Judgment aggregation in non-classical logics

11/10/2017 ∙ by Daniele Porello, et al. ∙ 0

This work contributes to the theory of judgment aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgment aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgment aggregation in non-classical logics is that they offer a number of modelling choices to represent agents' reasoning in aggregation problems. By studying judgment aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems.

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1. Introduction

Judgement aggregation (JA) has recently become a significant topic at the intersection of themes in social choice theory, multiagent systems, and knowledge representation. The reason is that JA provides a general theory for studying the procedures to aggregate agents’ possibly heterogeneous attitudes into a collective attitude that reflects, as close as possible, the individual views. The original applications of judgment aggregation were related to voting theory and to the modellisation of the decision-making processes of collegial courts, cf. List and Pettit (2002); Kornhauser and Sager (1993). In AI and multiagent systems, JA provides a sound methodology to define ascriptions of propositional contents to collective entities; for that reason, JA has been related for instance to belief merging, cf Pigozzi (2006); Konieczny and Pérez (2011), and to ontology merging, cf Porello and Endriss (2014).

As usual in formal modellings of agency, a modicum of rationality is presupposed in the understanding of agents, List and Pettit (2011). Since JA is about the aggregation of logically connected propositions, the notion of individual rationality of JA is essentially related to the concept of logical consistency. The concept of collective rationality is therefore intended as the preservation of individual consistency, via the aggregation procedure, at the collective level. As it is known from the beginning of the study of JA, the aggregation of individual consistent judgments may lead to inconsistent outcomes, cf. List and Pettit (2002). For that reason, a careful investigation of the properties of the aggregation procedures and of the extent to which they affect the preservation of consistency is required. This line of study has been developed into a quite sophisticated theory of aggregation procedures. A number of introductory readings to judgment aggregation is now available, for instance, Grossi and Pigozzi (2014), List and Polak (2010), List and Puppe (2009), and Endriss (2016).

Important aggregation procedures, with the significant example of the majority rule, do not preserve individual consistency once agents are allowed to express their judgments on any possible proposition, that is, once the agenda on which the agents express their views is allowed to be any subset of the logical language. Hence, one can roughly divide the JA approach to cope with collective inconsistency into two directions: the first is the investigation of the procedures that are indeed capable of preserving consistency; the second is the characterisations of the agendas that guarantee consistent collective outcomes.

In the standard view of JA, the logic that is used to model rationality is classical logic. Thus, the failure of preserving consistency that the results in judgment aggregation show concerns the notion of consistency defined by classical logic.

In this paper, I propose to extend the theory of judgment aggregation to a number of significant of non-classical logics. By non-classical logic, we shall mean here logical systems that reject one or more principles of classical logic and provide an alternative view of reasoning. A guiding example is intuitionistic logic, see Dummett (2000): by rejecting for instance the principle of the excluded middle, intuitionism provides a constructive account of reasoning. Many logics for JA have been investigated, designed, and discussed; however, they usually extend classical propositional logics, instead of investigating systems that are alternative to or weaker than classical logic. A few significant exceptions shall be discussed later on.

The motivation for studying judgment aggregation in non-classical logic are essentially three. Firstly, there is a theoretical interest in extending the theory of judgment aggregation to logics that, from a mathematical point of view, significantly differ from classical logic. Secondly, by studying non-classical logics, we shall approach weak inference systems; thus, it is worth investigating whether, by weakening the logic that models the concept of rationality, standard impossibility results still hold for weaker system, or whether we can actually circumvent collective inconsistencies by weakening classical logic. Finally, as usual in justifying the investigation of non-classical logics, the adequacy of classical connectives to model reasoning may be questioned. It is interesting to notice that arguments against the material conditional of classical logic have also been suggested within the literature on JA. In particular, by Dietrich (2010), where classical implication is replaced by subjunctive implications of Lewis conditional logic. By studying JA in non-classical logic, we enable the choice among a number of definitions of logical connectives that may be appropriate for certain aggregation problems. For this reason, we shall focus in this paper on a number of well-established non-classical logics with significant impact in philosophical logic or in computer science. For instance, intuitionistic logic provides a constructive view of reasoning that may suit an evidence-based view of inferences, while relevant logic allows for defining a conditional that better copes with the paradoxes of material implication, cf. Anderson et al. (1992). We shall start discussing JA in non-classical logic by dealing with intuitionistic logic. Then, we shall develop our analysis of non-classical logics by means of substructural logics, cf. Paoli (2002); Restall (2002). Substructural logics are a family of logics weaker than classical logic that reject one or more of the core principle of classical reasoning; for instance, the monotonicity of the entailment or the commutativity of logical connectives.

