1. Introduction
The rationality of collective propositional attitudes — such as beliefs, desires, and intentions — and of collective agency is a central issue in social choice theory and mathematical economics and it has also become an important topic in knowledge representation and in the foundations of multiagent systems. A collective propositional attitude is, generally speaking, a propositional attitude that is ascribed to a collective entity. A map of the most salient notions of collective attitudes was proposed by Christian List [1], who distinguished between three kinds of collective attitudes: aggregate, common, and corporate attitudes. Corporate attitudes presuppose that the collectivity to which they are ascribed is an agent in its own right, an agent who is distinguished from the mere individuals that compose the collectivity. We shall focus, for the main part of this work, on the other two kinds of collective attitudes. Common attitudes are ascribed to collectivities by requiring that every member of the group share the same attitude. Common attitudes have been presupposed for instance by the debate on joint action and collective intentionality [2, 1, 3]. In this view, possible divergences among the attitudes of the members of the group are excluded and the problem is to understand, for instance, whether an intention that is shared among the members of the group is indeed a collective intention of the group itself.
By contrast, aggregative attitudes do not presuppose that every member of the group share the same attitude. In this case, a propositional attitude can be ascribed to the collectivity by solving the possible disagreement by means of a voting procedure such as the majority rule. This view is appealing, since it seems to be capable of accounting for the perspective endowed by common attitudes, for which unanimity is demanded, but also for a number of situations in which it is reasonable to define a collective attitude without assuming that all the members of the group share the same attitudes. For instance, when we model the decisions of parliaments, organizations, or committees, we aim to ascribe collective decisions starting from a situation of initial disagreement. Observe that modelling an aggregative view of collective attitudes is conceptually simpler than modelling common group attitudes in a number of respects. Firstly, we do not need to assume joint intentionality nor a shared goal among the group of agents. By the definition of a majoritarian group, we are already assuming that individuals do have different goals and intentions [1]. For that reason, we are terming the attitudes of the group by group or collective attitudes and not by joint attitudes. For an aggregative view of collective attitudes, shared intentionality and shared goals do not need to enter the model for defining what a group attitude is [3].
Besides being descriptively adequate to a number of scenarios, nonunanimous collective attitudes are important also for defining, representing, and assessing group information. Consider the following situations involving artificial agents. Suppose three sensors have been placed in different locations of a room and they are designed to trigger a fire alarm in the case they detect smoke. By viewing the three sensors as a group, we may investigate what are the conditions that define the group action, in this case, “trigger the alarm”, and its dependency on the group “beliefs”. By forcing unanimity —by viewing group attitudes as common attitudes— we are assuming that the three sensors as a group trigger the alarm only in the case they all agree in detecting smoke. However, a unanimous view of group attitudes may lead to lose important bits of information: if the sensors disagree, the alarm is simply not triggered, even if the disagreement may be caused, for instance, by the fact that one of the three sensors is in a location that has not been reached by the smoke yet. Thus, there are reasons for abandoning common attitudes in modelling information merging [4]. An aggregative view provides the formal means to tailor the concept of collective information to the specific scenario, by selecting the appropriate aggregation method.
Although an aggregative view of group attitudes is desirable, several results in social choice theory and judgment aggregation show that many important aggregation procedures are not capable of guaranteeing a rational outcome [5]. A crucial example is the majority rule, that does not preserve the consistency of individual judgments, as the intriguing discursive dilemmas show [6]. As usual in the (beliefdesireintention) BDI approach to agency, at least a modicum of rationality has to be presupposed in order to define an agent. An agent cannot hold (synchronically) inconsistent attitudes, such as commitments or beliefs. Therefore, when the outcome of an aggregation procedure is inconsistent, as in the case of the majority rule, we simply cannot construe a majoritarian group as an agent. The solution that has been mainly pursued in the literature on judgment aggregation and social choice theory is to give up procedures such as the majority rule and to design aggregation procedures that guarantee consistency [5].
In this paper, I am interested in pursuing a different strategy: I investigate whether there is a viable notion of rationality with respect to which the outcome of a majoritarian aggregation can be deemed rational. The motivation is that, in several real scenarios, agents actually use the majority rule to settle disagreement. Besides that, the majority rule has a number of desirable features such as it is simple to understand and implement, preference aggregation is nonmanipulable (when consistent) [7], it has been associated to a suggestive epistemic virtue justified by the Condorcet’s jury theorem [8].
On a close inspection, as we shall see, the inconsistency of the majority rule is intertwined with the principles of classical logic. In [9, 10], a possibility result for the majority rule has been provided by means of a nonclassical logic, namely by means of linear logic [11, 12]. I will build on that in order to develop a number of logics for which majoritarian collective attitudes —that is, attitudes that are aggregatively defined by means of the majority rule— are consistent.
In principle, this proposal can be applied to any propositional attitudes such as beliefs, desires, intentions, or commitments. For the sake of example, I focus here on three types of propositional attitudes: actions, beliefs and obligations. Moreover, the proposed modelling can in principle be instantiated with a number of aggregation procedures, while I shall focus here on the significant case of the majority rule.
The methodology of this paper relies on two approaches. Firstly, at the propositional level, we use an important family of nonclassical logics, namely substructural logics —of which linear logic is an important example— since they enable a very refined analysis of the collective inconsistency under the majority rule. Secondly, the modelling of individual and collective propositional attitudes makes use of minimal modal logics. This is motivated by the idea of exploring a number of basic principles that govern the reasoning about individual and collective propositional attitudes. A presentation of minimal, or nonnormal, modalities is [13]. Nonnormal modal logics of actions, beliefs, and obligation that are based on classical propositional logic are extensively studied and discussed in the literature. An exhaustive overview is out of the scope of this paper, I shall only mention a few directly related work in the subsequent sections.
