## Abstract

In his classic monograph, *Social Choice and Individual Values*, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems. We present a complete logic for reasoning about decisive coalitions, along with formal proofs of Arrow’s and Wilson’s theorems. In addition, we prove the correctness of an algorithm for calculating, given any social rationality postulate of a certain form in the language of binary preference, the corresponding axiom in the language of decisive coalitions. These results suggest for social choice theory new perspectives and tools from logic.

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## Notes

- 1.
For instance, see Saari (2008) for a geometric perspective, Baigent (2010) for a topological perspective, Abramsky (2015) for a category theoretic perspective, Lauwers and Van Liedekerke (1995) and Herzberg and Eckert (2012) for model-theoretic perspectives, Fishburn and Rubinstein (1986) and Shelah (2005) for algebraic perspectives, and Bao and Halpern (2017) for a quantum perspective.

- 2.
Reasoning about decisive coalitions is also central in the theory of judgment aggregation (see, e.g., List and Polak 2010).

- 3.
In a chapter entitled “Origins of the Impossibility Theorem” (Arrow 2014), Arrow recalls: “As it happens, during my college years, I was fascinated by mathematical logic, a subject I read on my own until, by a curious set of chances, the great Polish logician, Alfred Tarski, taught one year at The City College (in New York), where I was a senior. He chose to give a course on the calculus of relations, and I was introduced to such topics as transitivity and orderings” (p. 154). Sen (2017, p. xvi) even reports having taught mathematical logic at Delhi University. For his views on the relation between logic and economics, see Sen (2017, p. 108).

- 4.
Earlier Murakami (1968) applied results about three-valued logic to the analysis of voting rules.

- 5.
This separation of the social choice theoretic content and the combinatorial content of Arrow’s theorem is also a theme in Makinson (1996).

- 6.
Strictly speaking, as Sen—like Arrow—takes the weak preference relation

*R*as primitive, Sen’s CCRs send a vector of reflexive, transitive, and complete relations on*X*to a binary relation on*X*. - 7.
- 8.
SPA is not as standard as the other conditions, but it has a natural connection to our main topic of decisive coalitions, explained at the end of this section.

- 9.
For the latter implication, if \(\mathbf {P},\mathbf {P}'\in L(X)\), then \(\mathbf {P}(x^\star ,y)=\mathbf {P}'(x^\star ,y)\) implies \(\mathbf {P}(y, x^\star )=\mathbf {P}'(y, x^\star )\) and hence \(\mathbf {P}_{\mid \{x^\star ,y\}}=\mathbf {P}'_{\mid \{x^\star ,y\}}\). So if \(\mathrm {dom}(f)=L(X)\), then the assumption of SPA implies that for all \(x,y\in X\setminus \{x^\star \}\), \(\mathbf {P}_{\mid \{x,y\}}=\mathbf {P}'_{\mid \{x,y\}}\) and \(\mathbf {P}_{\mid \{x^\star ,y\}}=\mathbf {P}'_{\mid \{x^\star ,y\}}\), which implies \(\mathbf {P}=\mathbf {P}'\), so \(f(\mathbf {P})=f(\mathbf {P}')\).

- 10.
In Taylor and Zwicker (1999), the definition of a ‘simple game’ requires that

*W*be closed under supersets, while the term ‘hypergraph’ is used for the more general structures. - 11.
The reason for this finiteness assumption is given in footnote 18.

- 12.
Note that ‘\(\equiv \)’ is a symbol in our formal language, which will be interpreted using the equality relation (Definition 4.4), which we denote by ‘\(=\)’.

- 13.
We use ‘\({:}{=}\)’ to mean “is by definition an abbreviation for.”

- 14.
A different approach would allow distinct alternative labels to stand for the same alternative, but would then add identity formulas \(x= y\) to our language, so that we could express a requirement of distinctness when needed by a formula of the form \(\lnot \;x =y\).

- 15.
In fact, we also have all the resources to show that the answer is

*yes*for the class of CCRs satisfying LD, IIA, and TR (resp. FR), but we do not want to overload the reader with variations on our results. - 16.
The axiom \(\lnot (0\equiv 1)\) reflects our assumption that the set

*V*of all voters is nonempty. - 17.
A lower bound is easy: since our logic includes propositional logic, deciding theoremhood is a co-NP-hard problem. The question is: what is an upper bound?

