When do our conditions on aggregation even implystrong dictatorship? For the gen-eral preference agenda, this cannot be the case: it is indeed well-known that Arrow’s axioms allow for non-strong dictatorships (in the form of ‘lexicographic dictatorships’, in which a ‘second-order dictator’ acts as tie-breaker wherever the ‘…rst-order dictator’
is indi¤erent).
A …rst observation is that ifall propositions are unambiguous, i.e., ifUR=X, as is for instance true for classical relevance, then a dictatorship is automatically strong, so that Theorem4 becomes a strong dictatorship result:
Corollary 5 If X is properly and irreversibly pathlinked and UR=X, every aggre-gation rule F :Jn! J satisfying IIP and UAP is strongly dictatorial.
The condition UR =X can in fact be weakened to the condition that all propo-sitions in X are disjunctions of unambiguous propositions. We call p 2 X the dis-junction of the set of propositions S X if accepting p is (rationally) equivalent to accepting at least one member of S (i.e., for any rational judgment set J 2 J, p2J ,S\J 6=?). For instance, the proposition ‘it rains or snows’ is presumably the disjunction off‘it rains’, ‘it snows’g.25 Finally, by a disjunctionof unambiguous propositions I of course mean a disjunction of some set S UR.
2 5Given a set S X, X need of course not contain a proposition that is its disjunction. The disjunction is unique as long asX contains no two equivalent propositions.
Theorem 5 If fp;:p :p 2 URg is properly and irreversibly pathlinked and all am-biguous propositions are disjunctions of unamam-biguous propositions, then every aggre-gation rule F :Jn! J satisfying IIP and UAP is strongly dictatorial.
I give two examples where the additional condition holds, and one where it fails:
The condition holds trivially ifUR=X (no ambiguous propositions), hence in particular if relevance is classical. So, for classical relevance Theorem5reduces, just like Theorem4, to the Arrow-like theorem in judgment aggregation stated in Corollary 2.
The condition also holds for the evaluation agenda of Example 4 (and thus for the illustrations of Section 8). Here, an ambiguous propositions is of type :vk, saying that visnot the value holding on matter k; this is the disjunction of the unambiguous propositions of type vk0, v0 6=v, saying that some other value v0 holds on matterk. In the illustration under Case 2 of Section 8 all premises of Theorem 5hold, so that strong dictatorships are the only ‘solutions’.
The condition fails for the general preference agenda: no ambiguous proposition (of type xRy) is a disjunction of unambiguous propositions (of type x0P y0 = :y0Rx0). This is why Arrow’s theorem in its general version is not a strong dictatorship result.
10 Conclusion
The relevance-based approach to judgment aggregation hopefully opens up new per-spectives, by overcoming proposition-wise independence without allowing for arbi-trariness. On the constructive side, I have generalized sequential-priority and premise-based aggregation towards an arbitrary priority structure, captured by a ‘priority graph’ over the propositions. On the axiomatic side, I have introduced more gen-eral, relevance-based axioms of independence and unanimity-preservation, and shown various impossibility theorems based on these axioms. In the special case of the classical relevance notion, the two axioms reduce to their classical counterparts, and the theorems reduce to familiar results such as the Arrow-like theorem in judgment aggregation.
This paper is a …rst step. Future research could focus on other relevance-based aggregation rules, axioms and theorems. It could also address a more normative ques-tion: how should the relevance relation be designed in the …rst place? For instance, when should relevance be transitive? Re‡exive? Acyclic? Should relevance connec-tions and logical connecconnec-tions be related in any systematic way? Under the priority interpretation of relevance, which propositions should have priority? Such questions are obviously di¢cult. Yet the relevance-based approach needs systematic criteria for designing the relevance relation.
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A Proofs
I now prove all results. Throughout,N is the set of individualsf1; :::; ng.
