Agenda manipulation under doctrinal paradox

Abstract

We study the aggregation of individuals’ beliefs into their collective judgment on a certain proposition, called the conclusion, which is logically related to other preliminary propositions. While the literature has highlighted the crucial dependency of voting outcomes on the selection of an agenda—i.e., the set of propositions voted on—the structure of this dependency is not well understood. We introduce two novel premise-based rules, defined through the conjunctive normal form and disjunctive normal form of the conclusion. Our theorem shows that for any profile of voters’ beliefs, these rules respectively yield the highest and lowest voting outcomes, with respect to a given total order over the set of truth values, within the class of all premise-based rules. This result clarifies the extent to which a voting outcome can be manipulated through the selection of an agenda.

1 Introduction

In this paper, we study the aggregation of individuals’ beliefs into their collective judgment on a certain proposition, called the conclusion, which is logically related to other preliminary propositions, called premises. This type of aggregation problem is a central focus in the field of judgment aggregation.

A classic example of judgment aggregation is the doctrinal paradox introduced by Kornhauser and Sager (1986), where three judges in a criminal court collectively decide whether a defendant is guilty,¹ With a slight modification of their original scenario, suppose a legal doctrine stipulates that the defendant is guilty if either of two premises—(i-a) the defendant did not perform a certain action and (i-b) the action was a contractual obligation—both hold, or if a single decisive premise—(ii) the defendant committed forgery in the contractual document—holds. Assume that the three judges differ in their beliefs about these premises, and consequently, about the conclusion, as follows:

	Premise (i-a)	Premise (i-b)	Premise (ii)	Conclusion
Judge 1	Yes	Yes	No	Yes
Judge 2	No	Yes	No	No
Judge 3	No	No	Yes	Yes

The key observation from this table is that the voting outcome depends on the choice of an agenda, the set of propositions judged in court. On one hand, if the judges implement a majority vote on the conclusion, the defendant is found guilty. On the other hand, if the judges vote on each of the three premises, they find that the action in question was a contractual obligation, yet the defendant neither violated the obligation nor forged the contract. As a result, the judges conclude that the defendant is innocent.

Building on this example, the literature on judgment aggregation has established several results that demonstrate how agenda selection can reverse voting outcomes. However, the structure of this dependency is not fully understood. For instance, if an agenda is chosen by a person or organization with preferences for a particular outcome (e.g., convicting the defendant), which agenda should they present to voters? Also, to what extent can the agenda setter manipulate the voting outcome by choosing different agendas? This paper addresses these questions in a formal voting model.

In our model, the conclusion is an arbitrary logical statement formed by combining multiple atomic propositions using logical conjunctions and/or disjunctions. To quantify the degree of voting outcome reversals, we allow individuals, as well as the collective body, to express intermediate beliefs, such as “partial (dis)agreement,” by enabling votes beyond binary choices.² We formally model this problem setting using the language of fuzzy logic, which extends standard Boolean logic by introducing a totally ordered set of truth values beyond the binary ones.

An aggregation rule is a function that maps each profile of individuals’ beliefs about propositions to their collective judgment on the conclusion. In particular, we focus on the class of premise-based rules, where the voting outcome is determined in two steps: First, individuals cast their votes on each premise in the agenda, yielding a collective judgment on each premise. Second, the collective judgment on the conclusion is derived from those on the premises by applying the deduction rule of fuzzy logic. In essence, each of these rules is defined by its associated agenda, which consists of the set of premises to be voted on. For instance, in the preceding example, the conclusion-based vote is a specific premise-based rule where the agenda consists only of the conclusion, while the atomic-premise-based vote involves an agenda containing the three atomic premises.³

We introduce a novel premise-based rule, where the agenda consists of all disjunctive clauses appearing in the conjunctive normal form of the conclusion. Our main result, Theorem 1, shows that for any profile of individuals’ beliefs, the outcome produced by this rule is always greater than the outcome of all other premise-based rules with respect to the total order over the set of truth values. Moreover, we show that the mathematical dual of this rule, defined according to the disjunctive normal form of the conclusion, is uniformly dominated by all other premise-based rules. Therefore, these rules serve as the uniform upper and lower bounds for the entire class of premise-based rules.

An implication of Theorem 1 can be best understood by viewing the doctrinal paradox as suggesting the possibility of manipulating voting outcomes through the selection of an agenda (Dietrich 2006, 2016). In light of this issue of agenda manipulation, our theorem clarifies the extent to which an agenda setter can influence the voting outcome. Specifically, since each rule in the theorem uniformly dominates or is dominated by all other premise-based rules, the outcomes under these rules define the full range of possible outcomes under any agenda selection, irrespective of the profile of individuals’ judgments. In addition, the result indicates which agenda should be chosen when an agenda setter has preferences for one of the extreme outcomes; in the context of the preceding example, if the agenda setter prefers conviction (resp. acquittal), our result recommends compiling the agenda based on the conjunctive (resp. disjunctive) normal form of the conclusion.

Since the seminal work by List and Pettit (2002), the literature on judgment aggregation has established many impossibility theorems that document the incompatibility of logically consistent aggregation with other desirable axioms (e.g., Nehring and Puppe (2008, 2010); Dietrich and Mongin (2010); Dokow and Holzman (2010a)). These results highlight the essential role of an agenda and showcase that a sort of inconsistency, as seen in our example, is inevitable for a wide range of agenda selections. Several papers, including Pauly and Van Hees (2006), Dokow and Holzman (2010b), Duddy and Piggins (2013), and Dietrich and List (2017a, 2017b), extend these insights to non-binary judgment settings, similar to the setting of this paper.

Motivated in part by the search for a better compromise in light of these impossibility results, another branch of the literature investigates the comparison of practically relevant aggregation rules, especially conclusion-based and atomic-premise-based rules. These rules are compared on the basis of truth-tracking properties à la Condorcet (Grofman and Feld 1988) in sincere voting settings by Grofman (1985), List (2005), Bovens and Rabinowicz (2006), Miyashita (2021), and Alabert and Farré (2022), as well as in strategic voting settings by Bozbay et al. (2014) and De Clippel and Eliaz (2015). While our analysis does not directly address the truth-tracking properties of aggregation rules, our theorem can provide insights into the lower bounds on false-positive and false-negative probabilities that the committee can achieve by varying the voting agenda, assuming sincere voting by committee members.⁴ This is because the aggregation rules offered in this paper provide uniform bounds on the outcomes of all premise-based rules that hold across all profiles of individual judgements.⁵

The rest of this paper is organized as follows. In Sect. 2, we illustrate our findings by extending the example from this introduction. After providing some needed fuzzy-logic terminology in Sect. 3, we introduce our voting model in Sect. 4. Section 5 presents the main result, which is proved in Appendix A. Finally, Appendix B provides additional remarks on our fuzzy-logic framework.

2 Illustrating example

Let us illustrate this paper’s findings by revisiting the example from Sect. 1, where the three judges must derive their collective judgment on the proposition $\phi$, read as “the defendant is guilty.” The judgment on $\phi$ is related to the judgments on the following three atomic propositions:

• $p_1$: “The defendant did not perform a certain action.”

• $p_2$: “The action was a contractual obligation.”

• $p_3$: “The defendant committed forgery in the contractual document.”