- 18.
Note that here we use the assumption that \(\mathsf {Alt}\) is finite (which follows from our assumptions that

*X*is finite and that \(|\mathsf {Alt}|=|X|\)), so that*EA*can be written as a finite conjunction in our language. - 19.
For \(\widehat{\mathbf {T}}\), we can drop the assumption that \(x\ne v\) for part 1 and the assumption that \(w\ne y\) for part 2; and we can drop the same assumptions for \(\overline{\mathbf {T}}\) if we add the assumption \(\lnot (a\equiv 0)\) to the conditionals.

- 20.
By ‘Boolean reasoning’, we mean a combined use of valid Boolean equations, Leibniz’s law, and propositional logic.

- 21.
*A*is*semi-decisive for**x**over**y**according to**f*if and only if for any \(\mathbf {P}\in \mathrm {dom}(f)\), if \(A\subseteq \mathbf {P}(x,y)\), then*not*\(y f(\mathbf {P})x\). - 22.
By \(WEA(\mathbf 1 )\), we mean the

*WEA*formula of Sect. 5.1 in which each \(c_{x,y}\) is 1. - 23.
- 24.
To make the following argument fully rigorous, we need to define the formal syntax of the language in which these postulates are given, as well as the formal syntax of the language into which we are translating them below. The former can be the language of predicate logic with a single binary predicate symbols

*P*, and the latter can be the language of (almost) decisiveness in Sect. 4. - 25.
Thanks to Mikayla Kelley for this way of distinguishing the logic of decisive coalitions for Condorcet and Borda.

- 26.
This is because there can be CCRs

*f*and \(f'\) such that:*f*and \(f'\) have the same (almost) decisive coalitions, i.e., for all \(x,y\in X\), \(\widehat{D}_f(x,y)=\widehat{D}_{f'}(x,y)\) and \(\overline{D}_f(x,y)=\overline{D}_{f'}(x,y)\);*f*does not satisfy non-null; and \(f'\) does satisfy non-null. For example, let*f*be the CCR such that for all profiles \(\mathbf {P}\) and \(x,y\in X\),*not*\(xf(\mathbf {P})y\). Given \(x,y\in X\), we will define a CCR \(f_{x,y}\) that differs from*f*only on the profile \(\mathbf {P}^\star \) such that for all \(x,y\in X\) and \(i\in V\),*not*\(x P_i y\); for this \(\mathbf {P}^\star \), let \(z f_{x,y}(\mathbf {P}^\star ) w\) if and only if either \(z=x\) and \(w\ne x\), or \(z\ne y\) and \(w=y\), so \(f_{x,y}(\mathbf {P}^\star )\) is a strict weak order. Then for all \(z,w\in X\), if \(z\ne w\), then \(\widehat{D}_f(z,w)=\widehat{D}_{f_{x,y}}(z,w)= \overline{D}_f(z,w)=\overline{D}_{f_{x,y}}(z,w)=\varnothing \), while if \(z=w\), then \(\widehat{D}_f(z,w)=\widehat{D}_{f_{x,y}}(z,w)=\wp (V)\) and \(\overline{D}_f(z,w)=\overline{D}_{f_{x,y}}(z,w)=\wp (V)\setminus \{\varnothing \}\);*f*does not satisfy non-null; and \(f_{x,y}\) does satisfy non-null.

## References

Abramsky S (2015) Arrow’s theorem by arrow theory. In: Villaveces A, Kossak R, Kontinen J, Hirvonen Å (eds) Logic without borders: essays on set theory, model theory, philosophical logic and philosophy of mathematics. De Gruyter, Berlin, pp 15–30

Agotnes T, van der Hoek W, Wooldridge M (2009) On the logic of preference and judgment aggregation. Auton Agents Multi-Agent Syst 22(1):4–30

Arrow KJ (1959) Rational choice functions and orderings. Economica 26(102):121–127

Arrow KJ (2012) Social choice and individual values, 3rd edn. Yale University Press, New Haven

