13.1From questions to propositions
Formalization begins with a question, but a question does not yet determine what agents are being asked to accept. A collective question becomes an agenda of propositions only after its terms are fixed and their logical relations made explicit. Take the question whether to accept both P and Q. For the present profile, the meaning of the combined proposition R is stipulated as P AND Q. This is not a discovered preference; it is the logical relation on which the judgments will be assessed.
The three agents produce rows (T,T,T), (T,F,F), and (F,T,F) for (P,Q,R). Each row satisfies R = P AND Q: the first accepts both inputs and the conjunction; the second rejects Q and R; the third rejects P and R. Whatever else may distinguish these people, their individual answers are internally consistent on the specified agenda.
Collective judgment then requires a rule for moving from rows to a single row. With a separate simple majority for each proposition, P receives T,T,F and so is accepted; Q receives T,F,T and so is accepted; while R receives T,F,F and so is rejected. The resulting collective profile is (T,T,F), which violates R = P AND Q. Thus internally consistent individual answers can yield a collective answer that contradicts a relation already built into the agenda.
The conflict is narrow. This profile supplies one explicit counterexample to a particular aggregation practice; it does not prove a universal impossibility theorem. Nor does fixing P, Q, and R settle every procedural choice: a premise-first rule that aggregates P and Q and derives R differs from voting directly on the conclusion R. Voting on all three separately is the issue-wise procedure that produced the inconsistent row. The agenda and its logical relations must therefore be specified when evaluating a rule. Otherwise there is no determinate standard against which collective consistency can even be evaluated.
Fixing meaning and logical relations is what makes aggregation answerable at all. In E02 the agenda is deliberately small: each person judges P, Q, and R, where R means P AND Q. The three profiles (T,T,T), (T,F,F), and (F,T,F) each satisfy that relation, while separate simple majorities return (T,T,F), which does not. The conflict is therefore exact and narrow: it does not depend on miscounting votes or on imputing inconsistent beliefs to anyone.
The precision is purchased by exclusion. A live discussion may not settle what P means, whether R is the right conclusion to draw, or whether an answer should be a truth value at all. The toy profile assumes crisp judgments, a fixed list of propositions, and agreement about their logical relations; it records none of the ambiguity over interpretation, relevance, or what counts as a reason. Even a procedural variation—such as removing the third person's vote on P and letting the two cast votes set that issue's denominator—belongs to a declared rule rather than to anyone's discovered intention. Such additions test proposed alternatives; they do not recover what the formalization has set aside.
That limit defines the application. E02 establishes that individually consistent answers can yield a collective answer that violates the shared conjunction, once the propositions and their relations are specified and separate simple majorities are taken. It is one explicit counterexample, not a proof that every open-ended dialogue will produce logical failure, nor a demonstration that one resolution—premise-first, conclusion-first, or another rule—is universally legitimate. If later cases retain the same meanings and logical relations, the same test can be applied to them; if they do not, the profile's verdict no longer transfers unchanged.
13.2Individual consistency
The E02 profile gives each of the three people a complete assignment of truth values to P, Q, and R, where R means P AND Q. Individual consistency here is therefore a matter of checking one row at a time against that connective.
The first person assigns (T,T,T). Since P is true and Q is true, their conjunction is true, so R is true as required. The second assigns (T,F,F). Here P is true but Q is false, making the conjunction false; R is correspondingly false. The third assigns (F,T,F). P is false and Q is true, so the conjunction is again false, and false is the correct value for R. Every row satisfies R = P AND Q. Nobody has asserted a conjunction while denying one of its parts.
That check matters because of what it removes from the explanation. The collective inconsistency is not inherited from an individual who reasoned illogically. Taking a separate simple majority for each proposition yields P = true, Q = true, and R = false: the majority profile (T,T,F) violates R = P AND Q even though no individual profile does. The conflict thus enters through the aggregation step, which counts each proposition independently and preserves the judgments of individuals only proposition by proposition.
The inferential limit is equally concrete. This is one explicit counterexample, not a proof that inconsistency is unavoidable for every agenda, electorate, or procedure. Nor does it show that an institution must satisfy every formal axiom; the agenda deliberately excludes the ambiguity and shifting interpretation that matter in open-ended discussion. The demonstration is narrow on purpose: once individual rows have been verified, the remaining question is what the procedure does with them.