The conceptual motivations for studying substructural logics are usually related to the idea of capturing a form of reasoning that better copes with the actual inferential practice of human or artificial agents. For instance, the weakest substructural logic that we discuss in this paper is the Lambek Calculus, developed by Lambek (1958), for which connectives are non-commutative. This entails that the ordering of the formulas is crucial for reasoning. The order-dependency of inferences can be used to model aggregation problems with temporal dependencies among the issues of the agenda. For instance, suppose and in this order have been accepted, then the constraint does not apply in this situation to infer .

Moreover, we approach linear logic, introduced by Girard (1987), that captures a form of resource-bounded reasoning. For example, suppose the proper axiom represents the inference “if I have one euro (), then I buy one coffee ()”. In classical logic, one can infer, by means of the contraction principle, that , namely, that I still have one euro, besides having the coffee. By dropping contraction, linear logic captures a the resource-sensitive aspect of causality: the antecedent has to be consumed during the inferential process so that the consequent may hold, cf. Girard (1995).

In philosophical logic, an important debate on the nature of logical implication emerged in the tradition of relevant logics: Anderson et al. (1992), Dunn and Restall (2002). The family of relevant logics rejects, in particular, the monotonicity of the entailment and design therefore logics for which true implications exhibit the relevance of the antecedent of the conditional to the consequent. In particular, relevant logics reject the axiom that means that whenever holds, can be implied by any , regardless of the relevance of for assessing . Relevant logics have also been motivated as logics for modelling epistemic agents and as logics that model inferences that depend on the amount of information available to cognitive agents, Mares (2004); Allo and Mares (2012); Masolo and Porello (2015).

We will also briefly discuss fuzzy logics that are defined within the substructural realm. We shall not discuss in details the case of paraconsistent logics because we are interested here in studying the preservation of consistency via aggregation procedures. However, paraconsistency is approached in this paper by presenting logics for which the principle of ex falso quodlibet does not hold (e.g. in the case of linear and relevant logics).

The methodology of this paper is proof-theoretical. This is motivated by the fact that we are going to introduce a number of logics with significantly different algebraic counterparts: the proof-theoretical methodology permits a compact treatment. Moreover, as we shall see, the proof-theoretical analysis allows for pinpointing the inferences principles that are responsible of the failures of preserving consistency in judgment aggregation.

The main contribution of this paper consists in the extension of the theory of judgment aggregation to the case of a number of non-classical logics. The choice of the logical systems in this paper is also motivated by the fact that, by dropping monotonicity and other principles of classical logic, it is possible to circumvent the collective inconsistency that threaten the judgment aggregation based on classical logics. Moreover, results for general monotonic logics have already been presented for instance by Dietrich (2007) and for nonmonotonic logics have been recently presented in Wen (2017). Another crucial aspect for preserving consistency that motivates the focus on linear and relevant logics is the distinction that this systems provide between additive and multiplicative, or extensional and intensional, logical connectives. As we shall see in Section 7 and in Section 8, the combination of the lack of monotonicity and the distinction between types of connectives allows for establishing positive results for the majority rule, that is, performing collective reasoning in those weak systems guarantees consistency. By relying on that, in Section 9.1 a strategy for circumventing judgment aggregation paradoxes is proposed. The idea is to assess individual reasoning with respect to classical logic, as usual in judgment aggregation, while assessing collective reasoning with respect to a weak logic for which consistency is ensured. To enable the assessment of individual and collective reasoning with respect to possibly distinct logics, we shall slightly rephrase the standard framework of judgment aggregation.

The remainder of this paper is organised as follows. Section 2 introduces the background on substructural logics. In particular, we recall the sequent calculi for classical and intuitionistic logic, for the Lambek calculus, and for linear logic. For the case of relevant logics, as we shall see, the sequent presentation is not satisfactory, thus we shall approach relevant logics by means of Hilbert systems. Section 3 introduces a model of judgment aggregation that is general enough to treat the case of the non-classical logics that we discuss here. In particular, we stress the interesting differences that the lacking of monotonicity, contraction, and commutativity entail on the model. Section 4 rephrases the standard results of JA in classical logic in the setting of this paper (i.e. in proof-theoretical terms) and discusses the case of conservative extensions of classical logic. Section 5 approaches JA for intuitionistic logic. Section 6 discusses JA for the Lambek calculus. Section 7 presents the case of linear logic. In particular, as we shall see, by focusing on a fragment of linear logic, general possibility results for judgment aggregation are achievable. Section 8 extends the previous analysis to the case of relevant logics. Section 9 discusses a number of extensions of the previous treatment to other logics and discusses how to circumvent the impossibility results based on agendas in classical logic by rephrasing classical collective rationality with substructural reasoning. Section 10 surveys related work, by focussing in particular on the use of non-classical logics in JA, on the use of proof-theoretical methods, and on the relationship with the algebraic approach to the study of general logics of JA. Section 11 concludes.