Arrow KJ (2014) Origins of the impossibility theorem. In: Maskin E, Sen A (eds) The Arrow impossibility theorem. Columbia University Press, New York, pp 143–148

Baigent N, Arrow KJ, Sen A, Suzumura K (2010) Topological theories of social choice. Handbook of social choice and welfare, vol 2. North-Holland, Amsterdam, pp 301–334

Bao N, Halpern NY (2017) Quantum voting and violation of Arrow’s impossibility theorem. Phys Rev A 95:062306. https://doi.org/10.1103/PhysRevA.95.062306

Barberá S (1980) Pivotal voters: a new proof of Arrow’s theorem. Econ Lett 6(1):13–16

Blair DH, Pollak RA (1979) Collective rationality and dictatorship: the scope of the Arrow theorem. J Econ Theory 21:186–194

Blair DH, Bordes G, Kelly JS, Suzumura K (1976) Impossibility theorems without collective rationality. J Econ Theory 13:361–379

Blau JH (1979) Semiorders and collective choice. J Econ Theory 21:195–206

Buchanan J (1954) Social choice, democracy and free markets. J Polit Econ 62(2):114–123

Campbell D (1992) Equity, efficiency and social choice. Clarendon Press, Oxford

Campbell DE, Kelly JS (2002) Impossibility theorems in the Arrovian framework. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, vol 1. North-Holland, Amsterdam, pp 35–94

Chagrov A, Zakharyaschev M (1997) Modal Logic. Clarendon Press, Oxford

Ciná G, Endriss U (2016) Proving classical theorems of social choice theory in modal logic. Auton Agents Multi-Agent Syst 30(5):963–989

Dryzek J, List C (2003) Social choice theory and deliberative democracy: a reconciliation. Br J Polit Sci 33(1):1–28

Endriss U (2011) Logic and social choice theory. In: Gupta A, Benthem J (eds) Logic and philosophy today. College Publications, London, pp 333–377

Fishburn P (1970) Arrow’s impossibility theorem: concise proof and infinitely many voters. J Econ Theory 2:103–106

Fishburn P, Rubinstein A (1986) Algebraic aggregation theory. J Econ Theory 38:63–77

Fleurbaey M, Mongin P (2005) The news of the death of welfare economics is greatly exaggerated. Soc Choice Welf 25:381–418

Geanakoplos J (2005) Three brief proofs of Arrow’s theorem. Econ Theory 26(1):211–215

Gibbard AF (2014) Intransitive social indifference and the Arrow dilemma. Rev Econ Des 18:3–10

Givant S, Halmos P (2009) Introduction to Boolean algebras. Springer, New York

Grandi U, Endriss U (2013) First-order logic formalisation of impossibility theorems in preference aggregation. J Philos Logic 42(4):595–618

Guha AS (1972) Neutrality, monotonicity, and the right of veto. Econometrica 40(5):821–826

Hammond P (1976) Equity, Arrow’s conditions, and Rawl’s difference principle. Econometrica 44:793–804

Hansson B (1976) The existence of group preference functions. Public Choice 28:89–98

Herzberg F, Eckert D (2012) The model-theoretic approach to aggregation: impossibility results for finite and infinite electorates. Math Soc Sci 64:41–47

Hurley S (1985) Supervenience and the possibility of coherence. Mind 94(276):501–525

Kalai E, Muller E, Satterthwaite M (1979) Social welfare functions when preferences are convex and continuous: impossibility results. Public Choice 34:87–97

Kelly JS (1988) Social Choice theory: an introduction. Springer, Berlin

Kirman AP, Sondermann D (1972) Arrow’s theorem, many agents, and invisible dictators. J Econ Theory 5:267–277

Kraft CH, Pratt JW, Seidenberg A (1959) Intuitive probability on finite sets. Ann Math Stat 30(2):408–419

Kroedel T, Huber F (2013) Counterfactual dependence and Arrow. Nous 47(3):453–466

Lauwers L, Van Liedekerke L (1995) Ultraproducts and aggregation. J Math Econ 24:217–237

