Democratic theory offers many substantive accounts of what democracy requires, yet few minimal, axiomatic characterizations of a “democratic procedure” as such. Existing models typically either analyse specific voting or aggregation rules within the framework of social choice theory, or articulate richer normative conceptions of democracy in political philosophy. Both strands are essential, but neither by itself isolates a bare procedural core of democracy in a simple mathematical form.
The central question addressed in this paper is: what is the weakest set of structural and individual-level conditions under which a collective decision process can be meaningfully described as democratic, purely in virtue of its procedural basis? The proposed answer abstracts from the content of decisions and from institutional detail, and focuses on three ingredients: a finite set of individuals, the expressions they produce in response to a given question, and the rule that maps a profile of expressions into a collective outcome.
A procedure is treated as democratic when two types of requirement are met: (1) a requirement on access to expression and the structure of the decision rule (each individual can express a judgement, and the rule depends only on how many times each admissible expression occurs), and (2) a requirement on the status of those expressions (they reflect individuals’ internal judgements, formed and expressed without coercion). This section situates the proposal within the existing literature.
A second body of work comes from political philosophy and democratic theory, where democracy is interpreted through richer normative lenses. Many accounts identify democracy with a form of collective self-rule among equals, emphasizing political equality (each person’s input has equal standing), autonomy or self-authorship (citizens are authors of the laws that bind them), and the illegitimacy of coercion or domination that undermines this autonomy (Christiano, 1996, 2008; Kolodny, 2014a, 2014b; Dahl, 1971, 1989).
Procedural accounts stress equal rights of participation, effective opportunities to influence outcomes, and the principle that collective decisions must be justifiable to those subject to them. Dahl’s (1971) concept of polyarchy identifies a set of institutional requirements, including freedom of expression, freedom of association, right to vote, eligibility for public office, and competitive elections, as procedural conditions for democratic governance. Christiano (1996, 2008) defends democracy on the basis that equal political participation publicly signals equal regard for each person’s interests, a requirement of distributive justice. Kolodny (2014a, 2014b) argues that social equality requires the absence of asymmetries in power and de facto authority, particularly in political contexts characterized by nonvoluntary subjection, final authority, and the use of force.
Epistemic accounts defend democracy as an effective method for tracking truth or justice, again under conditions that include freedom of expression and non-manipulated deliberation (List & Goodin, 2001; Landemore, 2013). Deliberative theories treat democracy as a communicative practice oriented towards mutual justification, explicitly opposing forms of coercion, manipulation, and deception that would corrupt the autonomy and moral equality of participants (Cohen, 1989; Gutmann & Thompson, 1996).
From this perspective, coercion and autonomy are not peripheral but central: being subjected to a rule without an opportunity to form and express one’s own judgement is paradigmatically non-democratic. The “coercion principle” in discussions of the boundary of the demos holds that those subject to systematic coercive power have a prima facie claim to participatory rights in the processes that determine that power (Song, 2012). Theories of freedom of speech connect the legitimacy of public discourse to the autonomy of individual speakers and the absence of coercive distortion of their expressed views (Scanlon, 1972; Cohen, 1993).
These contributions capture conceptual insights essential for any plausible understanding of democracy: that equal standing requires more than formal inclusion; that expression must be genuinely connected to internal judgement; and that coercion, domination, and manipulation undermine democratic legitimacy even when formal aggregative procedures are in place. What they largely lack, however, is a minimal, formally explicit account in which these ideas are distilled into a small set of axioms on a simple mathematical object, that is, a decision procedure mapping profiles of individual expressions into outcomes.
The present framework borrows its normative intuitions from this tradition, particularly from accounts that tie democracy to autonomy, moral equality of voice, and the rejection of coercive domination, but recasts them at a very high level of abstraction. The aim is not to model deliberation, justification, or public reasoning, but to capture, in the most economical way, the idea that a democratic procedure must allow each individual to form and freely express a judgement, and must treat only the multiplicity of those expressions, not the identities of the individuals, as procedurally relevant.
Within this landscape, the present paper offers a minimal axiomatic definition of democratic procedure. The key moves are: to treat a decision procedure as a function from profiles of expressions to an abstract set of rules or collective outcomes; to require universal expression, in the sense that each member of the relevant set of individuals has the possibility to produce an admissible expression (including possibly a symbol for non-expression); to require dependence on the count of expressions only, so that procedures that differ merely by permuting individuals but preserve counts are identified; and to introduce a simple distinction between internal judgements and public expressions, and impose two minimal freedom conditions, namely that each individual forms an internal judgement (autonomy of judgement), and that the public expression coincides with that judgement (absence of coercion).