2. Background on substructural logics

Sequent calculi were introduced by Gerhard Gentzen to study proofs in classical and intuitionistic logic. They provide an important theory in logic that allows for investigating the operational meaning of logical connectives, cf. Gentzen (1935).111For an introduction to sequent calculi, see Troelstra and Schwichtenberg (2000) and Negri et al. (2008). Besides providing a fine-grained tool to analyse inferences, sequent calculi can be used to model a number of logics in a uniform and elegant way. A sequent is an expressions of the form , where , the premises of the sequent, and , the conclusions of the sequent, are made out of formulas in a given logic. The structure of and , as we shall see, depends on the logic; for instance, in what follows, they may be sets, multisets, or lists of formulas.

The intuitive reading of a sequent expression is: “the conjunction of the formulas in entails the disjunction of the formulas in ”. A sequent calculus is specified by two classes of rules. The structural rules determine the structure of the sequent and define how to handle the hypotheses in a proof, the assumptions of reasoning; for instance, they entail that in classical logic and are sets of formulas. The logical rules define the operational meaning of the logical connectives. A fundamental insight concerning the meaning of the logical rules is due to the tradition of substructural logics and in particular to Jean Yves Girard (cf. Girard (1987)): The structural rules determine the behaviour of the logical connectives. The meaning of the structural rule for reasoning is the following: weakening (W) corresponds to the monotonicity of the consequence relation, contraction (C) amounts to assuming that multiple occurrences of the same formula can be identified, and exchange (E) forces the commutativity of conjunction and disjunction. When rejecting one or more structural rules, we simply cannot view the class of propositions about which reasoning is performed as a set; therefore, in the subsequent presentation, we shall define the structure of and as either a multiset or a list.

We introduce the language of classical propositional logic as usual. Assume a set of propositional variable P, the set of formulas of is defined as follows.

(1)

The sequent calculus for is presented in Table 1. The logical rules may be presented either in multiplicative form or in additive form. In the former case, the premises of the rule have possibly different contexts that are combined in the conclusion, in the latter, the rules presuppose that the premises of the sequent share the same context.222The terms additive and multiplicative have been introduced by Girard (1987). The distinction corresponds to the division between extensional and intensional connectives in relevant logics, cf. Paoli (2002). The dependency of logical rules on structural rules is evident by noticing that classical logic is imposed as soon as we assume (left and right versions of) weakening (W L, and W R), contraction (C R and C L), and exchange (E L and ER). In that case, the additive and the multiplicative versions of the rules of connectives are provably equivalent.

Identities

ax cut

Structural Rules

E L E R

C L
C R
W L W R

Negation

L R

Multiplicative presentation of logical connectives

R L
L
R
L R

Additive presentation of logical connectives

R L L

L
R R

L
R

R
Table 1. Sequent Calculus for Classical Logic (CL)

By disabling one or more structural rules, we enter the realm of substructural logics; for introductory readings cf. Paoli (2002); Restall (2002); Troelstra (1992).

Intuitionistic logic () can also be construed as a substructural logic. Gentzen surprisingly showed that intuitionistic logic can be captured by simply imposing upon the sequent calculus for that the right-hand side of the sequent contains at most one formula. This restriction is sufficient to prevent non-constructive principles, such as the excluded middle, to be provable. This restriction has been interpreted as a manner to disable structural rules locally by Girard et al. (1989): in intuitionistic logic the structural rules are prevented on the right-hand side of the sequent and fully permitted on the left-hand side. Define the language of intuitionistic logic () as follows. Let and the constant for false:

(2)

The sequent calculus of intuitionistic logic () is given by restricting the classical calculus to single-conclusion sequents. Negation is defined, as usual in intuitionistic logic, by means of and : .

2.1. Sequent Calculi for Substructural Logics

The weakest substructural logic that we discuss here is obtained by removing all structural rules. The resulting calculus is known as the Lambek calculus , developed by Lambek (1958) to capture syntactic parsing of natural language sentences along the tradition of categorial grammars. By rejecting exchange, the conjunction of , denoted by , is non-commutative. Moreover, there are two order-sensitive implications and . Note that is an intuitionistic logic, in the sense that at most one formula appears in the right-hand side of the sequent. We assume a minor extension of the Lambek calculus with the constant for false in order to define a form of intuitionistic negation in terms of the order sensitive implication (cf. Table 2) .333In fact, one can show that there are two negations definable in this way that depend on the order-sensitive implications: and . For our purposes, it is not worthy entering the details of the treatment of negation in . We refer to Wansing (2007), Restall (2006) for strong negations in and to Abrusci (1990) for the definition of the two negations for Lambek calculus.