Le Breton M, Weymark J (2010) Arrovian social choice theory on economic domains. In: Arrow KJ, Sen A, Suzumura K (eds) Handbook of social choice and welfare, vol 2. North-Holland, Amsterdam, pp 191–299

List C, Polak B (2010) Introduction to judgment aggregation. J Econ Theory 145(2):441–466

MacAskill W (2016) Normative uncertainty as a voting problem. Mind 124:967–1004

Mackie G (2003) Democracy defended. Cambridge University Press, Cambridge

Makinson D (1996) Combinatorial versus decision-theoretic components of impossibility theorems. Theory Decis 40:181–190

Mas-Colell A, Sonnenschein H (1972) General possibility theorems for group decisions. Rev Econ Stud 39(2):185–192

Monjardet B (1967) Remarques sur une classe de procédures de votes et les théorèmes de possibilité. In: La Décision. Colloque du CNRS, Aix-en-Provence, pp 117–184

Monjardet B (1978) Une autre preuve du théorème d’Arrow. R.A.I.R.O. Recherche Opérationnelle 12:291–296

Monjardet B (1983) On the use of ultrafilters in social choice theory. In: Pattanaik PK, Salles M (eds) Social choice and welfare. North-Holland, Amsterdam, pp 73–78

Morreau M (2015) Theory choice and social choice: Kuhn vindicated. Mind 123(493):239–262

Murakami Y (1968) Logic and social choice. Dover, New York

Nipkow T (2009) Social choice theory in HOL. J Autom Reason 43(3):289–304

Okasha S (2011) Theory choice and social choice: Kuhn versus Arrow. Mind 120(477):83–115

Pacuit E, Yang F (2016) Dependence and independence in social choice: Arrow’s theorem. In: Abramsky S, Kontinen J, Väänänen J, Vollmer H (eds) Dependence logic. Springer, Berlin, pp 235–260

Patty J, Penn EM (2014) Social choice and legitimacy: the possibilities of impossibility. Cambridge University Press, Cambridge

Reny PJ (2001) Arrow’s theorem and the Gibbard–Satterthwaite theorem: a unified approach. Econ Lett 70:99–105

Riker WH (1982) Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice. William H. Freeman, San Francisco

Rubinstein A (1984) The single profile analogues to multi profile theorems: mathematical logic’s approach. Int Econ Rev 25(3):719–730

Saari DG (2008) Disposing dictators, demystifying voting paradoxes: social choice analysis. Cambridge University Press, Cambridge

Salles M (2017) On quine on Arrow. Soc Choice Welf 48:877–886

Sen A (1969) Quasi-transitivity, rational choice and collective decisions. Rev Econ Stud 36(2):381–393

Sen A (1970) The impossibility of a Paretian liberal. J Polit Econ 78(1):152–157

Sen A (1979) Personal utilities and public judgements: or what’s wrong with welfare economics. Econ J 89(335):537–558

Sen A (1993) Internal consistency of choice. Econometrica 61(3):495–521

Sen A (2014) Arrow and the impossibility theorem. In: Maskin E, Sen A (eds) The Arrow impossibility theorem. Columbia University Press, New York, pp 29–42

Sen A (2017) Collective choice and social welfare: an expanded edition. Harvard University Press, Cambridge

Shelah S (2005) On the Arrow property. Adv Appl Math 34:217–251

Stegenga J (2013) An impossibility theorem for amalgamating evidence. Synthese 190(12):2391–2411

Suzumura K (1983) Rational choice, collective decisions, and social welfare. Cambridge University Press, Cambridge

Tang P, Lin F (2009) Computer-aided proofs of Arrow’s and other impossibility theorems. Artif Intell 173(11):1041–1053

Taylor A, Zwicker W (1992) A characterization of weighted voting. Proc Am Math Soc 115(4):1089–1094

Taylor AD, Zwicker WS (1999) Simple games: desirability relations, trading, pseudoweightings. Princeton University Press, Princeton

Troquard N, van der Hoek W, Wooldridge M (2011) Reasoning about social choice functions. J Philos Logic 40(4):473–498

von Neumann J, Morgenstern O (1947) Theory of games and economic behavior, 2nd edn. Princeton University Press, New Haven

Weymark JA (1984) Arrow’s theorem with social quasi-orderings. Public Choice 42:235–246

Wiedijk F (2007) Arrow’s impossibility theorem. Formaliz Math 15(4):171–174

Wilson R (1972) Social choice theory without the Pareto principle. J Econ Theory 5:478–486

Yu NN (2012) A one-shot proof of Arrow’s impossibility theorem. Econ Theory 50(2):523–525

## Author information

### Affiliations

### Corresponding author

## Additional information

We wish to thank the referees for *Social Choice and Welfare*, Mikayla Kelley, and Kotaro Suzumura for helpful comments.