The technical contribution of this paper summarised as follows. For the logical part, I define minimal modal logics of actions, beliefs, and obligations for individual and collective agents based on a significant fragment of substructural logics. I present the Hilbert systems for this logics, their semantics, and I show soundness and completeness. Moreover, I show that the majority rule preserves consistency for interesting fragments of such logics.
The conceptual contribution of this paper consists in the application of this logics to propose a consistent modelling of collective attitudes that may serve to the foundation of the status of collective agents.
A closely related approach is [14], the authors use judgment aggregation for modelling group attitudes by relying on (classical) modal logics of beliefs and goals. The main difference with respect to the present work is that their treatment applies to aggregation procedures that are known to guarantee consistency, e.g. the premise based procedure [5], whereas I am interested in approaching the problematic case of the plain majority rule. Another closely related work is the generalisation of the theory of judgment aggregation to capture general propositional attitudes [15]. The present work can be considered a contribution to the study of collective attitudes based on nonclassical propositional logics and on weak modalities.
The remainder of this paper is organised as follows. Section 2 discusses a famous case of doctrinal paradox [16] to highlight the problems of a consistent logical modelling of collective attitudes. Section 3 presents the framework of Judgment Aggregation [5]. In particular, we shall see how the majority rule fails in preserving the notion of consistency based on classical logic. Section 4 introduces and motivates a family of nonclassical logics, i.e. substructural logics, as a viable alternative to classical logic to define a notion of consistency that can be preserved by the majority rule at the collective level. Section 5, then, presents a possibility result for the majority rule with respect to substructural logics. Section 6 introduces the basics of the modal logics for modelling individual and collective attitudes. In Section 7, I introduce the proposal for modelling collective attitudes and I exemplify its applications to the doctrinal paradox. In particular, we shall spell out the individual and collective attitudes actually involved in the doctrinal paradox (i.e. beliefs, actions, and obligations). Finally, I will approach the treatment of corporate attitudes within the proposed framework, by discussing the nature of the corporate agent who is supposed to be the bearer of the collective attitudes. Section 8 concludes. The technical treatment of the logics introduced in this paper is presented in the Appendices.
2. The doctrinal paradox
To illustrate the problems of a theory of collective attitudes, we present the famous case of doctrinal paradox that actually emerged in the deliberative practice of the U.S. Supreme Court, namely the case of Arizona v Fulminante [16]. This case originally motivated the study of judgment aggregation, as well as an important debate on the legitimacy of collective decisions, cf. [16] and [6, 17]. The Court had to decide whether to revise a trial on the ground of the alleged coercion of the defendant’s confession. The legal doctrine prescribes that a trial must be revised if both the the confession was coerced and the confession affected the outcome of the trial. At the mere level of propositional logic, we formalise the propositions involved as follows: for “the confession was coerced”, for “the confession affected the outcome of the trial”, and for “the trial must be revised”. The legal doctrine is then captured by the formula of classical propositional logic . We only report the votes of three out of the nine Justices of the Supreme Court and we label them by , , and . Individual votes are faithfully exemplified by the following profile.
1  1  1  1  1  
1  0  0  1  0  
0  0  1  1  0  
maj.  1  0  1  1  0 
By defining the collective attitudes that we ascribe to the Court in an aggregative manner, that is by voting by majority, we obtain the following set of propositions: is accepted (because of agent 1 and 2), is accepted (because of 1 and 3), the legal doctrine is accepted (because it is unanimously accepted), and is rejected. By viewing the rejection of as the acceptance of , as usual in this setting, we can easily see that the set of collective attitudes is inconsistent. That means that, although each individual set of accepted propositions is consistent, the majority rule does not preserve consistency at the collective level.
List and Pettit [6] argued that the doctrinal paradox exhibits, besides the irrational outcome, a dilemma between a premisebased and a conclusionbased reading of the majoritarian aggregation. The premisebased reading let the individuals vote on the so called premises and , then it collectively infers and finally concludes, by , that is the case. By contrast, the conclusionbased reading let each individual draw the conclusions by reasoning autonomously on the propositions at issue, then it aggregates the sole conclusions, here , and in this case is rejected.
For the present applications, three points are worth noticing. Firstly, the notion of consistency that is not preserved by the majority rule is the notion of consistency defined with respect to classical logic. It is therefore interesting to investigate whether there are meaningful notions of nonclassical consistency that are preserved by the majority rule.
Secondly, the distinction between the premise and the conclusion based readings shows that in the doctrinal paradox there are inferences that are performed at the individual level and inferences that are performed at the collective level. In the premisebased reasoning, once , , and are accepted, the inference that draws is performed only by a minority of individuals: indeed, this reasoning step is performed only at the collective level on the propositions that have been accepted by majority. It is then interesting to investigate whether it is possible to make distinction between inferences performed at the individual level and inferences performed at the collective level visible, by means of the logical modelling.
Finally, the doctrinal paradox involves a number of individual and collective propositional attitudes such as individual and collective beliefs (concerning whether the confession was coerced) obligations (e.g. the legal doctrine), and actions (e.g. the revision of the trial). For that reason, a proper treatment of the reasoning principles for those attitudes shall be provided. In the remainder of this paper, I will develop a model of collective attitudes that addresses the previous points.
3. A model of Judgment Aggregation
We introduce the basic definitions of the judgement aggregation (JA) setting [5, 18], which provides the formal understanding of the aggregative view of collective attitudes [1]. I slightly rephrase the definitions for the present application, by referring to a variable logic which will be instantiated by a number of logics. Moreover, the definition are presented “syntactically”, by referring to the derivability relation of a given logic , rather than “semantically”, as it is usual in JA, in order to focus on the inference principles. In case is classical logic, the definitions match the standard presentation of JA [5].