These conditions deliver a notion of democratic procedure that is procedural (it does not appeal to the substantive correctness of outcomes), minimal (it imposes no structure on the space of rules and no specific form of majority, representation, or deliberation), compatible with existing social choice models (anonymity and related symmetry conditions can be seen as refinements of the basic count-dependence condition), and sensitive to freedom conditions (it explicitly encodes that what is being counted are free expressions of internal judgements).
Section 2 develops this framework in detail. It introduces the formal objects (individuals, expressions, decision procedures), defines the count function, and states the basic definition of democratic procedure in purely procedural terms. It then enriches the model with internal judgements and the two individual freedom conditions (autonomy of judgement and absence of coercion), leading to an augmented definition of democratic procedure with freedom conditions.
After reviewing related axiomatic approaches, we now fix notation and state a minimal definition.
Let \(A=\{a_1,\dots,a_n\}\) be a finite, nonempty set of human beings. Let \(V\) be the set of admissible expressions for a given question. We optionally allow a distinguished symbol \(\bot\) to represent non-expression.
An expression profile is a function \[ P: A \to V \cup \{\bot\}, \] where \(P(a)\) denotes the expression produced by individual \(a\).
Let \(D\) be an abstract set of rules. A decision procedure is a function \[ \Phi: (V\cup\{\bot\})^{A}\to D, \] which maps each expression profile into a rule.
Definition 1 (count). For each profile \(P\), define the count function \(C(P)\) by \[ C(P)(v)=\left|\{a\in A: P(a)=v\}\right| \quad \text{for each } v\in V\cup\{\bot\}. \] The object \(C(P)\) records the multiplicity of each admissible expression.
The present approach treats a process as democratic when two requirements hold. The first requirement concerns access to expression. The second concerns the operational basis of the collective outcome.
Definition 2 (democratic procedure, basic form). A decision procedure \(\Phi\) is democratic if both conditions hold:
universal expression: the profile \(P\) is defined on the whole set \(A\), so each \(a\in A\) can express an element of \(V\cup\{\bot\}\);
dependence on the count: for all profiles \(P\) and \(Q\), \[ C(P)=C(Q) \ \Rightarrow\ \Phi(P)=\Phi(Q). \] Condition (ii) states that the rule depends on the count of individual expressions, and not on the identity of those who expressed them.
Condition (ii) is equivalent to requiring the existence of a function \(\psi\) such that \[ \Phi=\psi\circ C. \] In this case, the procedure factors through the count function.
The basic definition above is not sufficient when expressions may fail to be free. We therefore distinguish (i) what an individual expresses, (ii) what the individual actually judges, and (iii) what the individual would judge under autonomy.
Fix a decision context \(s\). Define \(V_\bot := V \cup \{\bot\}\).
An expressed profile in context \(s\) is a function \[ P_s: A \to V_\bot, \] where \(P_s(a)\) is the expression produced by individual \(a\) in context \(s\).
An internal judgement profile in context \(s\) is a function \[ J_s: A \to V_\bot, \] where \(J_s(a)\) is the judgement effectively held by individual \(a\) in context \(s\).
An autonomous judgement profile in context \(s\) is a function \[ J_s^\ast: A \to V_\bot, \] where \(J_s^*(a)\) denotes the judgement that the same individual would hold in context \(s\) under autonomy of judgement.
For a given individual \(i\in A\), we may write \(p:=P_s(i)\), \(j:=J_s(i)\), and \(j^*:=J_s^*(i)\).
Condition 1 (autonomy of judgement, contextual). For each \(a\in A\), \[ J_s(a)=J_s^*(a). \]
Condition 2 (freedom of expression of judgement, contextual). For each \(a\in A\), \[ P_s(a)=J_s(a). \]
These two equalities induce four logical configurations at the individual level: \(p=j\) or \(p\neq j\) combined with \(j=j^*\) or \(j\neq j^*\).
Definition 3 (democratic procedure, with freedom conditions). A decision procedure \(\Phi\) is democratic in context \(s\) if it satisfies Definition 2 for the expressed profile \(P_s\) and, in addition, Conditions 1 and 2 hold for all \(a\in A\).