We are concerned with the legal doctrine stipulating that $\phi$ is accepted if either both $p_1$ and $p_2$ hold, or $p_3$ holds. This can be formulated by the following logic equation:

$$\begin{aligned} \phi = (p_1 \wedge p_2) \vee p_3. \end{aligned}$$

Assume that all judges agree on the above legal doctrine in use. However, they differ in their beliefs about each proposition, as will be summarized in Tables 1 and 2.

Table 1 Judges’ beliefs about $p_1$, $p_2$, $p_3$, and $\phi = (p_1 \wedge p_2) \vee p_3$

In the standard Boolean logic framework, a belief about each proposition is modeled as the assignment of a binary truth value $x \in \{0,1\}$ to the proposition, where 1 and 0 can be interpreted as “acceptance” and “rejection,” respectively. In order to make the example more interesting, we assume that the judges and committee can adopt intermediate beliefs of $\frac{2}{3}$ and $\frac{1}{3}$, which are interpreted as “partial acceptance” and “partial rejection,” respectively. Therefore, the set of truth values that can be assigned to each proposition is $\{0,\frac{2}{3},\frac{1}{3},1\}$, which is totally ordered with respect to the usual binary relation over scalars.

Unlike Boolean logic, multi-valued logic allows for multiple possible ways to assign a truth value to a compound proposition, such as $p \wedge q$ or $p \vee q$, in a manner consistent with the truth values assigned to its constituents p and q. For concreteness in this example, we assume that the truth value of $p \wedge q$ is the minimum, and the truth value of $p \vee q$ is the maximum, of the truth values assigned to p and q. This rule is known as Gödel logic and corresponds to a particular instance of our general framework.

Now, extending the example with binary judgments from Sect. 1, we assume that the three judges express their multi-valued judgments on each proposition, as described in Table 1. Specifically, based on their beliefs about each atomic proposition, each judge derives their judgment on the conclusion $\phi$ by applying the aforementioned Gödel logic. For instance, since the first judge assigns 1 to $p_1$ and $\frac{2}{3}$ to $p_2$, the minimum of these values, $\frac{2}{3}$, is assigned to $p_1 \wedge p_2$. The conclusion $\phi$ is then assigned the value $\frac{2}{3}$, which is the larger of the truth values assigned to $p_1 \wedge p_2$ and $p_3$. Thus, the first judge weakly accepts the conclusion.

We consider the aggregation of judges’ beliefs into their collective belief, implemented by using the median voting rule; that is, given a profile of individual beliefs about a proposition, the collective belief about the proposition is determined by its median value within the profile. We then observe the crucial dependence of voting outcomes on the choice of agenda. Specifically, if the judges take a median vote directly on the conclusion $\phi$, they assign it a truth value of $\frac{2}{3}$. In contrast, if they vote separately on each of the three atomic premises and derive the collective judgment on $\phi$, they assign it a truth value of $\frac{1}{3}$. These two aggregation procedures are known as the conclusion-based rule and atomic-premise-based rule, both of which have received significant attention in judgment aggregation due to their intuitive appeal (Kornhauser and Sager 1986; Nehring and Puppe 2008; Dietrich and Mongin 2010).

Table 2 Judges’ beliefs about the relevant clauses for $\phi$ and $\psi$, where $\psi = (p_1 \vee p_3) \wedge (p_2 \vee p_3)$ is a tautological rephrasing of $\phi$ via the distributive law

These rules can be seen as the two extreme premise-based rules, entailing the “coarsest” agenda $\{\phi \}$ and the “finest” agenda $\{p_1, p_2, p_3\}$, respectively, in terms of how the conclusion is decomposed into premises. Between these extremes, we can naturally consider “intermediate” premise-based rules, such as one entailing the agenda $\{p_1 \wedge p_2, p_3\}$ so that the voting outcome is determined in view of the collective judgments on two premises, $p_1 \wedge p_2$ and $p_3$. Applying this rule, the judges assign a truth value of 0 to both $p_1 \wedge p_2$ and $p_3$, resulting in the collective judgment of 0 on $\phi$. Alternatively, we can also consider another rule where the conclusion $\phi$ is tautologically rephrased into $\psi$:

$$\begin{aligned} \psi = (p_1 \vee p_3) \wedge (p_2 \vee p_3). \end{aligned}$$

The judges then vote on compound premises $p_1 \vee p_3$ and $p_2 \vee p_3$. Under this rule, they assign a truth value of 1 to $\phi$ (or equivalently, to its tautological restatement $\psi$). These results are summarized in Table 2.

To sum up, the four different premise-based rules yield distinct voting outcomes, as illustrated in Fig. 1. The diagram shows that the outcomes of the conclusion-based and atomic-premise-based rules lie between those of two novel rules: one based on an agenda consisting of disjunctive clauses (top) and the other on an agenda consisting of conjunctive clauses (bottom). We refer to these new rules as the CNF-based rule and the DNF-based rule, reflecting their reliance on the conjunctive normal form and disjunctive normal form of the logical statement $\phi$. Theorem 1 generalizes this leading example. Indeed, the outcomes of the conclusion-based, atomic-premise-based, and all other premise-based rules must always lie between those of the CNF-based and DNF-based rules.

Fig. 1

Summary of the example. The vertical direction captures the collective judgment on $\phi$, while the horizontal direction abstractly represents the level of coarseness in the decomposition of $\phi$ into premises

3 Fuzzy logics

In this section, we introduce terminology from fuzzy logic that will be used to formulate our voting model in the next section. Let there be a finite set $L_0$ with $|L_0| \ge 2$, whose elements are called atomic propositions. We define the set L of all propositions by closing $L_0$ under two logical connectives $\wedge$ (“and”) and $\vee$ (“or”). Formally, L is defined to be the smallest set satisfying the following two properties:

• i). $L_0 \subseteq L$.

• ii). If $\phi , \psi \in L$, then $\phi \wedge \psi \in L$ and $\phi \vee \psi \in L$.

Unlike other work in judgment aggregation, we do not consider the negation symbol $\lnot$ (“not”), thereby focusing on situations in which the conclusion depends positively on each premise. This restriction plays a substantial role in our analysis; we return to discussing this point in Sect. 5.3.

A proposition $\phi$ is said to be a constituent of $\phi '$ if (i) $\phi$ is $\phi '$ itself, or (ii) $\phi$ is used in the inductive process of constructing $\phi '$. In particular, $\phi$ and $\psi$ are called direct constituents of $\phi \wedge \psi$ and $\phi \vee \psi$. The degree of a proposition $\phi$, denoted by $d(\phi )$, refers to the number of occurrences of atomic propositions in its construction; formally, it is defined by setting $d(p) = 1$ for all $p \in L_0$, and recursively $d(\phi ) = d(\psi _1) + d(\psi _2)$ when $\phi = \psi _1 * \psi _2$ with $* \in \{\wedge , \vee \}$. A proposition $\phi$ with $d(\phi ) \ge 2$ is called a compound proposition.

We consider the quotient set $L/\!\simeq$ with respect to the equivalence relation $\simeq$ that represents the logical equivalence between two propositions. Specifically, we recall the following basic syntactic operations on propositions $\phi , \psi , \eta \in L$:

• Commutation: $\phi \wedge \psi \simeq \psi \wedge \phi$ and $\phi \vee \psi \simeq \psi \vee \phi$.

• Association: $\phi \wedge (\psi \wedge \eta ) \simeq (\phi \wedge \psi ) \wedge \eta$ and $\phi \vee (\psi \vee \eta ) \simeq (\phi \vee \psi ) \vee \eta$.