Let be a (finite) set of agents. Assume that and, as usual, and
is odd. An
agenda is a (finite) set of propositions in the language of a given logic that is closed under complements, i.e. nondouble negations, cf. [18]. Moreover, we assume that the agenda does not contain tautologies or contradictions, cf. [5].A judgement set is a subset of such that is (wrt ) consistent (), complete (for all , or ) and deductively closed (if and , ). Denote by the set of all judgement sets on . A profile of judgements sets J
is a vector
, where . An aggregator is then a function . The codomain of is the powerset , therefore we are admitting possibly inconsistent judgments sets. Let , the majority rule is the function such that .In JA, the collective set is also assumed to be consistent, complete, and deductively closed wrt. . We say that the aggregator is consistent, complete, or deductively closed, if is, for every J. The preservation of consistency and completeness with respect to classical logic under any profile defines the standard notion of collective rationality in JA. The seminal result in JA can be phrased by saying that the majority rule is not collectively rational [6]. That means that there exists an agenda and a profile of judgment sets such that is not consistent. A significant example is the doctrinal paradox that we have previously encountered. The case of the majority rule generalises to a theorem that applies to any aggregation procedure that satisfies a number of desirable conditions, cf [5]; that is, many worthy aggregation procedures fail in preserving consistency.
We formally define the distinction between the premisebased reading and the conclusionbased reading by defining two aggregation procedures. Suppose that the agenda is partitioned into two disjoint sets , the premises of the agenda, and , its conclusions. The premisebased procedure is a function that firstly aggregates the premises by majority, obtaining the set of accepted premises, then it infers from the conclusions ; i.e. the output of is . The conclusionbased procedure simply votes by majority on the conclusions , i.e. those formulas that are inferred individually.
4. Background on Substructural Logics
We place our analysis in the realm of substructural logics [19, 20] to show that the majority rule is in this case quite surprisingly consistent. Substructural logics are a family of logics that reject the (global) validity of the classical structural principle of contraction (C) and weakening (W).^{1}^{1}1The principles are labelled structural as they correspond to the structural rules of the classical sequent calculus [19]. Another structural principle is the exchange, which entails the commutativity of the conjunction and of the disjunction. This principles are captured in classical logic by the axioms (W) and (C) . Rejecting (W) amounts to preventing the monotonicity of the entailment, while rejecting (C) blocks the possibility of identifying several copies of the assumptions when drawing inferences.
An important family of substructural logics is that of relevant logics which are traditionally motivated in philosophical logic by the aim of capturing informative inferences [21, 22]. Another important family of substructural logics is that of linear logics, which are motivated mainly as logics of computation [12, 23]. The linear implication, denoted by , keeps track of the amount of formulas actually used in the deduction: for instance, modus ponens is valid only if the right amount of assumptions is given, so that . Resourcesensitivity is thus achieved by rejecting (C) and preventing the identification of two occurrences of . The resourcesensitivity of this logics has been used in applications to a number of topics in knowledge representation and multiagent systems such as planning [24], preference representation and resource allocation [25, 26, 27], social choice theory [9], action modelling [28, 29].
A crucial observation that motivates the use of substructural logic for modelling collective attitudes is that, by rejecting (W) and (C), we are led to split the classical connectives into two classes: the multiplicatives and the additives.^{2}^{2}2In the tradition of relevant logics, the distinction between multiplicatives and additives corresponds, respectively, to the distinction between intensional and extensional connectives, cf. [19]. For instance, the classical conjunction splits into two distinct operators: the multiplicative
(“tensor”) and the additive
(“with”) with distinct operational meaning [11, 12]. The inferential behaviour of the two conjunctions in fact differs: one can prove that and that , that is, the tensor case demands the combination of hypotheses , whereas the with case requires that the same hypothesis has been used to infer and . In presence of (W) and (C), and are provably equivalent, therefore they are indiscernible in classical logic. For this reason, the classical conjunction is more powerful and permits both inferential patterns. Analogous distinction can be made for disjunctions.For the purpose of this paper, the distinction between multiplicatives and additives is curiously related to an important distinction between the truth makers of a proposition in an aggregative setting: we will see that a multiplicative formula is made true by two possibly different coalitions of agents, one that supports and one that supports , whereas shall be made true by a single coalition of agents that supports both and , cf. [9]. This distinction is crucial for the preservation of consistency under the majority rule and we shall use it for distinguishing the reasoning steps that are performed at the individual level from those performed at the collective level.
The basic logic that we are going to use throughout this paper is the intuitionistic version of multiplicative additive linear logic.^{3}^{3}3Linear logic is capable of retrieving the structural rules in a controlled manner by means of the exponential operators [11, 12]. Here we confine ourselves to the fragment of multiplicative additive intuitionistic linear logic (exponentialfree). We also leave a proper comparison with the families of substructural and relevant logics for future work. The motivation for using the intuitionistic version is mainly technical: intuitionistic linear logic allows for an easy Kripkelike semantics which in turns provides an easy way to define the semantics of modalities. In principle, the analysis of collective attitudes of this article can be performed by means of the classical version of linear logic. In particular, intuitionistic and classical logic do no differ with respect to the aggregation of propositions under the majority rule [30]. For this reason, endorsing the intuitionistic restriction does not entail a substantial loss of generality of the approach in this case. We preferred, for the sake of simplification, to stick to the intuitionistic version of linear logic, leaving a proper definition of the semantics of modalities for classical linear logic for a future dedicated work.^{4}^{4}4The semantics of classical linear logic is provided by means of phase spaces, [31]. In order to define modalities for classical linear logic, we have to extend phase spaces by introducing neighbourhood functions on those structures and adapting the completeness proof for this case. This requires a dedicated work. Moreover, another possibility is to work with a fragment of relevant logic which admits the same type of semantics [19, 20].