Condition (i) in Definition 2 requires that the expression profile \(P\) be defined on the whole set \(A\), so that each \(a \in A\) can produce an element of \(V \cup \{\bot\}\). This condition is the formal counterpart of a minimal requirement of inclusive participation: every individual in the relevant population has the possibility to take part in the procedure through an admissible expression.
In the present framework, universal expression has three immediate implications.
First, the requirement concerns access, not actual participation. The symbol \(\bot\) can be interpreted as non‑expression or abstention. A profile in which some individuals are mapped to \(\bot\) is still admissible, provided that for each \(a \in A\) the expression \(P(a)\) is defined. Thus, the model distinguishes lack of access (which would violate condition (i)) from voluntary non‑expression within a context where access is guaranteed.
Second, universal expression is formulated relative to a fixed set \(A\). The condition states that, given a specified set of individuals, each member of that set has the possibility to express a judgement in the procedure. The problem of how the set \(A\) itself is determined, that is, the boundary of the demos, is not addressed at this level of abstraction. The present definition is therefore compatible with different answers to the boundary problem, but requires, for any given answer, that no member of the resulting population be excluded from expression.
Third, universal expression is logically independent of the aggregation rule. The condition does not constrain how expressions are translated into outcomes, only that the decision procedure takes as input a profile defined on all of \(A\). Rules that ignore some subset of individuals, either by construction or through a structural veto on their expressions, violate condition (i). In this sense, universal expression rules out procedures that are formally aggregative but de facto restricted to a proper subset of the population.
These observations motivate viewing condition (i) as a necessary procedural requirement for democracy in the present sense. A process in which some individuals lack the possibility to express a judgement about the decision, even though they belong to the relevant set \(A\), falls short of the minimal standard. Conversely, universal expression does not, by itself, secure any form of equality in the treatment of those expressions, nor any guarantee that expressions are free, non‑coerced manifestations of internal judgements. Those further requirements are captured by the remaining conditions of the model.
Condition (ii) in Definition 2 requires that, for all profiles \(P\) and \(Q\), \[ C(P) = C(Q) \Rightarrow \Phi(P) = \Phi(Q). \] That is, the decision procedure depends only on the count of individual expressions, and not on the identity of those who expressed them. Equivalently, there exists a function \(\psi\) such that \(\Phi = \psi \circ C\). The procedure thus factors through the count function.
In the present framework, this requirement plays the role of a minimal symmetry condition on the treatment of individuals. Two features are worth emphasizing.
First, dependence on the count can be read as a very simple form of anonymity. If two profiles differ only by a permutation of individuals in \(A\), then the corresponding count functions coincide, and therefore the collective outcome is the same. Individual labels are irrelevant, only the multiplicity of each admissible expression in \(V \cup \{\bot\}\) matters. The model does not specify which particular count patterns are mapped to which rules in \(D\), it only requires that any such mapping be invariant under permutations of individuals.
Second, the condition is formulated at the level of expressions, not preferences or utilities. The count function \(C(P)\) records how many times each admissible expression occurs in the profile, including the symbol \(\bot\). For example, in a binary choice with \(V = \{\text{yes}, \text{no}\}\), the count specifies the number of “yes”, the number of “no”, and the number of non‑expressions. A procedure may then be majority based, quota based, or follow any other rule that can be defined as a function of these counts. Procedures that attach different weights to different individuals, holding the counts fixed, are excluded by condition (ii).
This requirement has several immediate consequences.
First, it rules out fixed‑weight schemes where some individuals’ expressions systematically carry more influence than others, for example through weights that are not encoded in the set of admissible expressions but attached directly to individuals. Such schemes can be represented as functions \(\Phi\) that cannot be written as \(\psi \circ C\), since profiles with identical counts but different assignments of expressions to individuals may lead to different outcomes.
Second, it distinguishes procedural equality of input from equality of opportunity to express. Universal expression ensures that every individual can produce an expression. Dependence on the count adds that, once expressed, all expressions of the same type are treated symmetrically, regardless of who produced them. The combination of the two conditions yields a weak but nontrivial sense in which each individual has the same formal standing in the procedure.
Third, the condition is silent on the content of \(D\) and on the interpretive meaning of the rules. The same count pattern may correspond to very different substantive outcomes, depending on the choice of \(\psi\). The present model does not privilege any particular democratic decision rule, such as simple majority or proportional representation, but requires that any admissible rule be implementable as a function of counts alone.