(3)
L R
L R
L R
Table 2. Sequent calculus for the Lambek Calculus

By adding exchange to the Lambek calculus, we obtain linear logic (). That is, linear logic rejects the global validity of (W) and (C) both on the left and on the right hand side of the sequent. In , sequents are multisets of formulas. In Table 1, we presented two ways of defining logical rules: an additive version and a multiplicative version. The two formulations are redundant in classical logic because of (W) and (C), that are sufficient to prove that the additive formulation and the multiplicative formulation are equivalent. If we drop weakening and contraction, additives and multiplicatives are no longer equivalent, hence we have to account for two different types of conjunctions and disjunctions with distinct operational meanings. This operators are not visible in classical logic, because of the structural rules. Accordingly, in there are two different types of conjunction, a multiplicative conjunction

(tensor) and an additive conjunction

(with), and two types of disjunctions, multiplicative (parallel) and additive (plus). Implications can be defined by means of disjunctions and negations as usual, in the multiplicative implication is and the additive implication .444For the additive implication, whose status as an implication is in fact questionable, we refer to Troelstra and Schwichtenberg (2000). The reason why the additive implication is not satisfactory is that, in a categorical jargon, the adjunction w.r.t. additive conjunctions does not hold, that is is not equivalent to , cf Pym (2013). Given a set of propositional atoms P, the language of multiplicative additive linear logic is defined as follows.555We focus on the multiplicative-additive fragment of . Another important part of LL is given by the exponentials, that allow for retrieving the usual classical inferences in a controlled way. Therefore, instead of being yet another non-classical logic, linear logic is motivated at least as an analysis of proofs in classical and intuitionistic logic. We leave a discussion of the exponential for JA for future work.

(4)

By dropping weakening and contraction, the units of the logic have to be distinguished in multiplicatives 1 and (which are units for and respectivley) and additives ( and 0, which are units for and respectively).666Without W and C, also negation behaves differently. For example, the ex falso quodlibet principle is no longer globally valid in linear logic, for the multiplicative false constant . Thus, linear logic, beside being non-monotonic, is also a paraconsistent logic, at least in the weak sense of invalidating ex falso quodlibet. For the sake of simplicity of presentation, we shall use a single notation for negations in various logics. The sequent calculus for is presented in Table 3.

Identities

ax cut

Negation

Multiplicatives

R L
L
R
L R

Multiplicative units

1L 1 R
L R

Additives

R L

L
R
L R

Additive units

no rule for ( L ) R
0 L no rule for (0 R )
Table 3. Sequent calculus for

We label by the multiplicative fragment and by the additive fragment of . The intuitionistic version of , label it by , can again be defined by forcing the sequents to contain at most one formula on the right. Moreover, plus weakening is also known as affine logic, see Kopylov (1995).

2.2. Distributive Substructural logics

Since we used the notations decided by Girard (1987) to introduce linear logic, we keep this notation also for the other substructural logics, although the scholars in that area usually deploys different notation. A comparison with the notation used in substructural logics is presented in Table 4.

Girard (1987) Restall (2002)
1 1
0
0
Table 4. Notations

One of the peculiarity of linear logic with respect to other substructural logics is that in linear logic additive connectives are not distributive, that is the formulas and are not equivalent in . By contrast, relevant logics are in general distributive. This makes a critical difference form the point of view of semantics, cf. Troelstra (1992), and it has also serious consequences on the sequent calculus presentation. One may be tempted to define a rule of the sequent calculus that entails distributivity of the additives. Unfortunately, this is in general not possible without also assuming weakening and contraction. This is one of the important limitation of the sequent calculus, cf Ciabattoni et al. (2012). There is a number of ways to extend sequent calculi to cope with that. For instance, one may introduce hypersequents or display calculi Paoli (2002). For the present purposes, a presentation of the extensions of via Hilbert system suffices.

We start by presenting the Hilbert system for , label it by (cf. Troelstra (1992)). Then, we introduce its extensions. The concept of deduction of requires a tree-structure in order to handle the hypothesis in the correct resource-sensitive way.

  1. ,

  2. ,

  3. ,

Table 5. Axioms of

The notion of derivation in the Hilbert system for is the following.

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 5 (base cases).
(ii) We denote by a deduction tree with conclusion . If and are deduction trees, then the following are deduction trees (inductive steps).