• Distribution: $\phi \wedge (\psi \vee \eta ) \simeq (\phi \wedge \psi ) \vee (\phi \wedge \eta )$ and $\phi \vee (\psi \wedge \eta ) \simeq (\phi \vee \psi ) \wedge (\phi \vee \eta )$.

• Simplification: $\phi \wedge (\phi \wedge \psi ) \simeq \phi \wedge \psi$ and $\phi \vee (\phi \vee \psi ) \simeq \phi \vee \psi$.

We then extend $\simeq$ and say that two propositions $\phi$ and $\psi$ are equivalent, written as $\phi \simeq \psi$, if one can be obtained from the other through finitely many applications of these operations. Let $[\phi ]$ denote the equivalence class of $\phi$ in the quotient set $L/\!\simeq$. Note that two equivalent propositions can have different degrees.

A set of truth values is an arbitrary totally order set $(T,\ge )$ with $|T| \ge 2$. For concreteness, assume that $\{0,1\} \subseteq T \subseteq [0,1]$ and $\ge$ is the usual ordering over scalars, while our analysis is applicable to general T. Here, the extreme values 0 and 1 correspond to the Boolean variables of “falsity” and “truth,” respectively, and values strictly between them represent intermediate semantic values. Further interpretations of these truth values will be discussed later in this section. In the meantime, for $x,y \in T$ with $x \ge y$, we interpret a proposition assigned x as weakly closer to “truth” than a proposition assigned y.

A valuation is a function $v: L \rightarrow T$ that assigns a truth value $v(\phi ) \in T$ to each proposition $\phi \in L$, which necessitates natural properties that reflect the interpretations of the logical connectives. The formal definition follows.

Definition 1

A valuation is a function $v: L \rightarrow T$ satisfying the following two requirements for any $\phi , \psi \in L$:

• Invariance: $v(\phi ) = v(\psi )$ whenever $\phi \simeq \psi$.

• Entailment: $v(\phi \wedge \psi ) \le \min \{v(\phi ), v(\psi )\}$ and $v(\phi \vee \psi ) \ge \max \{v(\phi ),v(\psi )\}$.

The invariance condition simply requires that two logically equivalent propositions be assigned the same truth value. The entailment condition is motivated by the usual interpretations of $\wedge$ and $\vee$. Specifically, reading $\wedge$ as “and,” the compound proposition $\phi \wedge \psi$ logically entails each of $\phi$ and $\psi$ in the sense that whenever $\phi \wedge \psi$ is “true” to some extent, both $\phi$ and $\psi$ must be true to at least that extent. This is reflected in the requirement that $v(\phi \wedge \psi )$ be no greater than both $v(\phi )$ and $v(\psi )$. Likewise, reading $\vee$ as “or,” each of $\phi$ and $\psi$ entails $\phi \vee \psi$, so the compound proposition $\phi \vee \psi$ must be assigned a truth value no less than both $v(\phi )$ and $v(\psi )$.

In our voting model, each individual i has their own valuation $v^i$. This valuation represents the individual’s multi-valued judgments on relevant propositions and may differ across individuals in two ways. First, each individual’s valuation over atomic propositions $v^i_0$—i.e., the restriction of $v^i$ to $L_0$—can differ across individuals, meaning they may assign different truth values to some atomic propositions. Second, even when $v^i_0$ and $v^j_0$ coincide, $v^i$ and $v^j$ may still differ since the entailment condition need not pin down the truth value of a compound proposition from those of its constituents. We impose no assumptions on individual valuations beyond requiring that each satisfies Definition 1, so both sources of heterogeneity are permitted.

Example 1

The Gödel logic introduced in Sect. 2 offers semantics for $\wedge$ and $\vee$ via minimum and maximum truth value assignments, respectively. Specifically, a Gödelian valuation is a function $v:L \rightarrow T$ such that: (i) given any function $v_0:L_0 \rightarrow T$, set $v(p) = v_0(p)$ for each atomic proposition $p \in L_0$, and (ii) the value assigned to a compound proposition is recursively determined as

$$\begin{aligned} v(\phi \wedge \psi ) = \min \{v(\phi ), v(\psi )\} \quad \textrm{and} \quad v(\phi \vee \psi ) = \max \{v(\phi ), v(\psi )\}. \end{aligned}$$

(1)

When $T = \{0,1\}$, this reduces to the standard Boolean logic, where we have $v(\phi \wedge \psi ) = 1$ if and only if $v(\phi ) = v(\psi ) = 1$, and $v(\phi \vee \psi ) = 1$ if and only if $v(\phi )=1$ or $v(\psi )=1$. $\triangle$

A Gödelian valuation is constructed by first assigning arbitrary truth values to atomic propositions via a function $v_0: L_0 \rightarrow T$, and then extending these values to compound propositions using the min–max rule (1). No restrictions are imposed on $v_0$, reflecting the unrestricted nature of truth value assignments to atomic propositions, whereas the truth value of each compound proposition is uniquely determined once $v_0$ is fixed. In fact, this property is characteristic of the class of t-norm fuzzy logics, of which Gödel logic is a canonical example. As shown in Appendix B, however, Gödel logic is the only t-norm fuzzy logic that satisfies invariance with respect to simplification.⁶

In general, two valuations may assign different truth values to a compound proposition even when they agree on the truth values of its direct constituents. This is because the entailment condition imposes only minimal restrictions that the truth value of $\phi \wedge \psi$ must be no greater than those of $\phi$ and $\psi$, and the truth value of $\phi \vee \psi$ must be no less than them. Gödel logic’s min–max rule (1) is an extreme case in which these inequalities hold with equality. The next example illustrates a case where the truth value of a compound proposition is not pinned down by those of its direct constituents.

Example 2

Consider the set of states à la Savage (1972), exhausting all possible resolutions of uncertainty associated with the decision problem faced by individuals. Formally, we set up the state space $\Omega = \{\texttt {t},\texttt {f}\}^m$, with a generic element $\omega = (\omega _1,\ldots ,\omega _m)$ specifying a possible pattern of assigning Boolean variables to each atomic proposition. Then, we identify each proposition $\phi \in L$ with a subset of $\Omega$, called an event, as follows:

• i). Identify each atomic proposition $p_k$ with $\{\omega \in \Omega : \omega _k = \texttt {t}\}$.

• ii). Then, recursively identify compound propositions $\phi \wedge \psi$ and $\phi \vee \psi$ with $\phi \cap \psi$ and $\phi \cup \psi$, respectively.

In other words, each proposition is identified with the subset of states in $\Omega$ in which the proposition is assigned the Boolean variable $\texttt {t}$.⁷ Under this identification, an individual’s valuation $v:\phi \mapsto v(\phi )$ is given as a function that maps each event to a number in [0, 1], representing the individual’s belief regarding the occurrence of the states in which the proposition holds true. Note that the invariance property in Definition 1 is trivially satisfied due to the corresponding laws of set operations.