In this sense, dependence on the count is not a sufficient condition for democracy, but it is taken as a necessary structural feature of any procedure that qualifies as democratic in the present minimal sense. It captures the idea that, given universal expression, the procedure must treat like expressions alike, and must not encode inequalities between individuals at the level of the mapping from profiles to outcomes.
In the present framework, autonomy of judgement concerns the formation of the internal judgement, not its public expression. We therefore distinguish the internal judgement profile in context \(s\), \[ J_s: A \to V_\bot, \] from the autonomous judgement profile, \[ J_s^\ast: A \to V_\bot, \] where \(J_s^\ast(a)\) denotes the judgement that the same individual would hold in context \(s\) under autonomy of judgement.
Condition 1 (autonomy of judgement, contextual). Autonomy holds in context \(s\) if, for each \(a\in A\), \[ J_s(a)=J_s^\ast(a). \]
For a given individual \(i\in A\), letting \(j:=J_s(i)\) and \(j^\ast:=J_s^\ast(i)\), Condition 1 is equivalent to \(j=j^\ast\). Autonomy may fail even when the individual is free to express what is judged. In that case, the expressed response can still coincide with the internal judgement, while the internal judgement differs from the autonomous one.
In other words, within the model, each individual is represented as having both an effective judgement \(J_s(a)\), formed in the actual context, and a corresponding autonomous judgement \(J_s^*(a)\), and autonomy of judgement obtains when these coincide.
This condition formalizes a minimal requirement of individual autonomy at the level of judgement formation. The model does not describe the processes by which \(J_s(a)\) and \(J_s^*(a)\) are formed, nor does it specify the substantive features of the autonomous benchmark. It only requires that, for the purposes of the procedure, the effective internal stance of each individual in context \(s\) match the stance that the same individual would hold under autonomy in that context.
Two aspects of this condition are central. First, autonomy of judgement is defined contextually. The profiles \(J_s\) and \(J_s^*\) are indexed by the decision context \(s\), and are part of the model for that context, not introduced a posteriori to match observed expressions. Second, autonomy of judgement is conceptually prior to expression. The internal profiles \(J_s\) and \(J_s^*\) exist independently of the expressed profile \(P_s\). The relationship between internal and expressed judgements is specified by Condition 2. This separation allows the model to distinguish, at a minimal level, between (i) the autonomy of the judgements that individuals hold in a given context and (ii) the freedom with which those judgements are expressed.
Autonomy of judgement, in this sense, is a necessary background condition for treating a procedure as democratic in the present framework. If, in a given context, some individuals’ effective judgements systematically diverge from their autonomous judgements, the idea that they participate as autonomous agents in that context would be undermined, even if their expressions perfectly match their effective judgements.
Condition 2 requires that, in each decision context \(s\), the expressed profile \(P_s\) coincide with the internal judgement profile \(J_s\), that is, \[ \forall a\in A,\ P_s(a) = J_s(a). \] This condition states that, for every individual, the expression recorded by the procedure in context \(s\) is exactly the internal judgement that the individual holds in that context. In the terminology of Section 2.3, it can be described as freedom of expression of judgement in context \(s\); equivalently, it can be viewed as a minimal condition of absence of coercion at the level of expression.
Within the present framework, absence of coercion is interpreted in a minimal, procedural sense. The model does not attempt to classify or explain different mechanisms of influence, such as persuasion, manipulation, or structural pressure. It introduces only a formal criterion for when a procedure treats expressions as free: expressions count as free when they match the internal judgements that individuals are assumed to have formed autonomously in the relevant context.
Two points follow immediately.
First, the condition defines coercion relative to the discrepancy between \(J_s\) and \(P_s\). Whenever an individual is represented as expressing something different from her internal judgement, the equality \(P_s(a) = J_s(a)\) fails, and the procedure is not democratic in context \(s\) according to Definition 3. This includes cases in which an individual is induced to express a view she does not hold, as well as cases in which an individual is prevented from expressing a view she does hold, for instance by being forced to express \(\bot\) instead of a substantive element of \(V\).
Second, the requirement is context sensitive. A given decision procedure \(\Phi\) may satisfy Definition 2 at the level of expressions, yet fail to be democratic in some contexts because the equality between \(P_s\) and \(J_s\) does not hold. Definition 3 therefore classifies democracy not only in terms of the structural features of the rule, but also in terms of the match between internal and expressed profiles in the context in which the rule is applied.
