ The cases of AI, E and EE are similar The meaning of P areto is the following: for any profile and pair (x, y), if all voters prefer x to y, then society prefers x to y The formula IIA expresses that society s preference of x over y should not be affected by changes of voters preferences regarding alternatives other than x and y The formula Dictatorial says that there is a voter whose preferences coincide with society s preferences under any profile While these expressions are arguably very clear in the presented abbreviated form, they are exponential in the number of voters, which might be a problem for the implementation, since number of voters generally can be large Now we can state Arrow s Theorem Theorem 2 (Arrow 1951) Let M 3 Then (P areto IIA Dictatorial) First we formally prove that a coalition (a subset of N) is decisive on any pair whenever it is weakly decisive on some pair (cf (Endriss 2011)) This property is formalized by the formula Decisive C := (p C σ) A( i C p i σ), where C N So, weak decisiveness means that for a fixed pair (x, y), society will prefer x to y in any profile under which C is exactly the set of voters with that preference The formula Decisive C says that this implies much stronger property, that whenever all voters from C agree on any preference, the society will have this preference In the following lemma we prove that this holds, assuming Pareto condition and the independence of irrelevant alternatives After this, in Lemma 3 we prove that each decisive coalition has a decisive proper subset This is formalized as Contraction C :=A( i C p i σ) D C A( i D p i σ) Lemma 2 Let C N Then P areto IIA Decisive C Proof In the following proof, the first appearance of vertical dots is for a straightforward, but lengthy derivation of R, (x, y ) : D C p D from R, (x, y ) : i C p i The second occurrence of vertical dots depicts the repetition of previous arguments for each D C, in order to apply the disjunction elimination rule Finally, the last vertical dots stand for the repetition of a similar argument to infer all R, (a, b) : i C p i σ, where a or b, or both, are equal to x or y, as required by the AI rule Without this, the proof covers only the case x, y, x, y are all distinct Later we append R, (x, y) : P areto IIA Decisive C freely, instead of repeating the following proof as a subproof A proof of Arrow s Theorem The ingredients of Arrow s Theorem are the Pareto condition, independence of irrelevant alternatives and dictatorship These are expressed by the following formulas: P areto: A(p 1 p n σ) IIA: A C N (p C σ (p C σ)) Dictatorial: i N A(p i σ) where C N and p C is the conjunction of all p k for k C and all p k for k / C

4 R, (x, y ) : P areto IIA R, (x, y ) : (p C σ) R, (x, y ) : i C p i R, (x, y ) : D C p D R, (x, y ) : p D R, (x, y ) : i C p i R, (y, x ) : i N\C p i R, (x, y ) : i D p i R, (y, x ) : i N\D p i R, (y, y ) : i N p i R, (x, x ) : i N p i R, (x, y ) : p C R, (x, y ) : p C σ R, (x, y ) : σ R, (y, y ) : i N p i σ R, (y, y ) : σ R, (x, x ) : i N p i σ R, (x, x ) : σ R, (x, y ) : σ R, (x, y ) : p D R, (x, y ) : p D σ R, (x, y ): C N (p C σ (p C σ)) R, (x, y ) : p D σ (p D σ) R, (x, y ) : (p D σ) R, (x, y ) : p D σ R, (x, y ) : σ R, (x, y ) : σ R, (x, y ) : i C p i σ R, (x, y ) : A( i C p i σ) R, (x, y ) : Decisive C R, (x, y ) : P areto IIA Decisive C R, (x, y) : A(P areto IIA Decisive C ) Lemma 3 Let M 3 and let C N Then P areto IIA Contraction C Proof R, (x, y ) : P areto IIA R (x, y ) : A( i C p i σ) R, (x, y ) : D C A( i D p i σ) R, (x, y ) : D C E( i D p i σ) R, (x, y ) : E( i D p i σ), R, (x, y ) : i D p i σ R, (x, y ) : i D p i R, (x, y ) : σ R, (x, y ) : i N\(C\D) p i R, (y, x ) : i C\D p i R, (y, z ) : i C p i R, (z, y ) : i N\C p i R, (x, z ) : i D p i R, (z, x ) : i N\D p i R, (y, x ) : p C\D R, (y, z ) : p C R, (x, z ) : p D R, (y, z ) : i C p i σ R, (y, z ) : σ R, (x, z ) : σ R, (x, z ) : p D σ D C R, (x, z ) : C N (p C σ (p C σ)) R, (x, z ) : p D σ (p D σ) R, (x, z ) : (p D σ) (use previous lemma) R, (x, z ) : A( i D p i σ) R, (x, y ) : i D p i σ R, (x, y ) : σ R, (x, y ) : R, (x, z ) : σ R, (z, x ) : σ R, (y, x ) : σ (as before, using C \ D) R, (x, y ) : R, (x, y ) : D C A( i D p i σ) R, (x, y ) : D C A( i D p i σ) R, (x, y ) : Contraction C R, (x, y ) : IIA Contraction C R, (x, y) : A(P areto IIA Contraction C )

5 Note that we used the assumption M 3 when we introduced three new alternatives by the UD-rule Proof of Arrow s Theorem R, (x, y ) : P areto IIA Dictatorial R, (x, y ) : i N A(p i σ) (lengthy, but straightforward) R, (x, y ) : i N E (p i σ) R, (x, y ) : P areto IIA Contraction N R, (x, y ) : Contraction N R, (x, y ) : C N A( i C p i σ) R, (x, y ) : A(p k σ) (C = {p k }) R, (x, y ) : E (p k σ) R, (x, y ) : (p k σ) R, (x, y ) : p k σ R, (x, y ) : Can we define a natural deduction system for the full language of the modal logic of judgment aggregation? If so, the problem of completeness would then be reduced to proving the axioms and simulating the inference rules from (Ågotnes et al 2011) in our system Some steps towards this are a work in progress Questions regarding complexity, implementation etc References Ågotnes, T; van der Hoek, W; and Wooldridge, M 2011 On the Logic of Preference and Judgment Aggregation Autonomous Agents and Multi-Agent Systems, 22(1): 4 30 Arrow, KJ 1951 Social Choice and Individual Values: John Wiley and Sons Endriss, U Logic and Social Choice Theory In Gupta, A; and van Benthem, J eds 2011 Logic and Philosophy Today: College Publications Sen, AK Social Choice Theory In Arrow, KJ; and Intriligator, MD eds 1986 Handbook of Mathematical Economics, Volume 3: North-Holland (similar boxes for all singleton C N) R, (x, y ) : A( i C p i σ) R, (x, y ) : P areto IIA Contraction C R, (x, y ) : Contraction C R, (x, y ) : D C A( i D p i σ) (boxes for all singleton D C, as before) R, (x, y ) : A( i D p i σ) R, (x, y ) : P areto IIA Contraction D R, (x, y ) : Contraction D (nest boxes until only singletons remain) (similar boxes for all D C st D > 1) (similar boxes for all C N st C > 1) R, (x, y ) : (P areto IIA Dictatorial) R, (x, y) : A (P areto IIA Dictatorial) Further work The following questions remain open: Is the system complete?