Note that Hilbert system and the sequent calculus for are equivalent as expected: is provable in the sequent calculus iff the (multiplicative) disjunction of the formulas in is derivable from in .

Moreover, for the deduction theorem holds, that is if , then (cf. Troelstra (1992)). Note that we can also present plus weakening or contraction by adding to the suitable axioms. From linear logic, relevant logic is obtained by adding contraction (C) and distributivity of the additives (D1) and (D2). Therefore, the Hilbert system for is given by axioms 1- 20 plus the following (C), (D1) and (D2), cf. Paoli (2002); Restall (2002). The definition of derivation in simply extends our previous Definition 1.

  1. (W):

  2. (C):

  3. (D1): ,

  4. (D2):

We label the additive fragement of by , which includes the additive connectives of . We summarise the relationships between the logics that we have discussed in Figure 1. The Lambek calculus is the weakest logic that we discuss in this paper. By adding E, we obtain linear logic: by assuming exchange E, the conjunction becomes commutative and the two order-sensitive implications become equivalent. By adding contraction C and distributivity, we obtain relevant logic and by adding also weakening W, we obtain classical logic. By W and C, additives and multiplicatives become equivalent.

Figure 1. Classical and Substructural logics

3. A model of judgment aggregation for general logics

In this section, we adapt the model of Judgment Aggregation (see List and Puppe (2009); Endriss et al. (2012)) to cope with the logics that we have introduced. We have seen that reasoning about a number of propositions expressed in classical logics amounts to view the them as collected into a set. When discussing logics that lack weakening, contraction, or exchange, we have to replace the notion of a set of propositions (or judgments) of the standard JA with the notions of multiset or list of judgments. Recall that a multiset is a pair where is a set and is a function that assigns to each element of its multiplicity (i.e. a natural number). A list is multiset endowed with a strict total order : . is a submultiset of iff and for every , . is a sublist of iff it is a submultiset of and is the restriction of to . Moreover, we denote by the powerset of a multiset (list) , that is, the set of all multisets (lists) that are included in the multiset (list) .777Possible applications of viewing individual and collective judgments as multisets or lists are the following. Lists may encode propositions in an agenda with temporal dependency or, more generally, with a priority. The case of multisets is motivated as follows. As we discussed in Section 2, if we want to keep track of resource-sensitivity by means of logical reasoning, we have to drop contraction (C). That amounts to assuming that judgments form multisets. For instance, suppose that agents have to express their opinions on possible deals to trade goods (cf. Porello and Endriss (2010a); Porello and Endriss (2010b)). A deal that trades a single occurrence of a good with a single occurrence can be represented in linear logic by means of a formula . The implication states that one token of can be traded for one token of . In case we want to allow agents to express their opinions about “more” deals between s and s, we may add to the agenda a sufficient number of copies of goods and and of the “deal” .

Let

be a (finite) set of agents. We shall assume throughout this article that the number of individuals is odd and is bigger than 3. For every formula

, we define the complement of a formula as follows: , if is not negated, , if is negated and .888Note that, in intuitionistic logics, double negations do not cancel out. Even in case we were to assume that an agenda in intuitionistic logic is closed under (intuitionistic) negation, the condition of closure under complements does not rule out the presence of doubly negated formulas nor it makes equivalent to . An agenda is a (finite) set (multiset, list) of propositions in the language of a given logic (among those that we have previously introduced) that is closed under complements. Moreover, we assume that the agenda does not contain tautologies nor contradictions (in particular, it does not contain the units of the logic ), as usual in the JA literature (cf. List and Puppe (2009)).

Definition 2.

A judgement set (multiset, list) is a (finite) set (multiset, list) of elements of .

We slightly rephrase the usual rationality conditions on judgment sets in terms of derivability in a logic . With respect to a logic , we say that is complement-free if does not contain both and the complement of ; we say that is consistent iff 999In case the logic has a formula for expressing absurd, , this condition is reformulated as ; is complete iff for all , or ; and deductive closed iff whenever and , .

In JA based on classical logic, the notion of complement-freeness captures a weak form of consistency that is preserved by many aggregation procedures, viz. the majority rule.101010Complement-freeness has been introduced in Endriss et al. (2012) and corresponds to the property of weak consistency in Dietrich and List (2007). In classical logic, consistency entails complement-freeness. When dealing with JA in general logics, a significant departure from the standard JA framework is that consistency does not entail complement-freeness any longer. This is caused by the possible lack of weakening. Let be a logic where weakening holds, if a set (multiset) is inconsistent, then any (mutliset) such that is also inconsistent. This can be easily shown as follows: if and , then by weakening we infer . Thus, the lack of weakening permits that a consistent set may have inconsistent subsets, which may violate complement-freeness, even in the case of consistency of . As we shall see, this fact has significant consequences on the study of the aggregation of judgments. For this reason, we assume also the following condition.