The entailment condition is satisfied if v aligns with the definition of non-additive probability measures (a.k.a. capacities) in Schmeidler (1989). To see this, recall that a function $v: 2^\Omega \rightarrow [0,1]$ is called a capacity if it satisfies monotonicity—$v(\phi _1) \le v(\phi _2)$ whenever $\phi _1 \subseteq \phi _2 \subseteq \Omega$—along with the normalization conditions $v(\emptyset )=0$ and $v(\Omega )=1$. Since $\phi _1 \cap \phi _2 \subseteq \phi _k \subseteq \phi _1 \cup \phi _2$ for each $k \in \{1,2\}$, the monotonicity of v implies

$$\begin{aligned} v(\phi _1 \cap \phi _2) \le \min \{v(\phi _1),v(\phi _2)\} \le \max \{v(\phi _1),v(\phi _2)\} \le v(\phi _1 \cup \phi _2), \end{aligned}$$

which implies entailment. This argument relies only on monotonicity but not on other properties of subjective beliefs. For instance, v need not be convex or concave, thereby Definition 1 permitting for a valuation to exhibit arbitrary attitudes toward uncertainty (Schmeidler 1989). Moreover, since v can be a probability measure over $\Omega$ (while generally not being additive), our framework aligns with the setting of probabilistic opinion pooling (Dietrich and List 2017a, b), which concerns the aggregation of individuals’ subjective beliefs over events into a collective probability measure. $\triangle$

To conclude this section, we provide a few possible interpretations of our fuzzy-logic framework, with special emphasis on the aforementioned (in)determinacy of the truth value of compound propositions from those of their constituents.

One possible interpretation is that multi-valued judgments represent the quantification of “outcomes” associated with each proposition in question. Taking an example from De Clippel and Eliaz (2015), suppose a recruiting committee in academia must decide whether to hire a candidate. Their hiring decision may depend on criteria such as “the candidate is distinguished in terms of research ability” and/or “the candidate is good at teaching,” etc. It is not hard to imagine that each committee member’s assessment of the candidate based on these criteria comes in degree. For instance, the candidate’s research ability can be quantified by the number of quality publications, and teaching ability may be reflected in past teaching evaluations with a multi-level grading scale. Moreover, the conclusion can also be multi-valued, as the university can offer different packages varying in compensation. In line with this interpretation, the fuzziness of judgments stems from the objective outcome associated with a proposition, rather than the subjective degree of agreement on the proposition.

Alternatively, we can interpret multi-valued judgments as representing the degree of subjective confidence that each individual holds regarding a proposition. In the example of Sect. 2, a judge may regard the statement “the defendant is guilty” as more likely true than false, yet hesitate to assert that it is absolutely true. This idea of confidence in judgements can be formalized, as in Example 2, by viewing valuations as probabilistic beliefs over propositions. These varying degrees of confidence can then be aggregated to form a multi-valued judgment on the conclusion, rather than a simple binary judgment. This interpretation aligns well with real-world decision-making processes, where certainty is often a matter of degree rather than an absolute.

To summarize, our valuations can be interpreted from both objective and subjective perspectives. From the objective perspective, multi-valued judgments represent quantifiable outcomes; e.g., in academic hiring decisions, research and teaching evaluations are mapped to compensation according to a well-defined university policy. Under this interpretation, it might be natural to assume that all committee members adhere to the same rule, such as the min–max rule (1), when deriving conclusions from premises, as it reflects a shared institutional framework. On the other hand, from the subjective perspective, valuations capture individuals’ subjective beliefs, allowing for greater flexibility in forming personal judgments on compound propositions. This flexibility mirrors the nature of subjective probabilities, where the probability assigned to the intersection or union of two events is not necessarily determined by the probabilities of the individual events.

Our model accommodates both perspectives and the analysis leaves open the choice of interpretation. Importantly, Theorem 1 establishes uniform bounds on possible outcomes that hold across all profiles of valuations, and these bounds remain valid even when the domain of an aggregation rule is restricted to certain types of valuations. Specifically, the result continues to hold when individuals are required to have a certain class of valuations—such as Gödelian valuations—while retaining flexibility in their judgments on atomic propositions. Also, if one prefers to interpret our result within standard binary logic, that remains entirely feasible, as Boolean logic emerges as a special case of our model.

4 Voting model

This section presents our voting model, based on the mathematical notions in Sect. 3.

4.1 Decision problem

We consider an aggregation problem with finitely many atomic propositions and a compound proposition generated by them. Let $L_0 = \{p_1,\ldots ,p_m\}$ be a finite set of atomic propositions with $m \ge 2$. As before, let L be the set of propositions obtained by closing $L_0$ under $\wedge$ and $\vee$. The conclusion, which is also called a decision problem, is given as a compound proposition $\phi \in L$. Without loss of generality, assume that all elements in $L_0$ are irreducible in the sense that every $p_k \in L_0$ occurs at least once in the construction of $\phi$. We say that a decision problem is conjunctive (resp. disjunctive) if only $\wedge$ (resp. $\vee$) is used to construct $\phi$. Let V denote the set of all valuations defined on L.

4.2 Aggregation rules

A committee consists of finitely many individuals $I = \{1,\ldots , n\}$, each holding an individual valuation $v^i \in V$. Let $v^i_0$ denote the restriction of $v^i$ to $L_0$. A generic profile of valuations is written as $\varvec{v} = (v^1,\ldots ,v^n) \in V^I$.

We are interested in the aggregation of each profile of individuals’ valuations $\varvec{v}$ into the collective judgment on the conclusion $\phi$. An aggregation rule is defined as a function $F: V^I \rightarrow T$ that maps each profile of valuations $\varvec{v}$ to the truth value that the committee assigns to the conclusion. Also, we consider a function $f: T^I \rightarrow T$, called a voting rule, that is used to aggregate individuals’ beliefs about each proposition. Throughout, we assume that a voting rule f is monotone, meaning $f(x^1,\ldots ,x^n) \ge f(y^1,\ldots ,y^n)$ whenever $x^i \ge y^i$ for all $i \in I$, which is a compelling property of voting rules used in practice.

Among all aggregation rules, the two most important ones are conclusion-based and atomic-premise-based rules, which were introduced in Sect. 2 through the median voting rule f. The following provides their general definitions.

Definition 2

Let $\phi$ be a decision problem, and let $f:T^I \rightarrow T$ be a monotone voting rule.

• i). A conclusion-based rule $C_f: V^I \rightarrow T$ is defined by $C_f(\varvec{v}) = f(v^1(\phi ), \ldots , v^n (\phi ))$.

• ii). An atomic-premise-based rule $A_f: V^I \rightarrow T$ is an aggregation rule defined as follows: First, set $v^*_0 (p) = f(v^1(p),\ldots , v^n(p))$ for each $p \in L_0$. Second, set $A_f(\varvec{v}) = v^* (\phi )$, where $v^*: V \rightarrow T$ is the unique extension of $v^*_0$ to L obtained through (1).

In other words, $C_f$ is implemented by directly running a vote on the conclusion by using a given voting rule f. In contrast, $A_f$ is implemented by holding a vote on each atomic proposition in $L_0$ to indirectly aggregate individuals’ valuations. Then, the collective judgment on the conclusion is entailed in view of these preliminary votes by applying the min–max rule (1). Here, we impose a restriction that the committee adheres to this specific rule, which will also be adopted when defining the general class of premise-based rules in Sect. 5, while each individual is free to have any valuation. Though this is a technical assumption needed to derive our main result, it holds plausibility in the context of collective decision-making because Gödel logic produces the most “conservative” rule that yields more intermediate judgments on compound propositions than others. In this sense, adopting this logic reflects a “precautionary principle” in decision-making: When faced with uncertainty and potential harm from a decision, the group, which holds ultimate decision-making authority, tends to be more cautious than any individual would be, thus tending to select a less extreme judgment.⁸

A voting rule f is fixed (yet arbitrary) in the subsequence analysis. Throughout, we maintain the assumption that the same voting rule f is used for the aggregation of beliefs about different propositions. Although this may not always be a sensible assumption in practice, it allows us to make meaningful comparisons across different premise-based rules.⁹ This assumption enables us to isolate the effect of the selection of an agenda on outcomes, separately from the effect of the selection of a voting rule.