A.1 Theorem 1 on priority rules
To prove this theorem, letF F(Dp)p2X0 be a priority rule relative to a priority graph Ron a …nite agenda X. The set of R-maximal (resp. R-minimal) elements of a set S X is denoted maxR(S) (resp. minR(S)) and contains those s 2 S for which there is no r 2Snfsg such that sRr (resp. rRs). As R is acyclic onX0 and as X0 is …nite,
maxR S6=? and min
R S 6=?for all ?6=S X0: (10)
(a) LetRbe transitive. Suppose for a contradiction that IIP is violated. Then not allp2X0 have the property that, for all (J1; :::; Jn);(J10; :::; Jn0)2 Jn, if Ji\ R(p) = Ji0\ R(p) for all ithen
F(J1; :::; Jn)\ f pg=F(J10; :::; Jn0)\ f pg: (11) Letp2X0 beR-minimal such that this property fails. Pick(J1; :::; Jn);(J10; :::; Jn0)2 Jn such that Ji \ R(p) = Ji0 \ R(p) for all i and (11) is violated. Choose any q 2 R(p)nf pg. By R’s transitivity R(q) R(p), and so Ji\ R(q) = Ji0\ R(q) for all i. By p’s minimality property, (11) holds for q instead of p. As this is so for all q2 R(p)nf pg,
F(J1; :::; Jn)\ R(p)nf pg=F(J10; :::; Jn0)\ R(p)nf pg: (12) Now letY :=f~p2 f pg:the set (12)entails pg.~
Case 1: Y 6=?. Then, by de…nition of the priority rule, we have F(J1; :::; Jn)\ f pg=Y and F(J10; :::; Jn0)\ f pg=Y. This implies(11), contradicting the choice ofp.
Case 2: Y = ?. Then, again by de…nition of the priority rule, F(J1; :::; Jn)\ f pg=Dp(J1\f pg; :::; Jn\f pg)andF(J10; :::; Jn0)\f pg=Dp(J10\f pg; :::; Jn0\ f pg). These two sets are distinct as (11) is violated. So there is an i such that Ji\ f pg 6= Ji0 \ f pg. So, as Ji \ R(p) = Ji0 \ R(p), R(p) cannot contain both of p, hence contains none of p by negation-invariance. In other words, R(p) = R(p)nf pg. So the set (12) equals F(J1; :::; Jn)\ R(p), which contains a member of each pair r 2 R(p) and thus entails p or :p by non-underdetermination. This contradicts thatY =?.
(b) Assume the transitivity and quasi-independence conditions. For allp2X, put Rp :=R(p)[ f pg andRp :=R(p)nf pg. Let(J1; :::; Jn)2 Jn. The consistency of J :=F(J1; :::; Jn)follows from three claims:
Claim 1: X=[p2maxRX0Rp; hence,J =[p2maxRX0(J\ Rp).
Claim 2: for any pairwise irrelevant propositions(pi)i2I;the sets(Rpi)i2Iare logi-cally quasi-independent; hence, the sets(Rp)p2maxRX0are logically quasi-independent.
Claim 3: J\ Rp is consistent for eachp2X0, hence also for eachp2maxRX0. Proof of Claim 1. For a contradiction, suppose Xn [p2maxRX0 Rp 6= ?. Then, by negation-invariance, X0n [p2maxRX0 Rp 6= ?. Hence by (10) there is a q 2 maxR(X0n [p2maxRX0 Rp). As q =2 [p2maxRX0Rp, we have q =2 maxRX0. So q is relevant to someq0 2X0nfqg. As q is maximal inX0n [p2maxRX0Rp and relevant toq0, it does not belong toX0n [p2maxRX0Rp. Soq0 2 [p2maxRX0Rp. Hence, asRis transitive,q is relevant to some p2maxRX0, a contradiction asq =2 [p2maxRX0Rp.
Proof of Claim 2. Consider pairwise irrelevant propositions(pi)i2I and consistent sets Bi Rpi (i 2 I) whose union contains no pair p. I show that [i2IBi is consistent. Without loss of generality let each Bi contain a member of each pair p2 Rpi (otherwise extend the Bi’s to consistent setsBi Rpi with this property;
the present proof shows the consistency of[i2IBi, hence that of[i2IBi). As the sets R(pi) (i2I) are logically quasi-independent, (*) [i2I(Bi\ R(pi)) is consistent. By non-underdetermination, (**) each Bi\ R(pi) entails a p~i 2 f pig. Since each Bi
entailsp~i (2 f pg) and by de…nition equals(Bi\R(pi))[fpigor(Bi\R(~pi))[f:pig, it must by consistency equal (Bi\ R(pi))[ f~pig. So, taking the union, [i2IBi = [i2I((Bi\ R(pi))[ f~pig). This set is consistent by (*) and (**).
Proof of Claim 3. Suppose the contrary: there is a p 2 X0 for which J \ Rp is inconsistent. By (10), we can pick a p 2 X0 that is R-maximal subject to J \ Rp being inconsistent. By an argument similar to that made for Claim 1,
Rp =[q2maxR(X0\Rp)Rq; hence J\ Rp =[q2maxR(X0\Rp)(J\ Rq). (13) By Claim 2, the setsRq; q2maxR(X0\ Rp), are logically quasi-independent. Hence, as each J \ Rq in (13) is consistent (by the maximality of p), the set J \ Rp = J \ R(p)nf pg is consistent. By construction of priority rules, the consistency is inherited to the augmented set(J\R(p)nf pg)[(J\f pg)(see the restated de…nition in footnote 11). This set equals J \ Rp. The consistency of J \ Rp contradicts the choice ofp.