Definition 3.

We say that a set (multiset, list) is robustly consistent if is consistent and every proper subset (submutliset, sublist) of is.

Robust consistency always entails consistency and it entails complement-freeness also for logics without weakening. In order to meet the standard treatment of JA, we will always assume that individual judgments sets are robustly consistent, while we shall state explicitly in case we assume that the judgments are also complete. The reason for dropping completeness is that in a number of non-classical logics, e.g. intuitionistic logic and in logics with no classical negation, is not appropriate to express the duality between acceptance and rejection by means of negation. The reason is that positive information and negative information have different statuses in constructive systems.111111Similar motivations for relaxing completeness and closure under complements have been introduced for the case of Description Logics in Porello and Endriss (2014).

Denote the set of all robustly consistent judgement sets (multisets, lists) on . In case we assume that individual judgment sets are also complete, we denote by the set of all robustly consistent and complete judgement sets (multisets, lists) on . A profile of judgements sets J

is a vector

, where

. In the remainder of the paper, we will occasionally use the characteristic function of a judgment set (multiset or list), with values in

to visualise profiles by means of a table. By slightly abusing the notation, we will do so also for multisets and lists.

3.1. Aggregation procedures

We intend to model aggregators that take profiles of judgments that are rational according to a given logic and return judgements which can be evaluated with respect to a (possibly) different logic . In case and are the same, we are in the standard situation in JA. When the languages of and are different, we would need to define a translation function from the language of into the language of . We mainly discuss embeddings of the outcome of an aggregation procedure into weaker or stronger logics that share the same language, with the exception of the discussion in Section 9.1.

Definition 4.

An aggregator procedure is a function .

This definition is quite general and allows in principles for associating profiles of any structure (sets, multisets, lists) to any type of structure (set, multiset, or list). In practice, as we shall see in the next sections and as we discussed in Section 2, the type of structure is determined by the logic that is used to assess individual and collective judgments. For instance, the standard JA model associates profiles of sets to sets of judgments. The same shall hold for the case of intuitionistic logic. When discussing JA for the Lambek calculus, aggregation procedures shall associate profiles of lists to lists of judgments and, in the case of linear and relevant logics, the aggregation procedures shall associate multisets to multisets. In Section 9.1, we shall discuss the case of aggregators that connect distinct logics, and in particular we focus on aggregators that associate sets of judgments to multisets of judgments.

Note that our definition of aggregation procedure allows for aggregators that return sets (mutlisets, lists) of judgments that are inconsistent w.r.t. . This is motivated by the fact that we want to assess the outcome of an aggregation procedure w.r.t. possibly different logics. That is why the codomain of is defined by the set of all sets (multisets, lists) .

In the standard JA framework, an aggregation function depends on the choice of an agenda , that determines the class of judgment sets, besides on the number of individuals. Here, in the general case, an aggregation function depends on two agendas: the agenda w.r.t. which individuals make their judgments and the agenda w.r.t. which the rationality properties of the collective set are evaluated . We denote by the class of aggregation functions from judgments defined on to judgments defined in , i.e. . Moreover, we use the notation when we assume that individual judgments are also complete, i.e. . Denote by the set , the majority rule for is defined as follows.

Definition 5.

The majority rule is defined by iff .

We also define the following class of aggregation functions that is a simple generalization of the majority rule, the class consists of the uniform quota rules, discussed by Dietrich and List (2007).

Definition 6.

Let , the uniform quota rule defined with quota is the aggregation procedure such that iff .

The usual properties of aggregation functions can be rephrased as follows. We start by listing those properties that are usually intended to model fairness desiderata of aggregation procedures.

  • Anonymity (A): For any profile J and permutation , .

  • Neutrality (N): For any and in and profile J, if for all , then .

  • Independence (I): For any and profiles , if , then .

  • Monotonicity (M): For any and profiles and if and , then

  • Acceptance-rejection neutrality (arN): For any and any profile , we have that if for all agents , then .

(A) states that the aggregator does not favour any particular agent, while (N) implies that it does not favour any particular proposition. (I) means that the outcome of w.r.t. a proposition in two different profiles only depends on the patterns of acceptance in the two profiles. (M) implies that, by increasing the support of a proposition, does not change its acceptance. Acceptance-rejection neutrality (arN) has been introduced in Dietrich and List (2008, 2009) in order to characterise aggregators in case of weak assumptions concerining individual rationality, namely in case individual judgements sets are just assumed to be consistent. (arN) means that the aggregator is not biased either for or against the acceptance of any proposition.121212This version of acceptance-rejection neutrality is due to Endriss et al. (2012).