4.3 Uniform ordering between $A_f$ and $C_f$

We present an auxiliary result, establishing the uniform ordering between $A_f$ and $C_f$ when a decision problem is either conjunctive or disjunctive. This result extends the “if part” of Theorem 1 of Miyashita (2021) to non-binary settings and will be used to deal with the “base step” in mathematical induction when proving Theorem 1.

Lemma 1

If a decision problem $\phi$ is conjunctive, then $A_f \ge C_f$. Conversely, if $\phi$ is disjunctive, then $C_f \ge A_f$.

Proof

Suppose that a decision problem is conjunctive. Fix any $\varvec{v} \in V^I$. Since only $\wedge$ occurs in $\phi$, the law of entailment implies $v^i(\phi ) \le \min \{v^i(p): p \in L_0\}$, which in turn implies $v^i (\phi ) \le v^i(p)$ for all $p \in L_0$. Then, by the monotonicity of f, we have

$$\begin{aligned} f(v^1 (\phi ), \ldots , v^n(\phi )) \le f(v^1 (p), \ldots , v^n(p)), \quad \forall p \in L_0. \end{aligned}$$

The left-hand side is equal to $C_f (\varvec{v})$. Moreover, because the decision problem is conjunctive and the committee is assumed to employ the min–max rule (1), the minimum of the right-hand side over all atomic propositions $p \in L_0$ is equal to $A_f(\varvec{v})$. Thus, we obtain $A_f (\varvec{v}) \ge C_f (\varvec{v})$. The second assertion of the lemma is proved symmetrically. $\square$

This lemma follows directly from the observation that $\wedge$ and $\vee$ respectively reflect the minimum and maximum judgments on premises into the judgment on the conclusion. Consequently, when $\phi$ is conjunctive (resp. disjunctive), we confirm that $C_f$ aggregates individuals’ premise-wise minimum (resp. maximum) judgments, which must be less (resp. greater) than the aggregation of individuals’ judgments on any atomic premise.

5 Agenda manipulation

In this section, we define the class of premise-based rules, introduce two novel rules within this class, and present our main result.

While the notion of premise-based rules is intuitive, the formal definition involves some cumbersome notation. To clarify the central idea, let us consider a concrete example by revisiting a scenario where an agenda setter chooses the set of propositions that individuals vote on. Given a decision problem $\phi$ in question, we assume our agenda setter is able to (i) rephrase $\phi$ into any equivalent proposition $\phi ' \in [\phi ]$, and (ii) determine a set of constituents of $\phi '$ on which committee members vote, subject to the constraint that the chosen set of propositions is sufficient to derive the judgment on $\phi$.

For concreteness, suppose that the decision problem is given as the following compound proposition:

$$\begin{aligned} \phi = ((p_1 \wedge p_2) \vee p_3) \wedge (p_4 \vee p_5), \end{aligned}$$

(2)

which is generated from five atomic propositions. Our agenda setter can, for instance, rephrase $\phi$ into an equivalent statement,

$$\begin{aligned} \phi ' = ((p_1 \vee p_3) \wedge (p_2 \vee p_3)) \wedge (p_4 \vee p_5), \end{aligned}$$

by using the distributive law. The agenda setter can then choose which constituents of $\phi$ or $\phi '$ to ask individuals to vote on to derive their collective judgement on the conclusion. The agenda setter’s possible choices include $\{\phi \}$, $\{p_1, p_2, p_3, p_4, p_5\}$, $\{p_1 \wedge p_2, p_3, p_4 \vee p_5\}$, and $\{p_1 \vee p_3, p_2 \vee p_3, p_4 \vee p_5\}$. In contrast, $\{p_1 \wedge p_2, p_3\}$ is not a valid agenda because the judgments on $p_1 \wedge p_2$ and $p_3$ are not sufficient since the judgment on $\phi$ remains indeterminate without the judgment on $p_4 \vee p_5$.

5.1 Premise-based rules

For each proposition $\phi \in L$, let $Q_{\phi } \subseteq L$ denote the set of all constituents of $\phi$. We then define $\mathcal {P}_{\phi }$ as the collection of all subsets $P \subseteq Q_\phi$ satisfying the following properties:

• a) $\phi$ belong to the set obtained by closing P under $\wedge$ and $\vee$.

• b) There exist no distinct $\psi , \psi ' \in P$ such that $\psi$ is a constituent of $\psi '$.

For example, if $\phi$ is given as in (2), then $Q_\phi$ and $\mathcal {P}_\phi$ are given as follows:

$$\begin{aligned}&Q_\phi = \bigl \{p_1,\, p_2,\, p_3,\, p_4,\, p_5,\, p_1 \wedge p_2,\, p_4 \vee p_5,\, (p_1 \wedge p_2) \vee p_3,\, \phi \bigr \}, \\&\mathcal {P}_\phi = \bigl \{ \{p_1,p_2,p_3,p_4,p_5\},\, \{p_1 \wedge p_2, p_3,p_4,p_5 \},\, \{ (p_1 \wedge p_2) \vee p_3,p_4,p_5 \} \\&\hspace{38pt} \{p_1,p_2,p_3,p_4 \vee p_5\},\, \{p_1 \wedge p_2, p_3,p_4 \vee p_5 \},\, \{ (p_1 \wedge p_2) \vee p_3,p_4 \vee p_5 \},\, \{\phi \} \bigr \}. \end{aligned}$$

The set of agendas for a decision problem $\phi$ is defined as the union of the collections $\mathcal {P}_{\phi '}$ across all $\phi '$ equivalent to $\phi$. We denote this set by

$$ \mathcal {P}_{[\phi ]} = \bigcup _{\phi ' \in [\phi ]} \mathcal {P}_{\phi '}. $$

A premise-based rule is then defined for each agenda P by extending the two-step procedure used in the atomic-premise-based rule. The formal definition is presented below.

Definition 3

An agenda for a decision problem $\phi$ is any element of the family $\mathcal {P}_{[\phi ]}$. An aggregation rule F is called a premise-based rule if there exists an agenda $P \in \mathcal {P}_{[\phi ]}$ such that F is defined as follows: First, set $v_P^*(\psi ) = f(v^1(\psi ),\ldots ,v^n(\psi ))$ for each $\psi \in P$. Second, set $F(\varvec{v}) = v^*(\phi )$, where $v^*: V \rightarrow T$ is the unique extension of $v_P^*$ to the closure of P obtained through (1).

A premise-based rule F is defined through an agenda $P \in \mathcal {P}_{[\phi ]}$, which specifies the set of propositions to be asked in preliminary votes. The aggregation procedure is implemented as follows: First, derive the collective judgment on each proposition $\psi$ in agenda P by holding a vote on $\psi$ using the pre-specified voting rule f:

Second, deduce the collective judgment on the conclusion $\phi$ in view of these preliminary votes by applying the min–max rule (1). In doing so, the property (a) from the construction of the agenda P ensures that the collective judgment on $\phi$ is logically determined once the committee has reached judgments on all propositions in P. The property (b) is used here simply to rule out redundant propositions in P.