Individual consistency is a constraint on the relations among one person's judgments. It does not require the three judges to agree with one another about P or about Q. In the profile, the first judge accepts both conjuncts, the second accepts P and rejects Q, and the third rejects P while accepting Q. Those differences concern which propositions hold; they are not defects in anyone's reasoning. Each judge's attitude toward R follows from that judge's own attitudes toward the two conjuncts.
The distinction fixes the scope of the requirement. Consistency is assessed person by person: ask whether the same judge's verdict on R matches the truth condition imposed on P and Q by that judge's own acceptances and rejections, and stop there. Agreement would be a different condition altogether, asking whether the three judges share attitudes toward P, Q, and R. The profile satisfies the first condition and, by construction, not the second. Any attempt to diagnose a collective defect must therefore begin with this test rather than with interpersonal harmony.
Applied to the counting of these judgments, the problem is consequently located in aggregation rather than in individual reasoning. Separate simple majorities on P, Q, and R yield the pattern (T,T,F), which does not obey R = P ∧ Q even though every contributing individual profile does. The conflict is real but narrow: it shows that majority counting on each proposition separately can produce a pattern no individual holds. It does not show that collective judgment must always fail this way, that aggregation impossibility is universal, or that every institution is obligated to satisfy every formal axiom. Open-ended dialogue may rarely display conflicts as tidy as this one, and institutional design answers questions the profile leaves untouched. What the example establishes is a scope claim: individual consistency constrains each person's package of attitudes, and compliance with that constraint does not by itself guarantee a consistent result once the attitudes are counted together.
13.3A three-person conjunction example
To see the difficulty concretely, consider a fully specified profile of three people judging three propositions: P, Q, and R, where R is defined to hold exactly when P and Q both hold. The first person accepts P, accepts Q, and accordingly accepts R. The second accepts P but rejects Q, and therefore rejects R. The third rejects P while accepting Q, and likewise rejects R. Each of these three profiles is internally consistent: in every case the attitude toward R matches the truth condition imposed on the conjunction. Thus the disagreement among the three concerns which propositions are true, not whether anyone reasons about them incoherently.
Now form a collective judgment by taking a separate simple majority for each proposition. Two of the three people accept P, so P carries by 2/3. Two of the three accept Q, so Q also carries by 2/3. Only the first person accepts R; the other two reject it, so the majority on R is 2/3 against. The resulting collective profile affirms P, affirms Q, and denies R, even though R was defined as the conjunction of exactly those two affirmed propositions. The collective outcome is therefore inconsistent under the given definition, despite every individual outcome being consistent.
What this example establishes is deliberately narrow. It supplies one explicit case in which proposition-by-proposition majority voting, applied to individually consistent judgment profiles, yields a collective profile that violates a stated logical relation. It does not show that every aggregation method must fail, that such conflicts arise with any frequency in open-ended discussion, or that no institutional arrangement can handle them. The calculation is a local toy construction executed here for illustration, not evidence about any population or deployed procedure. List and Pettit similarly observe that consistency of individual judgments on interconnected propositions does not by itself settle how a consistent collective judgment should be formed (List and Pettit, 2002), though the present example proves nothing beyond its own three-person case.
What the profile licenses is a claim about a rule's logical safety, not about the world. Separate simple majorities are not guaranteed to preserve a stated conjunction: here they turn three individually consistent profiles—(T,T,T), (T,F,F), and (F,T,F), where R means P and Q—into the collective profile (T,T,F). The inference is therefore existential and negative. Because one fully specified case defeats the guarantee, no general assurance can rest on the bare consistency of individual judgments plus proposition-by-proposition voting. But the inference stops there. A single counterexample establishes possibility, not prevalence; it gives no estimate of how often such conflicts arise in open-ended dialogue, where agents may not divide their commitments into independently votable propositions at all.
Nor does the case rule out institutional responses. Once the conflict is displayed, a designer can choose where to apply majority rule. A premise-first procedure counts P and Q separately and derives R; a conclusion-first procedure takes a vote on R directly. These are different procedural commitments, and the profile shows that they can return different answers to the same profile—it does not show that either is universally legitimate. Incomplete judgments introduce a further procedural choice, addressed below; the present comparison keeps all three rows fixed.