A.2 Constrained entailment and (semi-)decisive coalitions: prepara-tory lemmas
Before proving the impossibility theorems, I show some lemmas that help us under-stand constrained entailment and its e¤ect on (semi-)winning coalitions.
First, as this de…nition of constrained entailment is symmetric inp and :q, con-strained entailment satis…es contraposition, as the reader checks easily:
Lemma 1 (contraposition) For all p; q 2X and all Y UR, p `R;Y q , :q `R;Y :p.
I now give a su¢cient condition for when a constrained entailment reduces to an unconditional entailment.
Lemma 2 For all p; q2X withR(p) R(:q)or R(:q) R(p),p`Rq if and only ifp`q.
Proof. Let p; q be as speci…ed. Obviously, p `q implies p`R;? q. Suppose for a contradiction that p `R q, say p `R;Y q, but p 6` q. Then fp;:qg is consistent. So there is aB 2 J containing pand :q. Then
the set B\ fr;:r:rRpg is an explanation ofp;
the set B\ fr;:r:rR:qgis an explanation of :q.
One of these two sets is a superset of the other one, asR(p) R(:q)orR(:q) R(p);
call this superset J. As p `R;Y q, Y [J is consistent. So, as J entails both p and :q, also Y [J [ fp;:qg is consistent. In particular, Y [ fp;:qg is consistent, in contradiction to the fact thatp`R;Y q.
The following fact helps in choosing the setY in a constrained entailment.
Lemma 3 For all p; q2X, if p`R q, then p`R;Y q for some set Y URn(R(p)[ R(:q)).
Proof. Letp; q2X, and assumep`Rq, sayp`R;Y q. The proof is done by show-ing thatp`R;Yn(R(p)[R(:q))q. Suppose for a contradiction that notp`R;Yn(R(p)[R(:q))
q. Then
(*)fp;:qg [Yn(R(p)[ R(:q))is consistent.
I show that
(**)p`p0 for allp02Y \ R(p)and :q `q0 for allq0 2Y \ R(:q),
which together with (*) implies thatfp;:qg [Y is consistent, a contradiction since p `R;Y q. Let p0 2 Y \ R(p). For a contradiction suppose p 6` p0. Then there is a B2 J containingpand:p0. The setJ :=B\fr;:r:rRpgdoes not entail:p, hence is an explanation ofp (by de…nition of a relevance relation). So J [Y is consistent (asp`R;Y q), a contradiction sinceJ[Y contains both p0 and :p0. This shows that p`p0. For analogous reasons, :q6`q0 for all q0 2Y \Xl.
Now I introduce notions of decisive and semi-decisive coalitions, and I show that semi-decisiveness is preserved along paths of constrained entailments.
De…nition 14 A possibly empty coalition C N is decisive (respectively, semi-decisive) for p 2 X if its members have judgment sets Ji 2 J, i 2 C, containing p, such that p 2 F(J1; :::; Jn) for all Ji 2 J, i2 NnC (respectively, for all Ji 2 J, i2NnC, not containing p).
While a decisive coalition for p can always enforce p (by using appropriate judg-ment sets), a semi-decisive coalition can enforcepprovided all other individuals reject p. LetW(p) and C(p) be the sets of decisive and semi-decisive coalitions forp 2X, respectively. The next lemma shows that semi-decisiveness is ‘contagious’ along con-strained entailments. The lemma parallels many other ‘contagion lemmas’ in social choice theory; indeed most standard proofs of Arrow’s Theorem use a contagion lemma (see, e.g., Gaertner’s [22] textbook).
Lemma 4 (contagion lemma) For all p; q 2 X, if p `R q then C(p) C(q). In particular, ifZ X is pathlinked, allp2Z have the same semi-decisive coalitions.26 Proof. Suppose p; q 2X, and p`R q, say p `R;Y q, where by Lemma 3 without loss of generalityY\ R(p) =Y\ R(:q) =?. LetC2 C(p). So there are setsJi2 J, i2C, containingp, such that p2 F(J1; :::; Jn) for all Ji 2 J,i2 NnC, containing :p. By Y’s consistency with every explanation ofp, it is possible to change each Ji, i2C, into a set (still in J) that contains everyy2Y and has the same intersection withR(p) asJi; this change preserves the required properties, i.e., it preserves that p2Ji for alli2C (asRis a relevance relation), and preserves thatp2F(J1; :::; Jn) for all Ji 2 J, i 2 NnC, containing :p (by Y \ R(p) = ? and IIP). So we may assume without loss of generality thatY Ji for all i2 C. Hence, by fpg [Y `q, allJi,i2C, contain q.