Let $\mathcal {F}_{\phi }$ denote the set of all premise-based rules when $\phi$ is a decision problem faced by the committee. Clearly, $\mathcal {F}_{\phi } = \mathcal {F}_{\phi '}$ holds if $\phi$ and $\phi '$ are equivalent and therefore belong to the same element of $L/\!\simeq$, enabling us to write $\mathcal {F}_{[\phi ]} = \mathcal {F}_\phi = \mathcal {F}_{\phi '}$. Note that both conclusion-based and atomic-premise-based rules belong to $\mathcal {F}_{[\phi ]}$, as each of them is defined by agenda $\{\phi \}$ and $L_0$, respectively.

5.2 CNF-based and DNF-based Rules

A proposition $\phi$ is said to be in conjunctive normal form if it is written as a conjunction of one or more constituents,

$$\begin{aligned} \phi = \delta _1 \wedge \cdots \wedge \delta _k, \end{aligned}$$

where each of $\delta _1,\ldots ,\delta _k$, called a (disjunctive) clause, is a disjunction of one or more atomic propositions. Similarly, a proposition $\phi$ is said to be in disjunctive normal form if it is written as a disjunction of one or more constituents,

$$\begin{aligned} \phi = \gamma _1 \vee \cdots \vee \gamma _k, \end{aligned}$$

where each of $\gamma _1,\ldots ,\gamma _k$, called a (conjunctive) clause, is a conjunction of one or more atomic propositions.

The CNF (resp. DNF) of $\phi$, written as $\phi _\wedge$ (resp. $\phi _\vee$), refers to a proposition in conjunctive (resp. disjunctive) normal form that is equivalent to $\phi$. It is a well-known fact in propositional logic that any proposition $\phi$ admits CNF and DNF. For instance, the CNF and DNF of $\phi$ in (2) are given by

$$\begin{aligned} \phi _\wedge&= (p_1 \vee p_3) \wedge (p_2 \vee p_3) \wedge (p_4 \vee p_5), \\ \phi _\vee&= (p_1 \wedge p_2 \wedge p_4) \vee (p_1 \wedge p_2 \wedge p_5) \vee (p_3 \wedge p_4) \vee (p_3 \wedge p_5). \end{aligned}$$

Let $\widehat{P}_\phi$ (resp. ) denote the set of all disjunctive (resp. conjunctive) clauses appearing in the expression for $\phi _\wedge$ (resp. $\phi _\vee$). In the above example, we have

In general, a proposition can have multiple expressions for CNF and DNF, but the choice among them does not affect the definition of our aggregation rules. Also, by appropriately eliminating redundant clauses, $\widehat{P}_\phi$ and can be constructed to satisfy (b) of Sect. 5.1, ensuring that they belong to $\mathcal {P}_{[\phi ]}$.

Definition 4

Let $\phi$ be a decision problem.

• i). A CNF-based rule $\widehat{B}_f$ is a premise-based rule, defined through agenda $\widehat{P}_\phi$.

• ii). A DNF-based rule is a premise-based rule, defined through agenda .

The following lemma is immediate from definition.

Lemma 2

If a decision problem $\phi$ is conjunctive, then $A_f = \widehat{B}_f$ and . Conversely, if $\phi$ is disjunctive, then $C_f = \widehat{B}_f$ and .

Proof

Let $\phi$ be conjunctive. Since $\phi$ is in CNF with $\widehat{P}_\phi = \{p_1,\ldots ,p_m\}$, we have $A_f = \widehat{B}_f$. Moreover, since $\phi$ is in DNF with a single conjunctive clause $\phi$ itself, meaning that , we have . The second assertion is proved symmetrically. $\square$

We now present our main result, showing that every premise-based rule is uniformly dominated by the CNF-based rule and uniformly dominates the DNF-based rule with respect to the total order over the set of truth values. In other words, the voting outcome that can arise under any agenda selection must lie between the outcomes of $\widehat{B}_f$ and , irrespective of the profile of individuals’ valuations.

Theorem 1

For any decision problem $\phi$ and any monotone voting rule f, it holds that for all $F \in \mathcal {F}_{[\phi ]}$.

Fig. 2

The ordering among the four procedures is illustrated for a conjunctive decision problem (left), neither conjunctive nor disjunctive (center), and disjunctive (right)

The findings of Theorem 1 are illustrated in Fig. 2, where the vertical direction captures a uniform ranking of four aggregation rules: conclusion-based, atomic-premise-based, CNF-based, and DNF-based. The central diagram corresponds to the case of a general decision problem, with the CNF-based rule positioned at the top and the DNF-based rule at the bottom. In this case, while no uniform ranking exists between $A_f$ and $C_f$, all premise-based rules fall between $\widehat{B}_f$ and . For a purely conjunctive or disjunctive problem, the diagram simplifies, as Lemma 2 shows that each of $A_f$ and $C_f$ coincides with $\widehat{B}_f$ or . As a result, the four aggregation rules are linearly ordered, as shown in the left and right diagrams.

In light of the issue of agenda manipulation, Theorem 1 delineates the extent to which the agenda setter can influence voting outcomes through agenda selection. Specifically, since $\widehat{B}_f$ uniformly dominates and is uniformly dominated by all other premise-based rules, they determine the full range of possible outcomes that can arise under any agenda selection, irrespective of the profile of individuals’ judgments. Moreover, the theorem offers guidance on agenda choices for an agenda setter who prefers extreme outcomes: To maximize outcomes, one should conduct separate votes on each set of conjunctively connected propositions and a joint vote on each set of disjunctively connected propositions. Conversely, to minimize outcomes, this structure should be reversed.¹⁰

5.3 Extension to propositions with negation

Our analysis is restricted to propositions constructed from atomic propositions using only conjunctions and disjunctions. This restriction ensures that the conclusion depends positively on each preliminary issue. A natural question is what happens when we allow the use of negation ( $\lnot$), so that the conclusion may take a form such as $\lnot p_1 \vee p_2$, which corresponds to the material implication “ $p_1 \rightarrow p_2$.”

The limitation of excluding negation is not essential if we can treat the negation of an atomic proposition as a new atomic proposition. For instance, when the conclusion is given as $\lnot p_1 \vee p_2$, one might define $\tilde{p}_1 = \lnot p_1$ and regard $\tilde{p}_1$ as atomic. With this replacement, our analysis could appear to go through simply by replacing $p_1$ with $\tilde{p}_1$. However, this replacement is not necessarily innocuous since our framework assumes that the same aggregation rule f is applied to all propositions. Suppose, for example, that f is the 2/3-majority rule and is initially applied to aggregate individual judgments on $p_1$ and $p_2$, after which the committee deduces its judgment on $\lnot p_1 \vee p_2$. If we instead replace $p_1$ with $\tilde{p}_1 = \lnot p_1$ and apply f directly to $\tilde{p}_1$, then the judgment on $\tilde{p}_1$ will reflect only 1/3 support when $p_1$ is accepted by 2/3 of the committee. In effect, we are imposing a different aggregation threshold on $\tilde{p}_1$ than the one applied on $p_1$ and $p_2$, violating the assumption that the same f is used across all propositions.