The only addition to intuitionistic linear logic is the adjunction of a strong negation which is motivated for modelling agents’ acceptance and rejection of propositions in the agenda in a symmetric manner, as it is usual in the setting of JA.^{5}^{5}5The intuitionistic negation, which is definable by means of the implication and the symbol for absurdity (i.e. ), does not capture the intended meaning of the rejection of an item in the agenda. Linear logic with strong negation has been studied in particular in [32]. We term this system by .
The language of , , then is defined as follows. Assume a set of propositional atoms , , then:^{6}^{6}6We use the notation introduced by Girard [11], denoting the additive conjunction and disjunction by and respectively. In [32] and [33], additives are denoted by means of the classical notations and .
By splitting connectives into additives and multiplicatives, the units of the logic split as well: we have a multiplicative constant for truth 1, which is the neutral element for , and an additive constant for truth , which is neutral for . In the intuitionistic case, we only have one constant for falsity [32, 33].
4.1. Hilbert system for
The Hilbert system for intuitionistic (multiplicative additive) linear logic has been basically developed in [34], see also [31, 19]. We extend to by adding a strong negation as in [32]. We define the Hilbertstyle calculus by introducing a list of axioms in Table 1 and by defining the following notion of deduction. The concept of deduction of linear logic requires a treestructure in order to handle the hypothesis in the correct resourcesensitive way. Notice that, by dropping weakening and contraction, we have to consider multisets of occurrences of formulas. This entails that, in particular, in linear logic, every modus ponens application (cf. rule) applies to a single occurrence of and of .
The notion of proof in the Hilbert system is then defined as follows.
Definition 1 (Deduction in ).
A deduction tree in is inductively constructed as follows. (i) The leaves of the tree are assumptions , for , or
where is an axiom in Table 1 (base cases).
(ii) We denote by a deduction tree with conclusion . If and are deduction trees, then the following are deduction trees (inductive steps).


We say that is derivable from the multiset in , denoted by , iff there exists a deduction tree with conclusion . The semantics of is presented in Appendix A.
5. A possibility result for substructural logics
By assessing the outcome of the majority rule with respect to classical logic, collective rationality fails. In the situation radically changes. Recall that the multiplicative connectives are and and the additive connectives of are and . We denote the additive implication by and we define it by . We label by the additive fragment of whose Hilbert system is obtained by taking all the axioms that contain additive formulas plus the rule (therefore we drop the rule).
If is a formula in classical logic, we define its additive translation as follows: , for atomic, , and . The additive translation of the outcome of the majority rule is then simply .
As it is standard in JA, we assume that the individuals reason by means of classical logic (), however we assess the outcome of the majority rule with respect to the logic , by means of the additive translation. This returns a voting procedure from profiles of judgments defined on a certain classical agenda to sets of judgments defined in .^{7}^{7}7Observe that Theorem 2 does not depend on the logic being intuitionistic, it holds for a classical version of additive linear logic [9]. We present it for , since it is the logical setting of this paper.
Theorem 2.
For any agenda in classical logic, suppose that for each , is consistent and complete wrt classical logic, then for any profile J, is consistent and complete wrt .
Proof.
The proof is based on the fact that, in additive linear logic, every minimally inconsistent set has cardinality 2.^{8}^{8}8Recall that a minimally inconsistent set is an inconsistent set that does not contain inconsistent subsets. This follows from the fact that every deduction in the additive fragment of linear logic contains exactly two formulas , as it has been noticed in [35].^{9}^{9}9This condition surprisingly corresponds to the median property introduced by [36]. See also [5]. This condition in fact characterizes agendas for which the majority rule returns consistent sets of judgments. Suppose then by contradiction that is inconsistent. Then, there exists a set of two formulas such that is inconsistent. Since both and are in , this entails that and , which entails that there exists an individual such that and are in . If is inconsistent wrt , then , therefore is also inconsistent wrt classical logic, since classical logic derives more theorems than . Therefore, is inconsistent, against the assumption of (classical) consistency of individual judgment sets. Completeness of follows by noticing that the majority rule always preserves completeness [18]. ∎
Theorem 2 shows that, although the majority rule does not preserve the notion of consistency of classical logic, it is still capable of preserving the notion of consistency defined by means of .^{10}^{10}10A number of extensions of for which consistency is preserved by majority is discussed in [9].
Observe that the notion of consistency of is weaker than the notion of consistency of classical logic. Since in classical logic we can prove more theorems than in , if a set of formulas is consistent in classical logic — i.e. we cannot prove — then its translation in is consistent. By contrast, if a set of formulas is consistent in —i.e. it cannot prove — that does not mean that its classical counterpart is consistent, since classical logic is more powerful.
Take the case of the doctrinal paradox that we have previously discussed. The agenda of propositions in classical logic is given by the following set: . The salient part of the profile of judgments J involved in the doctrinal paradox is represented in Table 2.
1  yes  yes  yes  no  no  no  yes  yes  no 

2  no  no  yes  yes  yes  no  yes  no  yes 
3  yes  no  no  no  yes  yes  yes  no  yes 
maj.  yes  no  yes  no  yes  no  yes  no  yes 
In classical logic, the outcome of the majority rule is inconsistent since, for instance, the acceptance of and of entails the acceptance of which contradicts (i.e. ). Moreover, the premisebased reading and the conclusionbased reading provide contradictory outcomes. The premisebased reading accepts , , and , then infers ; by contrast, the conclusionbased procedure just votes on the conclusion ( and ) and accepts .
The additive translation of is . Firstly, notice that in , we cannot infer the additive conjunction from and , we could only infer the multiplicative conjunction (by Axiom 6). Thus, the (multi)set is not inconsistent in .^{11}^{11}11In , therefore, in , .