Absence of coercion, in this sense, links the formal apparatus to the normative intuition that democratic procedures must not merely collect expressions, but must collect them as genuine manifestations of individuals’ own judgements. Combined with autonomy of judgement, the condition ensures that what is being counted by the procedure are not arbitrary signals, but the internal stances that individuals hold in the given context. Without this requirement, a procedure could satisfy universal expression and dependence on the count, yet operate entirely on coerced or otherwise distorted expressions, and it would still qualify as democratic in the purely structural sense of Definition 2. Definition 3 rules out this possibility by making absence of coercion an explicit component of the minimal democratic standard.
This framework separates two failures that are often conflated in discussions of democratic legitimacy. The first concerns the relation between an individual’s effective judgement and the individual’s public expression. The second concerns the relation between an individual’s effective judgement and the individual’s autonomous baseline.
Fix a context \(s\) and an individual \(i\in A\). Let \(p := P_s(i)\) denote what the individual expresses. Let \(j := J_s(i)\) denote what the individual effectively judges. Let \(j^* := J_s^*(i)\) denote what the same individual would judge under autonomy in the same context.
Freedom of expression of judgement holds when \(p=j\). It fails when \(p\neq j\). In this case the individual’s expression is not a faithful report of the individual’s effective judgement. The defining feature is a distortion at the level of expression.
Autonomy of judgement holds when \(j=j^*\). It fails when \(j\neq j^*\). In this case the individual’s effective judgement does not match the individual’s autonomous baseline. The defining feature is a distortion at the level of judgement formation.
The distinction is structural. Coercion targets the channel from judgement to expression. Lack of autonomy targets the channel from autonomous judgement to effective judgement. The two failures can occur independently. They can also co-occur in the same context.
The four logical configurations are immediate:
Both conditions hold: \(p=j\) and \(j=j^*\). The individual expresses an autonomous judgement without distortion. This is the case captured by Definition 3.
Freedom fails, autonomy holds: \(p\neq j\) and \(j=j^*\). The individual forms an autonomous judgement but is prevented from expressing it faithfully.
Autonomy fails, freedom holds: \(p=j\) and \(j\neq j^*\). The individual freely expresses an effective judgement that does not correspond to the autonomous baseline.
Both conditions fail: \(p\neq j\) and \(j\neq j^*\). The judgement is not autonomous, and the expression does not match even the non-autonomous judgement.
This separation matters because the two failures call for different remedies. Freedom of expression concerns the conditions under which an individual can state what is effectively judged without incurring penalties. Autonomy of judgement concerns the conditions under which the judgement itself is formed without impairing influences.
Coercion is defined by its effect on the equality \(p=j\). It is any external influence that makes an individual express something other than what is effectively judged. The model does not require a specific psychological mechanism. It only requires that the expression diverge from the effective judgement.
Coercion includes direct threats and sanctions. It includes credible risks of retaliation tied to identifiable expressions. It includes institutional constraints on the act of expressing a judgement. It includes incentives that make expression costly unless it matches an imposed signal.
Coercion can therefore take different forms. It can be legal, economic, social, or reputational. It can be explicit or implicit. It can operate through surveillance and traceability. It can operate through control of the communication channel. The common feature is the production of \(p\neq j\).
In this sense, coercion is compatible with autonomy. An individual may form an autonomous judgement, \(j=j^*\), and yet be forced to express a different response, \(p\neq j\).
Example: An individual may privately support a policy proposal (\(j = v_1\)) but publicly express opposition (\(p = v_2\), where \(v_1, v_2 \in V\) and \(v_1 \neq v_2\)) because of credible threats to employment, social standing, or physical safety. If the individual’s private judgement matches what would be judged under autonomy (\(j = j^*\)), then autonomy holds but freedom of expression fails. In such cases the public profile is not informative about the distribution of effective judgements, even if those judgements were autonomously formed.
Lack of autonomy is defined by its effect on the equality \(j=j^*\). It is any influence that alters the formation of an effective judgement relative to the autonomous baseline.
Such influences can include deception and systematic distortion of relevant information. They can include covert manipulation designed to bypass reflective control. They can include stable processes of indoctrination that shape what is judged without meaningful revision. They can include structural conditions that narrow what is thinkable by attaching pervasive costs to certain beliefs. They can include cognitive overload or urgency that prevents reflective formation of judgement.
In all these cases, the individual may still be free to express what is effectively judged. The equality \(p=j\) may hold. The failure concerns the formation of judgement, not its expression.