This example's practical application is therefore diagnostic rather than legislative: it marks a juncture at which the aggregation rule must be chosen explicitly, because the logical relation at stake will not select one for a reader. The residual limitation is exactly one case wide—enough to refute a guarantee, too little to measure the phenomenon.
13.4Premise and conclusion procedures
In E02, the three individual profiles are \((T,T,T)\), \((T,F,F)\), and \((F,T,F)\), where \(R\) means \(P \land Q\). Each row satisfies the defining condition: the person who accepts both \(P\) and \(Q\) accepts \(R\), while each person who rejects one conjunct rejects \(R\). Under separate simple majorities, \(P\) receives two affirmative votes and is accepted, \(Q\) receives two affirmative votes and is accepted, and \(R\) receives only one affirmative vote and is rejected.
The conflict follows from a choice of procedure. A premise-first rule asks the group to decide each conjunct and then computes the conclusion from those decisions. Since majority rule returns true for both \(P\) and \(Q\), the rule entails \(R = P \land Q\), hence \(R\) should be true. A conclusion-first rule instead puts \(R\) itself to a vote. There the majority rejects \(R\), so the group's stated answer is false. Thus the same judgment profile yields opposite collective answers depending on whether the conclusion is derived from premise outcomes or determined by a direct ballot.
The inferential step is deliberately narrow. It shows that majority aggregation over the two premises and majority aggregation over the conclusion can disagree when the connective is conjunction; the disagreement is not evidence that any individual agent is inconsistent. Nor does it establish that premise-first voting is correct and conclusion-first voting incorrect. Each procedure is a proposed institutional choice, and the profile leaves their relative standing open.
The limitation is equally specific. E02 is one explicit counterexample exhibiting a logical aggregation conflict. It does not prove a universal impossibility theorem, nor does it estimate how often such conflicts arise in open-ended dialogue or in actual institutions.
The distinction appears once each rule is carried through to completion. Under the premise-first rule, P and Q are put to the separate majorities, and once both are settled as true, R is derived as true. The procedure therefore yields a determinate and logically closed answer. Under the conclusion-first rule, the separate vote on R is authoritative for that proposition, so the collective answer is that R is false. This rule is equally determinate. Indeterminacy is not what separates them: each reaches an answer without further negotiation, and neither leaves the three-person profile unfinished.
What separates the rules is which judgments are authoritative. Premise-first makes the components of the conjunction the decision points and lets logical closure supply the answer to R; in doing so it does not use the separate collective vote on R as an input. Conclusion-first reverses the priority: it protects a direct collective answer on the proposition at issue, but treats the majorities on P and Q as recorded information rather than premises from which R must follow. Note that the negative individual judgments on R in the profile are not arbitrary—under the stipulated conjunction, each falls out of the corresponding false conjunct—yet premise-first still arrives at a true R from the collectivized premises. Each rule can be defended on its own terms, and neither is uniquely legitimate.
Which priority a procedure should serve is therefore an institutional choice, not a consequence forced by the profile. Premise-first suits settings that want the components settled explicitly and checked for consistency; conclusion-first suits settings where the target proposition must register a direct vote. The qualification that matters is narrow: E02 is one explicit counterexample to treating the two procedures as interchangeable. It does not establish a universal aggregation impossibility, and it says little on its own about open-ended dialogue, where the relevant propositions may never be fixed. Any extension beyond this conjunction, these three agents, and simple majorities remains conditional.
13.5Abstention and incomplete judgments
The premise-first and conclusion-first procedures answer different procedural priorities; the preceding arithmetic does not establish either as uniquely legitimate. They also leave open what happens when a participant supplies no judgment at all. That gap should be introduced explicitly rather than inferred. In a variant of the three-person profile, suppose the third person, whose profile is (F,T,F), withholds judgment on P only. The first person's P-judgment is true and the second's is true; the third contributes no P-judgment. Under the declared rule for this illustration, the P denominator is two cast votes rather than three. This is a rule stipulated by this study, not a recovered intention or an observed abstention. The other propositions retain their original votes.