This inconsistency disappears if the aggregation rule f is neutral, meaning it treats acceptance and rejection symmetrically. For instance, the simple majority rule with an odd number of voters is neutral in this sense. As shown in Theorem 4 of Miyashita (2021), neutrality allows an extension of Lemma 1 to include negation, at least when truth values are binary.¹¹ Consequently, under a neutral voting rule, we can incorporate negation by replacing an atomic proposition with its negation wherever necessary. That said, this approach does not extend in general to non-neutral aggregation rules. Moreover, complications arise when the conclusion involves both a proposition and its negation simultaneously, for example in expressions such as $(p_1 \wedge p_2) \vee (\lnot p_2 \vee p_3)$. Extending the current framework to accommodate such logical forms is left for future research.

Appendices

Proof of Theorem 1

In this proof, we denote the CNF-based and DNF-based rules by $\widehat{B}_{\phi }$ and , respectively, to emphasize their dependence on the underlying decision problem while suppressing their dependence on the voting rule f. We prove the theorem by induction on the degree $d(\phi )$ of $\phi$. The complication is that a premise-based rule F appearing in the theorem may be defined from an agenda consisting of constituents of some $\phi '$ equivalent to $\phi$, while the degree of $\phi '$ need not coincide with that of $\phi$.

To address this issue, we introduce the set $\mathcal {G}_{\phi }$ of all premise-based rules whose agendas belong to $\mathcal {P}_{\phi }$, rather than $\mathcal {P}_{[\phi ]}$. In other words, each $G \in \mathcal {G}_{\phi }$ is defined from an agenda consisting solely of constituents of $\phi$. Note that $\mathcal {G}_{\phi } \subseteq \mathcal {F}_{[\phi ]}$ since $\mathcal {P}_{\phi } \subseteq \mathcal {P}_{[\phi ]}$. We will prove the following weaker version of the claim in the theorem:

(3)

In fact, once (3) is established for every proposition $\phi$ of arbitrary degree, the proof of Theorem 1 follows, because $\widehat{B}_{\phi } = \widehat{B}_{\phi '}$ and for all $\phi '$ equivalent to $\phi$. Thus, it suffices to show that (3) holds for every $\phi \in L$.

Base step. Suppose that $d(\phi ) = 2$. Then, since $\phi$ must be either $p_1 \wedge p_2$ or $p_1 \vee p_2$, we have $\mathcal {P}_{\phi } = \{\{\phi \}, \{p_1, p_2\}\}$. This implies that $\mathcal {G}_\phi$ consists solely of conclusion-based and atomic-premise-based rules. Hence, by Lemma 1 and Lemma 2, it follows that for every $G \in \mathcal {G}_{\phi }$.

Inductive step. Given any integer $k \ge 2$, suppose that (3) is satisfied for any proposition $\psi \in L$ with $d(\psi ) \le k-1$. Take any proposition $\phi \in L$ with $d (\phi ) = k$. By construction, there exist $\psi _1,\psi _2 \in L$ such that $\phi = \psi _1 * \psi _2$, where $* \in \{\wedge , \vee \}$. Since the subsequent arguments will be symmetric, assume without loss of generality that $\phi = \psi _1 \wedge \psi _2$. Note that $d(\psi _1)$ and $d(\psi _2)$ are at most $k-1$, and thus the inductive assumption (3) applies to $\psi _1$ and $\psi _2$. The next claim establishes one inequality in (3).

Claim 1

$G \le \widehat{B}_\phi$ for any $G \in \mathcal {G}_\phi$.

Proof

Fix any $\varvec{v} \in V^I$. Since $\phi = \psi _1 \wedge \psi _2$ and $\widehat{P}_{\psi _j}$ is the set of all disjunctive clauses in the CNF of $\psi _j$ for each $j \in \{1,2\}$, the associative law implies that

Let $\widehat{P}_\phi$ be the subset of $\widehat{P}_{\psi _1} \cup \widehat{P}_{\psi _2}$ that collects all $\delta$ such that there exists no other $\delta ' \in \widehat{P}_{\psi _1} \cup \widehat{P}_{\psi _2}$ of which $\delta$ is a constituent. Then $\widehat{P}_\phi$ belongs to $\mathcal {P}_\phi$, and the CNF of $\phi$ reduces to $\bigwedge \{\delta : \delta \in \widehat{P}_\phi \}$. Under the CNF-based rule, individuals vote on each premise $\delta \in \widehat{P}_\phi$, and the collective judgment on $\phi$ is obtained by applying the min–max rule (1). Therefore, the value of $\widehat{B}_\phi (\varvec{v})$ is

(4)

Fix any $G \in \mathcal {G}_\phi$, and let $P \in \mathcal {P}_\phi$ be an agenda defining G.

First, assume that P is a singleton set. By (a) from the construction of P, this necessitates that $P = \{\phi \}$, meaning G is the conclusion-based rule. Since $\phi \simeq \bigwedge \{\delta : \delta \in \widehat{P}_\phi \}$, it follows from the law of entailment that $v^i (\phi ) \le v^i(\delta )$ for all $\delta \in \widehat{P}_\phi$ and $i \in I$. By this and the monotonicity of f, we obtain

where the last equality is due to (4).

Next, assume that $|P| \ge 2$. Since P is included in the set $Q_\phi$ of all constituents of $\phi$, it follows from (a) that we can partition P into two subsets $P_1$ and $P_2$, each consisting of constituents of $\psi _1$ and $\psi _2$, respectively. Then, for each $j \in \{1,2\}$, we have $P_j \in \mathcal {P}_{\psi _j}$, which defines a premise-based rule $G_j$ for a decision problem $\psi _j$. In particular, since $\phi = \psi _1 \wedge \psi _2$, we have $G(\varvec{v}) = \min \{G_1(\varvec{v}), G_2(\varvec{v})\}$ by applying the min–max rule (1). Moreover, by the inductive hypothesis, we have $G_j(\varvec{v}) \le \widehat{B}_{\psi _j}(\varvec{v})$ for each $j \in \{1,2\}$. Putting these together, it follows that

where the second line holds by the construction of $\widehat{P}_\phi$. $\square$

The next claim establishes the other inequality in (3). The proof is similar to the last claim but not entirely symmetric since we assume $\phi = \psi _1 \wedge \psi _2$.

Claim 2

for any $G \in \mathcal {G}_\phi$.

Proof

Fix any $\varvec{v} \in V^I$. Since $\phi = \psi _1 \wedge \psi _2$ and is the set of all conjunctive clauses in the DNF of $\psi _j$ for each $j \in \{1,2\}$, the distributive law implies that

Let be the subset of that collects all $\gamma _1 \wedge \gamma _2$ such that there exists no other that is a constituent of $\gamma _1 \wedge \gamma _2$. Then belongs to $\mathcal {P}_\phi$, and the DNF of $\phi$ reduces to . Under the DNF-based rule, individuals vote on each premise , and the collective judgment on $\phi$ is obtained by applying the min–max rule (1). Therefore, the value of is

(5)

Fix any $G \in \mathcal {G}_\phi$, and let $P \in \mathcal {P}_\phi$ be an agenda defining G.

Similarly to the proof of Claim 1, we firstly consider the case where G is the conclusion-based rule. Since , the law of entailment implies that $v^i(\phi ) \ge \phi (\gamma )$ for all and $i \in I$. Thus, it follows that

where the last equality is due to (5).