Example: An individual subjected to sustained propaganda may effectively judge that a proposed measure is desirable (\(j = v_1\)) and freely express this judgement (\(p = j\)), yet would have judged differently (\(j^* = v_2\), where \(v_1 \neq v_2\)) under conditions allowing access to diverse information and reflective deliberation. Here freedom of expression holds (\(p = j\)) but autonomy of judgement fails (\(j \neq j^*\)). In such cases the public profile faithfully reflects the distribution of effective judgements, but those judgements are not autonomous in the sense captured by \(j=j^*\).
The present definition treats autonomy and freedom of expression as jointly necessary conditions. Each condition targets a different step of the individual contribution to the collective procedure. For this reason, neither condition should be reduced to the other.
If coercion is treated as a general label for all forms of influence, then the distinction collapses. The model becomes unable to separate failures of expression from failures of judgement formation. The proposed notation prevents this collapse. It forces the analyst to state whether the problem is \(p\neq j\), \(j\neq j^*\), or both.
This separation also clarifies how a democratic deficit can persist even when participation is formally universal and aggregation is count-based. A procedure can satisfy universal expression and dependence on the count, yet fail to be democratic because expressions do not track effective judgements (Condition 2 fails), or because effective judgements do not track autonomous ones (Condition 1 fails), or both. Under these failures, the procedure cannot be democratic in the relevant context, even if it remains a count-based aggregation of expressions over the whole set \(A\).
The present framework thus identifies two distinct loci of democratic failure, each of which must be addressed for a procedure to qualify as democratic in a given context.
The minimal definition proposed in this paper draws on, departs from, and extends existing work in social choice theory and democratic theory. This section clarifies these connections.
Dependence on the count and anonymity in social choice. Condition (ii) in Definition 2 states that the decision procedure depends only on the count of expressions, that is, \(\Phi = \psi \circ C\). This requirement is closely related to the principle of anonymity in social choice theory. Anonymity is standardly defined as permutation invariance: an aggregation rule is anonymous if, whenever two profiles differ only by a permutation of individuals, the outcome is the same (May, 1952; Maskin, 1995). In May’s theorem, anonymity is one of three axioms that jointly characterize simple majority rule for binary decisions, together with neutrality (symmetry between alternatives) and positive responsiveness (monotonicity) (May, 1952).
The relationship between dependence on the count and anonymity can be stated precisely. Any rule that factors through the count function is anonymous, since permuting individuals leaves the count unchanged. Conversely, for any anonymous rule defined on a finite set of individuals and a finite set of admissible expressions, there exists a function \(\psi\) such that the rule can be written as \(\psi \circ C\). Thus, on finite domains, dependence on the count and anonymity are equivalent.
The difference lies in formulation and emphasis. In the social choice literature, anonymity is imposed as a structural property of the aggregation rule over preference profiles. The present framework treats the count function \(C\) as a primitive object, and defines a democratic procedure as one that factors through \(C\). This makes explicit that what matters for the outcome is the multiplicity of each expression, not the identity of those who expressed it. The count-based formulation is therefore not a generalization of anonymity, but a reformulation that foregrounds the role of the count as the relevant informational basis for the decision.
Universal expression and minimal procedural conditions. Condition (i) in Definition 2 requires that each member of the set \(A\) can express an element of \(V \cup \{\bot\}\). This requirement has counterparts in procedural accounts of democracy, particularly in Dahl’s characterization of polyarchy (Dahl, 1971). Dahl identifies a set of institutional requirements for democratic governance, including inclusive suffrage, the right to run for office, freedom of expression, and free and fair elections. Universal expression, in the present sense, corresponds most closely to inclusive suffrage: the idea that no member of the relevant population is excluded from participation.
However, the present condition is formulated at a higher level of abstraction. Dahl’s account specifies institutional features of electoral democracies, such as competitive elections and alternative sources of information. The present model does not presuppose elections, representation, or any particular institutional form. It only requires that, given a fixed set \(A\), each individual can produce an admissible expression. In this respect, the framework is more minimal than Dahl’s procedural account, and compatible with a wider range of decision contexts, including referenda, deliberative assemblies, or direct decision procedures.
A similar comparison holds for the so-called minimalist conception of democracy, which defines democracy solely in terms of competitive elections (Przeworski, 1999; Schumpeter, 1942). The present model does not require competition, representation, or iterability of the decision. It isolates a lower bound: a procedure cannot be democratic if some individuals lack the possibility to express a judgement, but universal expression alone does not guarantee any particular form of democratic governance.