On that basis P still receives a majority of true among the votes cast. Q keeps its three judgments (true, false, true) and receives a majority of true. R keeps its three judgments (true, false, false) and receives a majority of false. The collective output is once more P true, Q true, R false, although R means P and Q; the conjunction remains inconsistent with the two propositions assigned true. The incomplete judgment changes the evidential base of one issue without repairing the conflict, which remains a relation among the collective values of P, Q, and R.
The inference here is narrow. Because the third person's P-judgment was false and was not needed for P to obtain a true majority among votes cast, removing that judgment leaves both the P verdict and the contradiction unchanged. The example therefore shows only that withholding judgment can alter the basis of a proposition's majority without altering the collective result, under this one declared counting rule. It is a single explicit counterexample, not a theorem that abstention can never resolve aggregation conflicts, and it says nothing about how often such logical conflicts arise in open-ended dialogue. Its evidential value is accordingly modest: the declared denominator makes the arithmetic checkable, while the scope of what follows from it remains confined to this profile and this stipulated rule.
Suppose the third person's judgment on P is withheld, as the stipulated example allows. That abstention has to be classified before any majority is computed, because the choice determines what counts as the issue's electorate. If abstentions are excluded, only cast votes enter the tally: the P votes are T and T, so two cast votes, not three, form that issue's denominator, and P is accepted. Q and R retain their original votes, so Q is accepted and R is rejected. The abstention itself disappears from the arithmetic, though it may still be reported descriptively.
Recording it separately preserves that information without deciding its force. The record can mark the third person as having withheld on P while still applying the exclusion rule to the tally; alternatively, the recorded abstention can be treated as making P unresolved. Under the second policy the collective output is incomplete: there is no verdict on P, and hence no complete majority profile to compare with the inconsistent (T,T,F). What must not happen is a silent shift of denominator, where a two-vote result is presented as if three persons had been counted.
These are procedural choices, not discoveries about the voter's intention. A premise-first rule decides P and Q from the available votes and then tests or derives R under R = P ∧ Q; a conclusion-first rule lets each proposition's verdict stand as voted and tolerates the resulting tension. The two rules answer different questions about what the aggregation is for. E02 remains one explicit counterexample showing that a narrow inconsistency can arise even when every individual row is consistent. It does not establish a universal aggregation impossibility, nor does it settle which abstention policy a legitimate procedure must adopt.
13.6Scope of impossibility arguments
A formal impossibility claim is not established by exhibiting one procedure that fails. It requires a precise statement of the agenda—the propositions to be combined—the admissible profiles of individual judgments, and the procedural conditions under which collective judgments are formed. Alter any one of these and a different claim is at issue.
The constructed profile supplies that precision for a narrow case. The agenda is P, Q, and R, with R meaning P and Q. Three agents report (T,T,T), (T,F,F), and (F,T,F), and every row satisfies R = P AND Q. Separate simple majorities then yield (T,T,F), which affirms P and Q while denying their conjunction. The inconsistency is therefore a feature of this profile under this majority rule, not a general verdict on aggregation.
Procedural conditions also govern how incomplete judgments enter. If the third person's vote on P alone is treated as an abstention, then P is decided by two cast votes while Q and R retain three; that is a declared rule, not a recovered intention. A rule that counts only cast votes, one that requires a quorum, and one that leaves a proposition unresolved are distinct procedures, each needing its own specification before any impossibility claim can be framed.
The evidential starting point is correspondingly modest: consistent individual judgments on interconnected propositions do not by themselves settle how to form a consistent collective judgment (List and Pettit, 2002). That observation motivates the search for sharper results, but it does not provide one.
This counterexample refutes the universal guarantee that the stated issue-wise majority rule preserves consistency for every admissible profile. It does not establish that no other admissible rule can preserve consistency on all such profiles. A failure of this rule at this profile does not establish that every allowed rule fails on some allowed profile. A broader impossibility argument must identify its class of rules and show why none meets the stated requirements throughout the permitted domain. How often this particular conflict arises in open-ended dialogue remains unmeasured here.
E02 establishes a narrow aggregation conflict and nothing more. Three agents each judge P, Q, and R, where R means P and Q; their rows (T,T,T), (T,F,F), and (F,T,F) each obey that conjunction. Taking a separate simple majority for each proposition yields (T,T,F), which violates R = P and Q. So even a profile whose every individual row is logically consistent can generate a collectively inconsistent outcome under this particular rule.