Moreover, the premisebased and the conclusionbased reading do not provide contradictory outcomes. The conclusionbased reading provides , whereas the premisebased reading does not infer any conclusion now, since in . Therefore, the analysis of the doctrinal paradox in terms of takes the conclusionbased horn of the dilemma. The plain majority rule provides the same outcome as the conclusionbased procedure.^{12}^{12}12Two arguments are usually raised against the conclusionbased procedure: Firstly, it is usually not complete, secondly the reasons of the collective decisions, i.e. the premises, are invisible in the collective set. For a definition of complete conclusionbased procedure, see [37]. The solution provided by means of substructural logics shows that the plain majority rule provides the conclusionbased solution, therefore, it is both complete and it shows in the collective sets the reason for the collective choice.
We see how the distinction between additives and multiplicatives of substructural logics allows for distinguishing inferences performed at the individual level and inferences performed at the collective level. Firstly, notice that since the Hilbert system of is sound and is consistent, the following fact holds.
Fact 3.
The deductive closure in of , ), is consistent.
Henceforth, we can safely reason about by means of full . By Fact 3, we can extend the majority rule , which is defined from profile in to subsets of an agenda in , to any agenda of formulas of full that contains the additive translation of the classical agenda .
(1) 
By means of the additive translation, we are viewing the formulas that have been accepted by majority as additive formulas. That is, the inferences that have been performed autonomously by (a majority of) the individuals are visible in as additive formulas. Multiplicatives then can only combine formulas that have already been accepted by majority. For this reason, we view multiplicatives as performing inferences at the collective level.
For instance, if we assume that contains the formula , we can infer from the acceptance of and in , the multiplicative conjunction . The difference between and in this setting is that requires a single majority of agents that support and , in order to be collectively accepted. By contrast, only requires that there exist two possibly distinct majorities of individuals, one that supports and one that supports .
The reason why the analysis of the doctrinal paradox by means of substructural logics takes the conclusionbased solution is that the legal doctrine has been interpreted in additive terms by means of the formula . This formulation entails that is accepted only if a majority of individuals autonomously infer .
5.1. Collective reasoning
In order to obtain the solution provided by the premisebased reading of the doctrinal paradox, as we argued, we have to introduce a reasoning step that is not performed by (a majority of) the individuals, that means a reasoning step that takes place only at the collective level. For this reason, the premisebased reading requires a multiplicative formulation. We can obtain the premisebased solution to the dilemma by dropping the additive formulation of the legal doctrine and replace it with a multiplicative formulation: . Notice that, by keeping both the formulations of the legal doctrine, the inconsistency between accepting and rejecting is back. This fact shows why in classical logic, that does not make the distinction between multiplicatives and additives, the dilemma between the premisebased and the conclusionbased procedures is challenging and corresponds to the inconsistency of the majority rule.
Since the multiplicative formulation of the legal doctrine applies as step of reasoning performed only at the collective level, it has to be modelled as a constraint that is external to the agenda of propositions about which the agents express their votes.^{13}^{13}13The view of constraints that are external to the agenda is close to the approach in [38]. We term this external constraints collective constraint and we define a set of collective constraints as a set of formulas in . A collective constraint norms the way in which the group has to reason once a number of propositions is accepted by majority. Collective constraints do not belong to the individual judgments sets and they are not used to perform inferences at the individual level. We can in principle assume that collective constraints are subject to aggregation, namely, that the individuals decide by voting how the group is supposed to reason about the collectively accepted propositions. To model that, we assume that the individual judgments are partitioned into two sets: a subset of the agenda (as usual) and a subset of collective constraints. We assume that, for each , is consistent wrt ; moreover, the collective constraints are compatible with the judgments of the individuals, once they are interpreted as additive formulas: that is, we assume that is consistent in . Denote by S the profile of individually accepted collective constraints. The majority rule is defined as usual from profiles of constraints to sets of constraints in .^{14}^{14}14We are proposing to model the decision about the collective constraints by means of the majority rule, in order to meet the standard view of the doctrinal paradox, where the legal doctrine was in fact a matter of voting. We define the following two steps procedure.
Definition 4.
Let an agenda in and an agenda in that is composed of three disjoint sets: the additive translation of , a set of collective constraint , and a set of conclusions . The two step procedure is defined as follows:
operates as follows: firstly, aggregates by majority the formulas in the individuals’ agenda (and translates them into additive formulas) and the collective constraints in ; secondly, infers possible conclusions in , by means of the logic and the constraints in .^{15}^{15}15Note that by construction, returns a set of formulas, not a multiset. is nonetheless evaluated wrt by viewing it as a multiset with multiplicity 1. Although multisets have no straightforward interpretation in an aggregative setting, it is the device that is necessary to really drop contraction. Once we assume that formulas form sets, then and become indistinguishable and that would enable contraction, which entails . In principle, contraction is not harmful for the majoritarian aggregation, cf. [9]. Thus we could add contraction and work with sets, rather then with multisets in this case. We preferred to assume multisets here, because they provide the standard means to define the Hilbert systems for linear logics. We leave a treatment for logics enabling contraction in an aggregative setting for future work.
If we do not assume any collective constraint, i.e. , then the procedure of Definition 4 coincide with the procedure defined in Equation 1. Thus, in that case, the consistency of is guaranteed. Unfortunately, by adding possibly arbitrary constraints in full , consistency is not ensured any longer.^{16}^{16}16This is standard for premisebased procedures in the case individuals are permitted to vote on nonatomic propositions. To guarantee the consistency of in case the constraints are expressed by means of any multiplicative formula, we have to assume that the following property holds: every subset of that is minimally inconsistent in has cardinality at most 2.^{17}^{17}17This is the median property instantiated for the logic , that provides necessary and sufficient conditions for the consistency of the majority rule, cf. [9]. See also [4] for the use of the median property under external constraints.