Next, assume that $|P| \ge 2$. As before, we partition P into $P_1$ and $P_2$, and let $G_1$ and $G_2$ be premise-based rules for $\psi _1$ and $\psi _2$, defined through $P_1$ and $P_2$, respectively. By $\phi = \psi _1 \wedge \psi _2$ and the inductive hypothesis, it follows that

(6)

We want to show that the above right-hand side is greater than (5). To this end, consider any proposition $\gamma = \gamma _1 \wedge \gamma _2$ with for $j \in \{1,2\}$. By the law of entailment, we have $v^i(\gamma ) \le v^i (\gamma _1)$ for all $i \in I$, from which the monotonicity of f yields

(7)

Similarly, we have

(8)

Combining (7) and (8), it follows that

In particular, since this inequality is satisfied for any , we confirm that the right-hand side of (6) is greater than (5), as desired. $\square$

By the preceding two claims, we obtain for any $G \in \mathcal {P}_\phi$ and proposition $\phi$ of degree k. Therefore, by the principle of mathematical induction, (3) is satisfied for any proposition $\phi \in L$, which concludes the proof of Theorem 1. Q.E.D.

Gödel logic and t-norm fuzzy logics

The t-norm fuzzy logics constitute a commonly considered class of multi-valued logics in which the truth value of a compound proposition is recursively determined from the truth values of its constituents.¹² Such logics are defined through two binary operators ${{\,\mathrm{\texttt {l}}\,}},{{\,\mathrm{\texttt {u}}\,}}: T^2 \rightarrow T$ that satisfy the following axioms for all $x,y,z \in T$:

• Commutation: ${{\,\mathrm{\texttt {l}}\,}}(x,y) = {{\,\mathrm{\texttt {l}}\,}}(y,x)$ and ${{\,\mathrm{\texttt {u}}\,}}(x,y) = {{\,\mathrm{\texttt {u}}\,}}(y,x)$.

• Association: ${{\,\mathrm{\texttt {l}}\,}}(x,{{\,\mathrm{\texttt {l}}\,}}(y,z)) = {{\,\mathrm{\texttt {l}}\,}}({{\,\mathrm{\texttt {l}}\,}}(x,y),z)$ and ${{\,\mathrm{\texttt {u}}\,}}(x,{{\,\mathrm{\texttt {u}}\,}}(y,z)) = {{\,\mathrm{\texttt {u}}\,}}({{\,\mathrm{\texttt {u}}\,}}(x,y),z)$.

• Monotonicity: if $x \ge z$ then ${{\,\mathrm{\texttt {l}}\,}}(x,y) \ge {{\,\mathrm{\texttt {l}}\,}}(z,y)$ and ${{\,\mathrm{\texttt {u}}\,}}(x,y) \ge {{\,\mathrm{\texttt {u}}\,}}(z,y)$.

• Identity: ${{\,\mathrm{\texttt {l}}\,}}(x,1) = x$ and ${{\,\mathrm{\texttt {u}}\,}}(x,0) = x$.

A t-norm valuation is a function $v:L \rightarrow T$ such that: (i) given any function $v_0: L_0 \rightarrow T$, set $v(p) = v_0 (p)$ for each atomic proposition $p \in L_0$, and (ii) the value assigned to a compound proposition is recursively determined as

The operators ${{\,\mathrm{\texttt {l}}\,}}$ and ${{\,\mathrm{\texttt {u}}\,}}$ are interpreted as providing lower and upper bounds, respectively, on the truth value of each compound proposition based on those of its direct constituents, and the above axioms serve as minimal requirements for these operators to meaningfully represent such bounds. For example, Gödel logic is a canonical example of t-norm fuzzy logics, with ${{\,\mathrm{\texttt {l}}\,}}(x,y) = \min \{x,y\}$ and ${{\,\mathrm{\texttt {u}}\,}}(x,y) = \max \{x,y\}$.

While these axioms imply that every t-norm valuation satisfies entailment,¹³ a t-norm valuation does not necessarily satisfy the invariance requirements imposed in Definition 1. Specifically, we additionally require invariance with respect to simplification, which is not a standard axiom in t-norm fuzzy logics:

• Simplification: ${{\,\mathrm{\texttt {l}}\,}}(x,{{\,\mathrm{\texttt {l}}\,}}(x,y)) = {{\,\mathrm{\texttt {l}}\,}}(x,y)$ and ${{\,\mathrm{\texttt {u}}\,}}(x,{{\,\mathrm{\texttt {u}}\,}}(x,y)) = {{\,\mathrm{\texttt {u}}\,}}(x,y)$.

This condition turns out to be extremely restrictive. As shown in the next two lemmas,¹⁴ requiring simplification forces the operators ${{\,\mathrm{\texttt {l}}\,}}$ and ${{\,\mathrm{\texttt {u}}\,}}$ to be idempotent, meaning ${{\,\mathrm{\texttt {l}}\,}}(x,x)=x$ and ${{\,\mathrm{\texttt {u}}\,}}(x,x)=x$ for all x, which in turn forces the operators to coincide with the Gödelian min–max operators. Hence, within the class of t-norm fuzzy logics, Gödel logic is the only one compatible with Definition 1.

Lemma 3

If ${{\,\mathrm{\texttt {l}}\,}}$ and ${{\,\mathrm{\texttt {u}}\,}}$ satisfy identity and simplification, then they satisfy idempotence.

Proof

Taking $y=1$, simplification implies ${{\,\mathrm{\texttt {l}}\,}}(x,{{\,\mathrm{\texttt {l}}\,}}(x,1)) = {{\,\mathrm{\texttt {l}}\,}}(x,1)$, while identity implies ${{\,\mathrm{\texttt {l}}\,}}(x,1) = x$. Thus it follows that ${{\,\mathrm{\texttt {l}}\,}}(x,x) = {{\,\mathrm{\texttt {l}}\,}}(x,{{\,\mathrm{\texttt {l}}\,}}(x,1)) = {{\,\mathrm{\texttt {l}}\,}}(x,1) = x$. The proof for ${{\,\mathrm{\texttt {u}}\,}}$ is similarly done by taking $y=0$. $\square$

Lemma 4

If ${{\,\mathrm{\texttt {l}}\,}}$ and ${{\,\mathrm{\texttt {u}}\,}}$ satisfy commutation, monotonicity, identity, and idempotence, then ${{\,\mathrm{\texttt {l}}\,}}(x,y)=\min \{x,y\}$ and ${{\,\mathrm{\texttt {u}}\,}}(x,y)=\max \{x,y\}$ for all $x,y \in T$.

Proof

Footnote 13 establishes ${{\,\mathrm{\texttt {l}}\,}}(x,y)\le \min \{x,y\}$. To prove the reverse inequality, assume without loss of generality that $x\le y$. Idempotence and monotonicity imply $x = {{\,\mathrm{\texttt {l}}\,}}(x,x) \le {{\,\mathrm{\texttt {l}}\,}}(x,y)$. Since $x = \min \{x,y\}$, we obtain $\min \{x,y\} \le {{\,\mathrm{\texttt {l}}\,}}(x,y)$, establishing ${{\,\mathrm{\texttt {l}}\,}}(x,y)=\min \{x,y\}$. The argument for ${{\,\mathrm{\texttt {u}}\,}}$ is analogous. $\square$