The notion of collective rationality that is standard in JA (cf. List and Puppe (2009); Endriss et al. (2012)) states that an aggregation procedure is collectively rational iff for every profile, the output of the aggregation procedure is consistent, complete, and deductively closed w.r.t. classical logic. Here, we make explicit the logic with respect to which the aggregation is assessed, that is, we define the rationality conditions with respect to the logic that is used to evaluate the output of the aggregation.

  • is complement-free iff for every J, is complement-free w.r.t. .

  • is consistent w.r.t. iff for every J, is consistent w.r.t. .

  • is deductively closed iff for every J, is deductively closed w.r.t. .

  • is complete iff for every J, is complete w.r.t.

  • is weakly rational (WR) iff for every J, is complete and complement-free w.r.t.

  • is robustly consistent iff for every J, is robustly consistent wr

In case we do not assume that individual judgments are complete, we confine ourselves to the study of the preservation of (robust) consistency.

Let be a list of axioms among those above. We denote by the set of aggregation functions defined with domain and range that satisfy the axioms in .

In Endriss et al. (2012), two classes of aggregation functions can be characterized in terms of the axioms above. The first class just contains the majority rule, hence it is the characterization of the majority rule, which adapts May’s theorem for the case of JA, cf. May (1952). The second class is obtained by dropping weak rationality (WR) and corresponds to the class of uniform quota rules. The following proof adapts the one in Endriss et al. (2012) for the case of the logics that we have introduced. The significant difference is that, in order to show that the majority rule satisfies (WR), we have to assume that the individual judgment sets are robustly consistent.

Theorem 7.

is the majority rule iff

Proof.

From left to right, the majority rule satisfies the axioms. We only show that the majority rule is weakly rational. Assume that . For complement-freeness, suppose, by contradiction, that there exists a in such that both and are in . Then and . This entails that there exists an individual such that and are in , against the assumption that each is robustly consistent.131313Assuming only consistency of the individual sets is not sufficient, for instance is consistent in but not complement-free.

The majority rule is also complete. Suppose by contradiction that neither nor are in . Then and , which entails that there exists an such that violates completeness.

From right to left. Assume satisfies the axioms above. Since satisfies ,,, the outcome of only depends on the cardinality of the set of individuals accepting (see also List and Puppe (2009)). That is, can be represented by a function such that iff . Since satisfies (M), if and , then also . Suppose then that is the minimum for which . Since satisfies (WR), must be complete, we get that , otherwise there are profiles that lead to incomplete judgment sets. Since has to be complement-free, we get , to avoid acceptance of a formula and its negation. Thus, , hence is the majority rule. ∎

Note that there is no mention of preserving (robust) consistency at this point, Theorem 7 only shows that the majority rule preserves complement-freeness and completeness. By assuming mere consistency instead of robust consistency of the individual judgments, as in the standard JA result, cf. List and Puppe (2009), the proof fails for the case of logics without weakening. For instance, suppose J is composed of copies of : each is consistent but not complement-free, hence the majority would violate complement-freeness as well.

The class of uniform quota rules is characterized as follows. Since the rationality conditions do not enter the proof, we can simply adapt the proof in Endriss et al. (2012) to the present framework.

Proposition 8.

is a uniform quota rule iff

We adapt now the concept of safety of an agenda, that is due to Endriss et al. (2012). Since we are assuming that the individual agenda and the collective agenda might differ, the concept of safety applies to a pair of agendas.

Definition 9.

For any set of axioms , a pair of agendas is safe for axioms iff every in is robustly consistent.

The safety of an agenda amounts to assuming that every judgment aggregation problem defined on that agenda that uses aggregators of the given class preserves (robustly) consistent outcomes in . In case we obtain the standard definition of safety Endriss et al. (2012).

Since we are assessing JA in a variety of logics, it is useful to present the following definition of safety of logics for sets of axioms.

Definition 10.

A pair of logics is safe for axioms iff every pair of agenda is safe for axioms (i.e. for every pair of agendas , every in is robustly consistent.)

The concept of safety of logics amounts to assuming that for every possible agenda defined by means of the language of the logic and for every profile of judgments, the aggregation function preserves robustly consistent judgments defined w.r.t logic when evaluated w.r.t. the logic .