## Appendix

### Appendix

In this Appendix, we sketch the proofs of the completeness theorems stated in Theorems 4.8.2, 4.8.4, and 7.4.2. A similar approach can be used to prove completeness theorems for logics that include inverse decisiveness as in Sect. 7.4.

For a logic \(\mathbf {L}\), a set \(\Sigma \) of formulas is \(\mathbf {L}\)-*inconsistent* if and only if there are \(\sigma _1,\dots ,\sigma _n\in \Sigma \) such that \(\vdash _\mathbf {L} (\sigma _1\wedge \dots \wedge \sigma _n)\rightarrow \bot \), where \(\bot \) is any formula of the form \(\varphi \wedge \lnot \varphi \). Otherwise \(\Sigma \) is \(\mathbf {L}\)-*consistent*. We say that a formula \(\varphi \) is \(\mathbf {L}\)-consistent if and only if \(\{\varphi \}\) is \(\mathbf {L}\)-consistent. Finally, a set \(\Sigma \) of formulas is *maximally*\(\mathbf {L}\)-*consistent* if and only if for every set \(\Delta \) of formulas, \(\Sigma \subsetneq \Delta \) implies that \(\Delta \) is \(\mathbf {L}\)-inconsistent.

To say that a logic \(\mathbf {L}\) is complete with respect to a class *K* of CCRs is to say that for any formula \(\psi \), if \(\psi \) is true of all CCRs from *K*, then \(\psi \) is a theorem of \(\mathbf {L}\). Equivalently, if \(\psi \) is *not* a theorem of \(\mathbf {L}\), then there is a CCR from *K* of which \(\psi \) is *not* true. Since by propositional logic, we have \(\not \vdash _\mathbf {L} \psi \) if and only if \(\not \vdash _\mathbf {L} \lnot \psi \rightarrow \bot \), the formula \(\psi \) not being a theorem of \(\mathbf {L}\) is equivalent to \(\lnot \psi \) being \(\mathbf {L}\)-consistent. Moreover, there being a CCR from *K* of which \(\psi \) is not true is equivalent to there being a CCR from *K* of which \(\lnot \psi \) is true. Thus, to prove that \(\mathbf {L}\) is complete with respect to *K*, it suffices to show that for any formula \(\varphi \), if \(\varphi \) is \(\mathbf {L}\)-consistent, then there is a CCR from *K* of which \(\varphi \) is true.

We will give a fairly detailed proof of Theorem 7.4.2, where \(\mathbf {L}\) is \(\overline{\widehat{\mathbf {T}}}\) (resp. \(\overline{\widehat{\mathbf {W}}}\)) and *K* is the class of CCRs satisfying LD, IIA, and TR (resp. FR). The proof where \(\mathbf {L}\) is \(\widehat{\mathbf {T}}\) or \(\overline{\mathbf {T}}\) (resp. \(\widehat{\mathbf {W}}\) or \(\overline{\mathbf {W}}\)) and *K* is the class of CCRs satisfying UD, IIA, and TR (resp. FR) is quite a bit simpler, as we will explain at the end.

In what follows by ‘consistent’ we mean ‘\(\overline{\widehat{\mathbf {T}}}\)-consistent’ or ‘\(\overline{\widehat{\mathbf {W}}}\)-consistent’, and by ‘\(\vdash \)’ we mean ‘\(\vdash _{\overline{\widehat{\mathbf {T}}}}\)’ or ‘\(\vdash _{\overline{\widehat{\mathbf {W}}}}\)’, depending on which case the reader has in mind. The proof is the same except at one point that we will flag. Indeed, the proof would be essentially the same for still other logics based on other rationality postulates (recall Sect. 8).