Autonomy, coercion, and normative democratic theory. Conditions 1 and 2 introduce a distinction between internal judgements and expressed profiles, and require that expressions coincide with internal judgements in a given context.A terminological clarification is needed. In the present framework, coercion is reserved for distortions of expression relative to effective judgement, that is cases in which \(P_s(a)\neq J_s(a)\). Deception, manipulation, and related influences typically operate on judgement formation and are therefore treated as failures of autonomy, that is cases in which \(J_s(a)\neq J_s^\ast(a)\), even when expression remains free (\(P_s(a)=J_s(a)\)). These conditions draw on normative accounts that link democratic legitimacy to autonomy and the absence of coercion (Christiano, 1996, 2008; Kolodny, 2014a, 2014b; Scanlon, 1972; Cohen, 1993).
In particular, theories of freedom of expression argue that legitimate collective decisions must be based on expressions that reflect individuals’ own judgements, not expressions that have been coerced, manipulated, or otherwise distorted by external forces (Cohen, 1993; Scanlon, 1972). The “coercion principle” in discussions of the boundary problem holds that those subject to coercive power have a claim to participatory rights in the processes that determine that power (Song, 2012). Deliberative theories stress that democratic procedures must allow individuals to form and express judgements autonomously, free from domination or manipulation (Gutmann & Thompson, 1996).
The present framework formalizes these ideas in a minimal way. The model does not describe mechanisms of coercion, does not distinguish persuasion from manipulation, and does not impose substantive rationality constraints on judgements. It only requires that, in a given context \(s\), autonomy of judgement hold, in the sense that \(J_s(a)=J_s^\ast(a)\) for each \(a\in A\), and that freedom of expression of judgement hold, in the sense that \(P_s(a)=J_s(a)\) for each \(a\in A\).
This yields a necessary condition for democracy: if expressions are coerced, in the sense that \(P_s(a) \neq J_s(a)\) for some \(a\), then the procedure is not democratic in that context, even if it satisfies universal expression and dependence on the count.
What is taken from established theories. The present model adopts three core ideas from the existing literature. First, from social choice theory, it adopts the idea that a democratic procedure must treat individuals symmetrically, in the sense captured by anonymity or, equivalently, by dependence on the count. Second, from procedural accounts of democracy, it adopts the idea that every member of the relevant population must have access to the decision procedure. Third, from normative democratic theory, it adopts the idea that democratic legitimacy requires that expressions be free, in the sense that they reflect individuals’ own judgements rather than externally imposed signals.
What departs from established theories. The present framework departs from existing work in two main respects. First, it does not presuppose any institutional structure, such as elections, representation, or deliberation. The model is compatible with these forms, but does not require them. In this sense, it is more minimal than standard procedural accounts, which define democracy in terms of specific institutional arrangements. Second, the model does not specify any particular decision rule. It does not privilege majority rule, supermajority rule, or any other aggregation scheme. It only requires that, whatever rule is used, it depend only on the count of expressions, and that those expressions be free.
What is new relative to established theories. Three features of the present framework are, to our knowledge, novel. First, the explicit introduction of the count function \(C\) as a primitive object, through which the decision procedure must factor. While the idea that democratic procedures depend only on counts is implicit in the anonymity condition, making the count function explicit allows a clearer statement of what information is procedurally relevant. Second, the contextual distinction between internal judgements \(J_s\) and expressed profiles \(P_s\), formulated at the same level of abstraction as the decision rule itself. This allows the model to state a minimal freedom condition without appealing to institutional or behavioral detail. Third, the integration of structural conditions (universal expression, dependence on the count) and freedom conditions (autonomy of judgement, absence of coercion) within a single axiomatic definition. Most existing work treats these two aspects separately: social choice theory focuses on structural properties of aggregation rules, while normative democratic theory focuses on freedom and autonomy. The present model combines both, and treats all four conditions as jointly necessary for a procedure to qualify as democratic in a given context.
This paper has proposed a minimal axiomatic definition of democratic procedure, built on four conditions: universal expression, dependence on the count, autonomy of judgement, and absence of coercion. The framework integrates structural requirements, drawn from social choice theory, with freedom requirements, drawn from normative democratic theory, and treats both as jointly necessary for a procedure to qualify as democratic in a given context.