The majority profile above supplies a counterexample, not a general impossibility. One explicit profile shows that separate simple majorities can break conjunction coherence; it does not show that every aggregation method fails, nor that every agenda will generate such a profile. The step from a claim that this rule can conflict here to a claim that no procedure can meet the relevant requirements needs a theorem rather than an example.
The full theorem is not supplied in the present evidence, so its application cannot be checked. Matching a cited impossibility result to a concrete case requires confirming that the theorem's exact agenda, admissible profiles, and procedural conditions cover the case at hand; without the full statement those hypotheses cannot be verified, and reconstructing them by guess would only manufacture support. Accordingly, the theorem cannot enlarge the conclusion available from the displayed profile. If a later audit of the source work supplied the full statement and confirmed that it covers this rule and this class of profiles, a broader claim would become possible; until then the scope remains bounded by the three-person example.
The same boundary answers the practical worry. Nothing here shows that tidy conjunction profiles are common in open-ended dialogue, and an institution need not satisfy every formal axiom, so the example cannot condemn institutional design as such. Its use is diagnostic: it marks one precise point where separate majority judgments can come apart. From there the question becomes which rule to adopt and which coherence or representational feature that rule gives up.
13.7Choosing a rule with its sacrifice visible
The three rows of E02 make the choice concrete. Each agent judges P, Q, and R = P ∧ Q as (T,T,T), (T,F,F), or (F,T,F), and every row respects the conjunction. Separate simple majorities nevertheless return P as true, Q as true, and R as false, a set of collective answers that neither a single row nor the stated relation can license.
A premise-first rule would aggregate P and Q and then derive R from those outcomes. It purchases consistency by construction, but it gives up independent majority support for the conclusion: the votes cast on R are never counted on their own terms. A conclusion-first rule runs the other way. It treats the majority recorded on R as the answer to the question actually put, and pays by admitting an inconsistency between the accepted conclusion and the accepted premises. Neither proposal is free of cost. The real difference between them is which judgment the procedure refuses to let the electorate deliver directly, and which logical property is protected at that judgment's expense.
The abstention variant shows how a change of denominator can resemble a repair without providing one. Remove the third person's vote on P alone: the remaining P votes are T and T, so two cast votes, not three, form that issue's denominator, while Q and R retain their original votes. P still comes out true and R false, so E02 is still inconsistent. What has changed is that the three propositions are no longer decided by quite the same body of voters. A rule that tolerates abstention therefore buys flexibility at the price of comparability across issues, without altering this profile's outcome.
The lesson stays local. E02 is one fully specified counterexample, not a proof that every aggregation procedure must fail some formal axiom, and an institution may accept a sacrifice deliberately. What a defensible choice cannot do is leave that sacrifice unnamed.
Suppose the institution takes each proposition separately and records simple majorities. With E02's rows (T,T,T), (T,F,F), and (F,T,F), the collective profile is P true, Q true, R false, so it breaks R = P and Q even though every individual row respects that connection. The sacrifice can now be named rather than hidden. A premise-first rule decides P and Q by majority and fixes R from them; it gives up independent aggregation of judgments about R, so the collective may affirm R when a majority has denied it. A conclusion-first rule takes the majority on R and revises the premises to fit; it gives up the premise majorities. Each rule discards something, and consistency alone does not decide which discard is acceptable.
An abstention rule changes the arithmetic without settling the disagreement. Remove the third person's vote on P only: P then has two cast votes, T and T, while Q and R retain their three original votes. The denominator for P is therefore two, not three. That is a declared procedural rule, not a recovered voter intention. In this case P remains true, Q true, and R false, so the inconsistency survives the redesign; what changes is the basis on which each issue was counted. Whether a two-vote denominator is proper is itself a choice requiring justification.
The inferential step is modest but decisive for scope: making the logical consequences transparent does not settle which sacrifice the institution may impose. Exactness about what each proposed rule loses makes the trade-off legible; it does not promote one rule to the authoritative one. Reasonable participants may still favor premise-first aggregation because derivations govern, or conclusion-first aggregation because recorded verdicts govern practice, and the profile underdetermines that preference. The limit matters: this is one explicit counterexample, not a proof of a universal aggregation impossibility, nor a claim that formal axioms must constrain every open-ended dialogue. The logic closes before the procedural choice does.