The premisebased reading of the doctrinal paradox can be obtained by instantiating as follows: , , and .^{18}^{18}18Note that in this case satisfies the median property. By assuming the same profile as in Table 2 on the judgments and by assuming unanimity on the collective constraint, the outcome of is then . The collective set shows that does not hold because there is no single majority of agents that supports both and , is accepted because there are indeed two majorities that support and , and that is sufficient to apply the collective constraint in , and conclude by modus ponens .
It is possible to obtain the conclusionbased outcome by means of by simply replacing with , that imposes the additive formulation of the legal doctrine.^{19}^{19}19Observe that in presence of weakening and contraction, as in classical logic, the two formulations of the legal doctrine are equivalent. In this view, the additive formulation forces the conclusionbased reading, whereas the multiplicative constraint entails the premisebased reading. From this perspective, the additive formulation is more demanding in terms of the cohesion of the group: it requires that the same majority accepts the premises in order to draw the conclusion, whereas the premisebased reading only requires that there exist possibly distinct majorities on all the premises.
provides an interesting propositional base to model collective attitudes. In the next Section, we introduce a number of modal extensions of in order to model the collective propositional attitudes involved in the doctrinal paradox.
6. Modal logics for modelling collective attitudes
We present three modal extensions of to reason about actions, beliefs, and obligations of individuals and collectives. We assume a set of coalitions C and to express individual propositional attitudes, we admit singleton coalitions; in that case the meaning of a coalition in C is .^{20}^{20}20This move is similar to the approach in [39] to discuss coalitional ability. We introduce three modalities , , and to express that coalition does , believes that, and is obliged to . We label this logics by , , and , for action, beliefs, and obligations, respectively.
We use minimal (or nonnormal) modalities in order to ensure a number of basic principles to reason about this modalities. The discussion of further principles is left for future work.
As we have seen in the previous section, in order to understand and possibly circumvent collective inconsistency, we crucially rely on the distinction between multiplicatives and additives. The role of modalities is to make the coalitions of agents that support a certain proposition explicit. Let be one of the modalities in . The distinction between multiplicatives and additives affects the modal extension in the so called combination axioms, which is usually admitted in the classical counterparts of this logics: . This axiom can be interpreted in substructural logics in two ways: a multiplicative and an additive way. The multiplicative combination of two propositions, in our interpretation, requires two possibly different winning coalitions that support each combined propositions, whereas the additive combination forces the same coalition to support both propositions. This distinction is reflected by the operational meanings of the two conjunctions.^{21}^{21}21This distinction is related to the distinction between group knowledge and distributed knowledge in epistemic logics, cf. for instance [40]. Therefore, the additive version of the axiom means that if the same coalition support and , then the same coalition supports the additive combination of and : . The multiplicative means that if a coalition supports and coalition supports then, the disjoint union of the two coalitions supports the combination . The condition of disjointness of and is crucial for modelling the collective attitudes in a consistent way. In particular, the condition shows that the individuals that are member of the coalition are all equally relevant to make the propositions accepted. Take for instance the case of and : if we enable the inference to , then we lose the information concerning the marginal contribution of agent 2 in both coalitions. For this reason, we assume that the set C of coalitions is closed under disjoint union.^{22}^{22}22We define the disjoint union of two coalitions by , if and , otherwise. The combination axiom is reminiscence of coalition logic [41]. Note that we do not assume any further axiom of coalition logic. For instance, we do not assume coalition monotonicity: , if . The motivation is that we are modelling profilereasoning, that is, we start by a fixed profile of individual attitudes and we want to capture, by means of the modalities , how the collectivity reasons about the propositions that have been accepted by majority in that profile. In this setting, given a profile of individual attitudes, there exists only one coalition that supports a proposition that has been accepted by majority. This is a different perspective wrt coalition logic and logic of coalitional ability [39]. The technicalities of this logics are presented in Appendix B and C.^{23}^{23}23We do not discuss here the computational complexity of the logics that we introduce. We simply summarise a few points. Intuitionistic linear logic is PSPACE complete [24] and the logic of action based on is proved to be in PSPACE in [29]. For the case of (i.e. with strong negation), we have decidability, since the version of based on sequent calculus enjoys cut elimination [32]. A number of normal modal logics based on are also proved to be decidable [33]. Therefore, in principle, it is possible to show that also our modal logics are decidable. Moreover, it is known that the fusion of decidable logics is decidable [42], thus we can in principle adapt the standard theory for the case of nonnormal modal logics based on .
6.1. Collective actions
The logic to reason about collective actions is based on the minimal logic of bringingitabout, which was traditionally developed on top of classical propositional logic [43, 44]. The principles of this logic aim to capture a very weak view of actions that, for instance, does not presuppose intentionality or explicit goals. We apply this minimal view to conceptualise collective actions. This is adequate for an aggregative perspective on collective actions, for which the collective is not assumed, in general, to have joint intentionality nor any shared goal, [1].
In [45], a version of bringingitabout based on was introduced for modelling resourcesensitive actions of a single agent. We propose here a version with coalitions, based on , we label it by (action). Four principles of agency are captured by the classical bringingitabout logic [43]. The first corresponds to the axiom of modal logics: , it states that if an action is brought about, then the action affects the state of the world, i.e. the formula that represents the effects of the action holds. The second principle corresponds to the axiom in classical bringingitabout logic. It amounts to assuming that agents cannot bring about tautologies. The motivation is that a tautology is always true, regardless what an agent does, so if acting is construed as something that affects the state of the world, tautologies are not apt to be the content of something that an agent actually does. The third principle corresponds to the axiom: . The fourth item allows for viewing bringing it about as a modality, validating the rule of equivalents: if then . (cf. Appendix B).
The language of , simply extends the definition of , by adding a formula for each coalition . The axioms of are presented in Table 3. The Hilbert system is defined by extending the notion of deduction in Definition 1 by means of the new axioms in Table 3 and of two new rules for building deduction trees, cf. Definition 6.1.