Suppose \(\varphi \) is consistent. Let \(T_1\) be the set of all terms generated by the coalition labels in \(\mathsf {Coal}(\varphi )=\{a\in \mathsf {Coal}\mid a\hbox { appears in }\varphi \}\) using −, \(\sqcap \), and \(\sqcup \) as in Definition 4.1. Define an equivalence relation *E* on \(T_1\) by: *sEt* if and only if \(s\equiv t\) is a valid equation of Boolean algebra. Let \(T_1/E\) be the quotient of \(T_1\) by *E*. It follows that \(T_1/E\) is a Boolean algebra in which the complement of [*t*] is \([-t]\), and the meet of [*s*] and [*t*] is \([s\sqcap t]\). Since \(T_1\) is the set of terms generated by \(\mathsf {Coal}(\varphi )\), it follows that the Boolean algebra \(T_1/E\) is generated by the set of equivalence classes of elements of \(\mathsf {Coal}(\varphi )\). Since the set of such equivalence classes is finite, \(T_1/E\) is finite by the well-known fact that any finitely generated Boolean algebra is finite (see, e.g., Givant and Halmos 2009, p. 82). For each \(x,y\in \mathsf {Alt}\) and \([t]\in T_1/E\), pick a coalition label \(c_{x,y,[t]}\in \mathsf {Coal}\setminus \mathsf {Coal}(\varphi )\) such that if \(\langle x,y,[t]\rangle \ne \langle x',y',[t']\rangle \), then \(c_{x,y,[t]}\ne c_{x',y',[t']}\). Since \(\mathsf {Alt}\) and \(T_1/E\) are finite, only finitely many coalitions labels \(c_{x,y,[t]}\) are needed. Crucially, \(c_{x,y,[t]}\) does not appear in \(\varphi \) or *t*. Now define

We claim that the set \(\{\varphi \}\cup \Theta \) is consistent. If not, then there are \(\theta _1,\dots ,\theta _{n+1}\in \Theta \) (\(n\ge 0\)) such that \(\{\varphi ,\theta _1,\dots ,\theta _{n+1}\}\) is inconsistent, but no proper subset is inconsistent. Thus, \(\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _{n+1})\rightarrow \bot \) and hence \(\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow \lnot \theta _{n+1}\) by propositional logic. The formula \(\theta _{n+1}\) is \((t\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))\) for some \(t\in T_1, x,y\in \mathsf {Alt}\). Thus, by propositional logic we have:

Not only does \(c_{x,y,[t]}\) not appear in \(\varphi \) or *t*, but also \(c_{x,y,[t]}\) does not appear in \(\theta _1,\dots ,\theta _n\). Toward a contradiction, suppose that \(c_{x,y,[t]}\) does appear in some \(\theta _i\), so \(\theta _i\) is the formula \((s\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(s)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))\) for some term \(s\in [t]\). As an axiom of our logic, we have \(s\equiv t \rightarrow (\theta _i [s/s]\leftrightarrow \theta _i[t/s])\), which is the same as \(s\equiv t\rightarrow (\theta _i\leftrightarrow \theta _{n+1})\). Since \(s\in [t]\), \(s\equiv t\) is also an axiom of our logic, which with the previous step implies that \(\theta _i\leftrightarrow \theta _{n+1}\) is a theorem of our logic. But then from the inconsistency of \(\{\theta _1,\dots ,\theta _{n+1}\}\), its proper subset \(\{\theta _1,\dots ,\theta _{n+1}\}\setminus \{\theta _i\}\) is also inconsistent, contradicting our assumption that no proper subset is inconsistent. Thus, we conclude that \(c_{x,y,[t]}\) does not appear in \(\theta _1,\dots ,\theta _n\).