Section 8 clarified how the present model relates to existing work. The count-based condition reformulates the anonymity principle of social choice theory by making the count function an explicit primitive. Universal expression corresponds to inclusive participation in procedural accounts, but is formulated without presupposing elections, representation, or specific institutional arrangements. The two freedom conditions formalize, in minimal terms, normative intuitions about autonomy and the absence of coercion that are central to deliberative and egalitarian theories of democracy, but are typically not incorporated into axiomatic models of aggregation. Section 7 showed that these two conditions are structurally independent: coercion distorts the channel from judgement to expression, while lack of autonomy distorts judgement formation itself.
Three limitations of the present framework point to directions for further work. First, the model is static. It treats a single decision in a fixed context \(s\), and does not address how judgements are formed, revised, or influenced over time. Extensions that incorporate dynamic aspects, such as learning, deliberation, or sequential decision making, would allow the framework to engage more directly with epistemic and deliberative accounts of democracy.
Second, the model is deterministic. It assumes that internal judgements \(J_s\) and expressed profiles \(P_s\) are fully specified, and that the condition \(P_s(a) = J_s(a)\) either holds or fails for each individual. In many realistic contexts, there may be uncertainty about whether expressions reflect internal judgements, or probabilistic distortions between the two. A probabilistic extension, similar to models of noisy preference revelation or response error in psychometrics, could provide a more flexible framework for assessing degrees of coercion or expressive freedom.
Third, the boundary of the set \(A\) is taken as exogenous. The model does not address the boundary problem, that is, the question of which individuals belong to the relevant population for a given decision. While this abstraction allows the framework to focus on procedural conditions internal to the decision process, it leaves open a foundational question in democratic theory. Future work could explore how boundary principles, such as the all-affected principle or the coercion principle, interact with the four conditions proposed here.
Despite these limitations, the framework offers a formal basis for analysing democratic procedures that separates access and equality of input from freedom of expression, and makes both explicit in a simple mathematical language. By treating the count function as a primitive, and by introducing a contextual distinction between internal judgements, autonomous baselines, and expressed profiles, the model provides tools for clarifying what it means, in procedural terms, for a decision to be democratic. A procedure qualifies as democratic only when what is counted are judgements formed in autonomy and expressed in freedom. It also suggests criteria for assessing how far concrete institutional arrangements approximate, or fall short of, the proposed minimal standard.
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1.1. Social choice theory and axiomatizations of voting rules
Social choice theory studies the aggregation of individual inputs into collective outputs (Arrow, 1951; Sen, 1970). Many authors have sought to characterize particular decision rules via sets of axioms, often interpreted as requirements of fairness, rationality, or procedural equality. A classic result is May’s theorem (May, 1952), which shows that for binary decisions, simple majority rule is the unique aggregation rule satisfying anonymity, neutrality, and positive responsiveness under universal domain assumptions. Related work characterizes other aggregation rules, such as supermajority rules, scoring rules, and utilitarian or other cardinal aggregation schemes, by suitable sets of axioms, often centered on symmetry, monotonicity, or independence conditions (Goodin & List, 2006; List & Pettit, 2011).
Two features of this axiomatic tradition are particularly relevant. First, many axioms encode a weak notion of political equality. Anonymity expresses the idea that what matters for the collective outcome is how many individuals support a given option, not who they are. Neutrality captures an analogous symmetry between options. These conditions are routinely motivated as democratic desiderata, yet they are imposed indirectly, as structural properties of the aggregation rule over preference profiles, rather than as explicit features of a minimal definition of a democratic procedure.
Second, standard models in social choice presuppose, but do not analyse, the formation and expression of individual judgements. Individuals’ preferences or votes are taken as given inputs to the aggregation rule. Issues such as whether each person can in fact express a judgement, whether expression is voluntary, and whether expressed preferences coincide with internal judgements formed without coercion, typically fall outside the formalism. The focus is on the mapping from profiles to outcomes, not on the conditions under which profiles themselves arise.
The social choice literature provides a sophisticated toolkit for studying the properties of aggregation rules and for justifying particular voting procedures as “fair” or “democratic” in a procedural sense (Maskin & Sen, 2017). However, it does not offer a minimal, stand-alone definition of “democratic procedure” that makes access to expression, count-based dependence on individual inputs, and freedom from coercion explicit and primitive components of the model. The present framework is complementary to social choice theory. It retains a highly abstract view of the decision rule and a count-based invariance condition reminiscent of anonymity, but explicitly elevates this and two individual freedom conditions to definitional status.