To establish that C 2 C(q), I consider any sets Ji 2 J,i 2NnC, all containing :q, and I show thatq 2F(J1; :::; Jn). We may assume without loss of generality that Y Jifor alli2NnC, by an argument like the one above (using thatY is consistent with any explanation of:q,Ris a relevance relation, Y \ R(:q) =?, and IIP). As f:qg [Y ` :p, all Ji, i 2 NnC, contain :p. Hence p 2 F(J1; :::; Jn). Moreover, Y F(J1; :::; Jn)by Y UR. So, asfpg [Y `q,q2F(J1; :::; Jn), as desired.
For any set S of coalitions C N, let S :=fC0 N :C C0 for someC 2 Sg.
2 6Constrained entailments preserve semi-decisiveness but usually not decisiveness.
Lemma 5 For all p; q2X,
(a) p`Rq irreversibly if and only if :q`R:p irreversibly;
(b) if p`Rq irreversibly then C(p) C(q).
Proof. Let p; q 2 X. Part (a) follows from Lemma 1 and the fact that, for all Y UR,fqg [Y 6`p if and only iff:pg [Y 6` :q.
Regarding (b), supposep`Rqirreversibly, sayp`R;Y qwithfqg[Y 6`p. We can assume without loss of generality that Y \ R(p) = Y \ R(:q) = ?, since otherwise we could replaceY byY0:=Yn(R(p)[ R(:q)), for which stillp`R;Y q(by the proof of Lemma 3) and fqg [Y0 6` p. To show C(p) C(q), consider any C0 2 C(p). So there is aC 2 C(p) with C C0. Hence there areJi 2 J, i2C, containing p, such thatp2F(J1; :::; Jn) for all Ji 2 J,i2NnC, containing :p. Like in earlier proofs, I may suppose without loss of generality that, for alli2C,Y Ji (using that Y is consistent with all explanations ofp,Ris a relevance relation, IIP, andY\R(p) =?);
hence, by fpg [Y `q, q2Ji for all i2C. Further, as f:p; qg [Y is consistent (by fqg [Y 6` p), there are sets Ji 2 J, i 2 C0nC, such that f:p; qg [Y Ji for all i2C0nC.
I have to show that q 2 F(J1; :::; Jn) for all Ji 2 J, i 2 NnC0, containing :q.
Consider such setsJi, i2 NnC0. Again, we may assume without loss of generality that for all i2NnC0, Y Ji (as Y is consistent with all explanations of :q, Ris a relevance relation, IIP, andY \ R(:q) = ?), which by f:qg [Y ` :p implies that :p2Ji for all i2NnC0. In summary then,
Ji 8
<
:
fp; qg [Y for alli2C f:p; qg [Y for alli2C0nC f:p;:qg [Y for alli2NnC0.
So p 2 F(J1; :::; Jn) (by the choice of the sets Ji, i 2C) and Y F(J1; :::; Jn) (by Y UR). Hence, as fpg [Y `q,q2F(J1; :::; Jn), as desired.
In the following characterisation of decisive coalitions it is crucial thatp2UR. Lemma 6 If p2UR, W(p) =fC N : all coalitions C0 C are in C(p)g.
Proof. Let p 2UR and C N. If C 2 W(p) then clearly all coalitions C0 C are inC(p). Conversely, suppose all coalitionsC0 C are inC(p). AsC2 C(p), there are sets Ji,i2 C, containing p, such that p2 F(J1; :::; Jn) for all sets Ji, i2NnC, not containingp. To show thatC2 W(p), consider any setsJi,i2NnC (containing or not containingp); I show thatp2F(J1; :::; Jn). LetC0 :=C[ fi2NnC:p2Jig.
By C C0, C0 2 C(p). So there are sets Bi, i 2 C0, containing p, such that p 2 F(B1; :::; Bn) for all sets Bi, i 2 NnC0, not containing p. As p has a single explanation, we have for all i2C0 Ji\ fr;:r :r 2 R(p)g=Bi\ fr;:r :r 2 R(p)g, henceJi\ R(p) =Bi\ R(p). So, by IIP and the de…nition of the setsBi,i2C0, and sincep62Ji for all i2NnC0,p2F(J1; :::; Jn), as desired.