A number of important differences are worth noticing, when discussing the principles of agency in linear logics. The first principle is captured by E1, that is, the linear version of T: . Since in linear logics all the tautologies are not provably equivalent, the second principle changes into an inference rule, that is ( nec) in Definition 6.1: if , then . Note that this also entails that agents do not bring about contradictions. Moreover, the rule (re) captures the fourth principle.
The principle for combining actions is crucial here: as we have discussed, it can be interpreted in substructural logic in a multiplicative way, by means of (axiom E3), and in an additive way, by means of (axiom E2).^{24}^{24}24The principles for combining actions have been criticised on the ground that coalitions and may have different goals, therefore it is not meaningful to view the action of as a joint action. However, the aggregative view of group actions defined by means of the majority rule presupposes that the group is not defined by means of a shared goal nor a shared intention. Therefore, Axioms E2 and E3 are legitimate from this point of view.
Definition 5 (Deduction in ).
A deduction tree in is inductively constructed as follows. (i) The leaves of the tree are assumptions , for , or where is an axiom in Table 3.
(ii) If and are deduction trees, then the trees in Definition 1 are also deduction trees in . Moreover, the following are deduction trees (inductive steps).
nec 
6.2. Collective beliefs
The logic for modelling collective beliefs is a minimal epistemic logic [46]. Nonnormal epistemic modalities have been applied also to modelling reasoning about epistemic states of strategic agents in [47, 48]. By using beliefs operators to model propositional attitudes, we are viewing the content of beliefs as sharable among the community of agents, in accordance with the motivations of the framework of judgment aggregation.^{25}^{25}25Alternatively, we could have epistemic attitudes of agents by means of acceptance logic [49]. This solution is the one adopted by [14]. The language of intuitionistic linear logic of beliefs with coalitions, label it by , extends the language by adding a doxastic modality . The Hilbert system for extends with the following axioms (cf. [46]). The notion of derivation extends that of by adding the rule of equivalents for the modalities .
Definition 6 (Deduction in ).
A deduction tree in is inductively constructed as follows. (i) The leaves of the tree are assumptions , for , or where is an axiom in Table 4 (base cases).
(ii) If and are deduction trees, then the trees in Definition 1 are also deduction trees in . Moreover, the following are deduction trees (inductive steps).
Note that the rule ( nec) that we used for actions is dropped for the case of epistemic logics, that is, we assume that agents may believe tautologies although we do not force them to do so. However, we assume that a modicum of rationality still applies and agents cannot believe contradictions, cf. axiom B1. We also assume the positive introspection axiom, B4, which exhibits in this context the public nature of the belief. A negative introspection axiom can be defined as well, but we do not discuss it in this paper. Moreover, we assume that for the additive conjunction, agents can infer their belief in from the belief in (axiom B5). The combination of beliefs is the original part here. Again, we distinguish between an additive and a multiplicative combination in order to keep track of the contribution of individuals to the acceptance of the collective belief.
6.3. Collective norms
The use of nonnormal modal logic to express deontic modalities was motivated in [13, 50]. Moreover, nonnormal deontic logics have been used to discuss institutional agency in [51] and to model weak permissions in [52]. Here we present a minimal version of deontic logic, for the sake of the exemplification, and we leave a proper treatment of deontic principles in for future work. We extend the language of by adding a number of modalities for obligations , for . Moreover, we also assume the respective dual modalities (permission) . We term this system by . The motivation for this formalisation is that we want to distinguish norms that constraint the behaviour of individuals, e.g. , to norms that applies to coalitions or collectives, e.g. .^{26}^{26}26For a taxonomy of norms that applies to groups, we refer to [53].
The Hilbert system for extends the case of by adding the axioms in Table 5.
Definition 7 (Deduction in ).
A deduction tree in is inductively constructed as follows. (i) The leaves of the tree are assumptions , for , or where is an axiom in Table 5 (base cases).
(ii) I (ii) If and are deduction trees, then the trees in Definition 1 are also deduction trees in . Moreover, the following are deduction trees (inductive steps).
7. A model of collective attitudes
We are ready now to introduce the model of collective attitudes. Individual attitudes shall be captured by formulas , where is an individual (coalition) and is one of the modalities that we have encountered. Collective attitudes are then modelled by formulas , where is a winning coalition in a given profile with respect to the majority rule.^{27}^{27}27The majority rule is not going to be interpreted by means of any logic formula, as for instance in [54, chapter 4] We focus in this paper on the majority rule, however the proposed model of collective attitudes can be applied in principle to any aggregation procedure.
In order to meet the standard hypothesis of JA, we assume that the individuals reason in classical logic. That is, we assume that the judgments of the individuals are expressed by means of the versions based on classical logic () of the logics , , and , which are known from the literature;^{28}^{28}28Note that the classical versions of the modal logics that we have introduced can be obtained by adding (W) and (C) to the lists of axioms. we label them by , , and , respectively. Let be one of , , and .
All of this logics are defined, besides by the axioms of classical propositional logic, by means of the classical versions of the modal axioms and rules that we have encountered. For instance, each satisfies a rule of equivalents (re), (re), (re). Observe that, the classical combination axioms, which dismiss the distinction between multiplicatives and additives, in this case have all the following form. Let and :
(2) 
The agenda is a subset of that is closed under negations. An individual judgment set is a subset of such that it is consistent and complete wrt ; moreover, we assume that for every modal operator , for , that occurs in formulas in , . That is, individual judgments are about individual beliefs/actions/obligations. This is motivated by the rationale of the framework of judgment aggregation, where individuals reason independently of each other on the matters of the agenda.