On the one hand, since \(\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow ((t\sqsubseteq c_{x,y,[t]})\rightarrow \widehat{D}_{x>y}(c_{x,y,[t]}))\), and \(c_{x,y,[t]}\) does not appear in \(\varphi \wedge \theta _1\wedge \dots \wedge \theta _n\) or *t*, it follows by the \(\overline{D}\)-generalization rule (Sect. 7.2) that

On the other hand, since

and \(c_{x,y,[t]}\) does not appear in \(\varphi \wedge \theta _1\wedge \dots \wedge \theta _n\wedge \overline{D}_{x>y}(t)\), we can apply the substitution rule to replace \(c_{x,y,[t]}\) in (2) by 1 to obtain \(\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n \wedge \overline{D}_{x>y}(t))\rightarrow \lnot (t\sqsubseteq 1)\). But then since \(t\sqsubseteq 1\) is a theorem of our logic, by propositional logic we obtain \(\vdash \lnot (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n \wedge \overline{D}_{x>y}(t))\) and then \(\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow \lnot \overline{D}_{x>y}(t)\). It follows, given (1), that \(\{\varphi ,\theta _1,\dots ,\theta _n\}\) is inconsistent, contradicting the fact that no proper subset of \(\{\varphi ,\theta _1,\dots ,\theta _{n+1}\}\) is inconsistent. Thus, we conclude that \(\{\varphi \}\cup \Theta \) is consistent.

We extend the consistent set \(\{\varphi \}\cup \Theta \) to a maximally consistent set \(\Gamma \) using a standard construction. Fixing an enumeration \(\varphi _0,\varphi _1,\dots \) of the formulas of our language, we define:

It is then a standard exercise to verify that \(\Gamma \) is a maximally consistent set.

For the next step of the proof, let \(T_2\) be the set of all terms (including 0 and 1) generated by the coalition labels from the finite set

Define an equivalence relation \(E_\Gamma \) on \(T_2\) by: \(sE_\Gamma t\) if and only if \(s\equiv t\in \Gamma \). Then since \(\Gamma \), as a maximally consistent set, contains all valid equations of Boolean algebra, it follows that the quotient \(T_2/E_\Gamma \) is a Boolean algebra in which the complement of [*t*] is \([-t]\), and the meet of [*s*] and [*t*] is \([s\sqcap t]\). Let \(\le \) be the order of this Boolean algebra, so \([s]\le [t]\) iff \([s]\sqcap [t]=[s]\). By the same argument used above to show that \(T_1/E\) is finite, we have that \(T_2/E_\Gamma \) is finite.

Since \(T_2/E_\Gamma \) is a finite Boolean algebra (with at least two elements, since \(\lnot (0\equiv 1)\) is an axiom), it is isomorphic to the powerset of a (nonempty) set *V*. Let *f* be the isomorphism sending elements of \(T_2/E_\Gamma \) to subsets of *V*. We take the set *X* of alternatives to be the set \(\mathsf {Alt}\) of alternative labels. We then define \(\widehat{D}:X^2\rightarrow \wp (\wp (V))\) by:

This is well defined, i.e., the choice of representative from [*t*] does not matter, because if \(\widehat{D}_{x>y}(s)\in \Gamma \) and \(s\equiv t\in \Gamma \), then it is easy to see using the maximal consistency of \(\Gamma \) and the principles of our logic that \(\widehat{D}_{x>y}(s)\leftrightarrow \widehat{D}_{x>y}(t)\in \Gamma \) and hence \(\widehat{D}_{x>y}(s)\in \Gamma \) if and only if \(\widehat{D}_{x>y}(t)\in \Gamma \). With this definition of *D*, the fact that \(\Gamma \) contains the axioms of \(\overline{\widehat{\mathbf {T}}}\) implies that *D* satisfies the properties of Theorem 7.3 for TR (resp. the fact that \(\Gamma \) contains the axioms of \(\overline{\widehat{\mathbf {W}}}\) implies that *D* satisfies the properties of Theorem 7.3 for FR).

Next we define \(\overline{D}:X^2\rightarrow \wp (\wp (V))\) by \(f([t])\in \overline{D}(x,y)\) if and only if \(\overline{D}_{x>y}(t)\in \Gamma \). Again this is well-defined, for the same reasons as before. The fact that \(\Gamma \) contains the axiom \((\overline{D}_{x>y}(a)\wedge (a\sqsubseteq b))\rightarrow \widehat{D}_{x>y}(b)\) implies that \(\langle \widehat{D},\overline{D}\rangle \) satisfies the left-to-right direction of property 6 of Theorem 7.3: if \(A\in \overline{D}(x,y)\), then for all \(B\supseteq A\), we have \(B\in \widehat{D}(x,y)\). Proving the converse implication was the reason for our introduction of the set \(\Theta \) of formulas. Suppose \(f([t])\not \in \overline{D}(x,y)\), so \(\overline{D}_{x>y}(t)\not \in \Gamma \) and hence \(\lnot \overline{D}_{x>y}(t)\in \Gamma \) by the maximal consistency of \(\Gamma \). Then given that \((t\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))\in \Gamma \), from \(\lnot \overline{D}_{x>y}(t)\in \Gamma \) we have \(\lnot \widehat{D}_{x>y}(c_{x,y,[t]})\in \Gamma \) by the maximal consistency of \(\Gamma \), whence \(\widehat{D}_{x>y}(c_{x,y,[t]})\not \in \Gamma \) by the consistency of \(\Gamma \). Thus, \(f([c_{x,y,[t]}])\not \in \widehat{D}(x,y)\). Since \(t\sqsubseteq c_{x,y,[t]}\in \Gamma \), we also have \([t]\le [c_{x,y,[t]}]\) and hence \(f([t])\subseteq f([c_{x,y,[t]}])\), which completes the proof of property 6.

Since \(\langle \widehat{D},\overline{D}\rangle \) satisfies the properties of Theorem 7.3 for TR (resp. FR), it is representable by a CCR satisfying LD, IIA, and TR (resp. FR). Let \(\alpha \) be the coalition labeling that sends each coalition label *a* to the image \(f([a])\subseteq V\) of the equivalence class of [*a*] in \(T_2/E_\Gamma \). Since the set of terms was defined inductively, one can prove by induction that the extended labeling \(\dot{\alpha }\) from Definition 4.2 satisfies

Let \(\beta \) be the alternative labeling that sends each alternative label *x* to itself, recalling from above that we take the set *X* of alternatives to be \(\mathsf {Alt}\).

We claim that for any formula \(\psi \), we have:

The set of formulas was defined inductively, so we prove the claim by induction. For formulas of the form \(t\equiv s\), we have the following equivalences:

For formulas of the form \(\widehat{D}_{x>y}(t)\), we have:

The proof for formulas of the form \(\overline{D}_{x>y}(t)\) is analogous. For formulas of the form \(\lnot \varphi \):

The proofs for formulas of the form \(\psi \wedge \chi \), \(\psi \vee \chi \), etc., also use the maximal consistency of \(\Gamma \), which ensures that \(\psi \wedge \chi \in \Gamma \) if and only if \(\psi ,\chi \in \Gamma \); \(\psi \vee \chi \in \Gamma \) if and only if \(\psi \in \Gamma \) or \(\chi \in \Gamma \); etc. Thus, (5) holds. Then since our initial consistent formula \(\varphi \) belongs to \(\Gamma \), we have \(f\models _{\alpha ,\beta } \varphi \). This completes the proof that every \(\overline{\widehat{\mathbf {T}}}\)-consistent (resp. every \(\overline{\widehat{\mathbf {W}}}\)-consistent) formula is true of some CCR satisfying LD, IIA, and TR (resp. FR).

The proof of Theorem or 4.8.2 or 4.8.4 is even simpler. In this case, without the interaction between \(\widehat{D}\) and \(\overline{D}\) to worry about, we can skip the step involving the new coalition labels \(c_{x,y,[t]}\) and the set \(\Theta \) of new formulas. Simply extend \(\varphi \) to a maximally consistent set \(\Gamma \) and continue with the rest of the proof as above. At the point where the representation result from Theorem 7.3 (for decisiveness regimes) was used, use the representation result from Theorem 3.5 (for almost decisiveness) or Theorem 3.6 (for decisiveness) instead.

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Holliday, W.H., Pacuit, E. Arrow’s decisive coalitions.
*Soc Choice Welf* **54, **463–505 (2020). https://doi.org/10.1007/s00355-018-1163-z

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