We want to associate the collective judgments accepted by majority with the coalitions of agents that support them. For this reason, we define the collective agenda that coincides with for the formulas that do not contain modalities and for any formula in that contains a modal operator, denoted by , we have for any such that . That is, contains all the possible collective attitudes that can be ascribed to winning coalitions of individuals.
We restate the definition of the majority rule as follows: is defined by
(3) 
That is, for non formulas the majority works as usual, for formulas, the majority aggregates them by merging the supporting individuals into the relevant winning coalition.
Note that, in case the codomain of is based on classical logic, may be inconsistent wrt to classical logic, therefore the codomain of is the power set of the agenda. For instance, take the following profile J of sets of formulas of :
 :

 :

 :

The majority rule returns the following set: . We can show that is inconsistent in as follows. From and , we can infer, by means of a classical combination axiom, . From and , we can infer, again by a combination axiom, that , which entails , since is equivalent to and satisfies the classical version of axiom (re). Therefore, is inconsistent since contradicts (a classical version of) axiom B1 .
We assess the collective attitudes in wrt the logics , or . We label by one among , , and . We extend the additive translation presented in Section 5 by . Hence, we can translate into its counterpart in the additive fragment of the modal logics that we have defined. This returns an aggregation procedure . We can now assess the outcome of the majority rule defined on profiles based on classical modal logics wrt an agenda of propositions of . We label the additive fragment of by .
By means of Theorem 2, the consistency result of the majority rule for can be extended to . Note that the additive logics , , or contain only the additive axioms for the modalities (i.e. they do not contain E3, B3 and O3).
Theorem 8.
For every agenda and profile , is consistent and complete wrt .
Proof.
Firstly, we show that also for , , or , it holds that every derivation contains at most two formulas. For any , notice that the rule (re) does not increase the number of formulas in a derivation. Moreover, ( nec) in does not increase the number of formulas in a derivation as well. The additive versions of the axioms in , , or cannot increase the number of formulas, since in the respective additive versions, we dropped the rule.
Therefore, by the argument of Theorem 2, is consistent and complete wrt modal . We show that also is consistent. Suppose by contradiction that it is not. Then, since every derivation in , , or contains at most two formulas, there exist and in such that . Suppose and , where possibly differs from . If and are in , then there exist at least individuals such that and at least individuals such that . From this we infer that there exists an individual such that and . To conclude, we show that if and are inconsistent, then and are. If , then the argument holds because it amounts to a simple relabelling of the indexes of the modalities. If , then the proof of may differ form the proof of only in that it may not use the instances of the axiom ; therefore if one can prove in the case of , with less axioms, one can do that by identifying and with . Therefore, is inconsistent in , thus it is inconsistent in their classical counterparts. ∎
For instance, take the profiles of judgments sets in that we have previously encountered. The majority returns now the collective set in : , which is not inconsistent in . The reason is that we cannot derive and for any , from the formulas and . This is due to the fact that the additive combination of beliefs is not applicable in this case.
By means of the additive connectives, we are only reasoning about propositions that are accepted because of a winning coalitions of agents; this means that the reasoning steps that are visible in as additive formulas are those that have already been performed autonomously by the individuals. In order to reason about propositions that are accepted by distinct winning coalitions and to model inferences performed at the collective level, we embed into the full logics , , and .
A version of Fact 3 easily holds also for the modal counterparts, thus, we can define the aggregation procedure that returns the deductive closure of with respect to an given agenda of formulas in that contains (cf. Equation 1).
To add collective constraints, that are external to the individual agenda, we restate the definition of the aggregation procedure in this setting (cf. Definition 4). A set of collective constraints is a set of formulas in such that for each modal formula , is either a winning coalition of agents or , where each is a winning coalition of agents.
We assume that, for each individual , accepts a set of constraints . Moreove, is compatible with the judgments of the individuals, once they are interpreted by means of the additive translation: is consistent in .
Definition 9.
Let an agenda in and an agenda in that is composed of three disjoint sets: the additive translation of , a set of constraints , and a set of conclusion . The two step procedure is defined as follows:
To guarantee the consistency of in case the constraints are expressed by means of any multiplicative formula, we have to assume that every minimally inconsistent subset of has cardinality at most 2.
7.1. Collective attitudes in the doctrinal paradox
We conclude our analysis by exemplifying our treatment of collective attitudes for the case of the doctrinal paradox, cf. Section 2. We use the combination of the collective attitudes that we have introduced for the sake of illustration. In fact, in order to define a logic that combines all the logics , we should extend the theory of fusion of modal logics to the case of nonnormal modalities and to substructural propositional axioms. This is in principle a viable solution, given that our modal logics are mutually independent. Let be the fusion of the logics . A version of Theorem 8 for holds. Note that, by putting all the modalities together, since no new inference rule nor axioms have been introduced, we still have that every derivation in such a system would contain at most two formulas. A detailed treatment of that goes beyond the scope of the present paper and it is left for future work.
We formalise the individual and collective propositional attitudes involved in the doctrinal paradox as follows. We use as a variable for coalitions of agents.

: believes that the confession was coerced.

: believes that the confession affected the outcome of the trial.

: is obliged to bring about that the trial is revised.
The issues to be decided can be represented by means of the agenda of propositions in the fusion of the classical modal logics , , and :
The s are here coalitions of agents in . We separate the individual agenda, for which , for , and the collective agenda where is any winning coalition wrt the majority rule.
The legal doctrine is here represented by means of the combination of the deontic operator and of the action operator that means, intuitively, that under the condition that the court believes , it is obligatory to bring about that the trial is revised, which captures the normative force of “the trial must be revised” (see for instance [51]).
The critical part of the profile J of the doctrinal paradox is represented in Table 6.
1  1  1  1  1  0  
1  0  0  1  0  1  
0  0  1  1  0  1  
maj.  1  0  1  1  0  1 
The outcome of in classical logic on the profile above is then the following set of collective attitudes:
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