In case the aggregation procedure is defined w.r.t. a single agenda and w.r.t. a single logic, we say that the (single) agenda is safe for the class of axioms and that the (single) logic is. Note that concept of safety still applies to classes of axioms that define a single procedure, e.g. to the case of the majority rule that is defined by the axioms WR, A, I, N, and M. In that case, a possibility result — that states the existence of a procedure that satisfies a certain number of axioms and preserves consistency — and safety results — that state that every function in a certain class preserve consistency — coincide. We label the class of axioms that characterise the majority rule by MAJ.

The last concepts of the standard theory of judgment aggregation that we rephrase for this setting is the following list of properties of agendas. Here we simply generalise it to cope with a variety of logics.

Recall that a set (multiset, list) of formulas of is inconsistent w.r.t. iff . is minimally inconsistent iff is inconsistent and every subset (submultiset, sublist) of is consistent.

Definition 11.

The following properties define classes of agendas:

  • An agenda has the median property (MP) iff every minimally inconsistent set (multiset, list) of formulas of has cardinality at most 2.

  • An agenda has the simplified median property (SMP) iff every (non-trivially) inconsistent set (multiset, list) of formulas of has a subset (submultiset, sublist) such that and are logically equivalent: and .

  • An agenda has the syntactic simplified median property (SSMP) iff every (non-trivially) inconsistent set (multiset, list) of formulas of has a subset (submultiset, sublist) .

  • An agenda has the -median property (kMP) iff every minimally inconsistent set (multiset, list) of formulas of has cardinality at most .

The median property is the case with of the -median property. Moreover, the SSMP entails SMP which in turn entails MP. The opposite directions do not hold. The median property is due to Nehring and Puppe (2007). As we shall see, the median property characterizes, in case of classical logic, the agendas that are safe for the majority rule. The other properties are adequate to characterize agendas for larger classes of aggregators, cf. Endriss et al. (2012); List and Puppe (2009).

3.2. Summary of results

We summarise in the following table the results that we are going to establish in the subsequent sections concerning the safety of logics and agendas for a set of axioms. The first line of Table 6 simply rephrases the results about classical logic and classical agendas known from JA.

MAJ (WR, A, I, N, M) Quota rules (A, I, N, M) (WR, A, N ,I) (WR, A , N) (WR, A, I)
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
always safe safe with safe iff SMP safe iff SMP safe iff SSMP
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
always safe safe with safe iff SMP safe iff SMP safe iff SSMP
safe iff MP safe iff kMP safe iff SMP safe iff SMP safe iff SSMP
always safe safe with safe iff SMP safe iff SMP safe iff SSMP
Table 6. Summary of results concerning the safety of agendas and logics for sets of axioms.

As we shall discuss, all the monotonic logics (i.e. the logics whose sequent calculus admits weakening W) that are listed in the table exhibit, regarding the safety of the agenda, a situation that is analogous to that of classical logic. This is expected, due to the results in Dietrich (2007). However, dropping monotonicity is not sufficient to achieve safety or possibility results, as the situation of a number of non-monotonic logic shows (cf. , , ). The case of and are significant here: since every agenda in or is safe for MAJ, we can claim that the logics and are indeed safe for those axioms. In particular, as we shall see, the majority rule is robustly consistent for any agenda in or . Those systems are indeed non-monotonic however, to achieve safety, as we shall see, we have to restrict to the additive fragment of those logics.

Observe that when an agenda is not safe for a certain class of axioms AX, this means that there exists an aggregation procedure in the class of functions defined by means of AX that is not robustly consistent. Thus, if an agenda is not safe for axioms AX, this entails that the agenda is not safe for any subset of AX. Moreover, if a logic is not safe for a certain class of axioms AX, it means that there exists an agenda , such that it exists an aggregation function that is not robustly consistent. Namely, a possible outcome of is inconsistent in . If is inconsistent in (that is ), then is inconsistent in any logic that is stronger (i.e. that proves more sequents) than (i.e. ). Therefore, if a logic is not safe for AX, then any stronger logic is not safe for AX.

4. Judgment Aggregation in extensions of classical logic

For classical logic , we assume that every judgment set is also complete. Moreover, robust consistency and consistency in this case coincide. In our setting, List and Pettit’s result can be rephrased as follows.

Theorem 12 (List and Pettit).

There are agendas defined in that are not safe for (i.e. for axioms (WR, A, I, N, M)).

For instance, the agenda provides the famous discursive dilemma (cf. List and Pettit (2002); Kornhauser and Sager (1993)). That is, on that agenda, there is in fact a profile J such that the majority rule returns an inconsistent set.

1 1 1 0 0 0
1 0 0 0 1 1
0 0 1 1 1 0
maj. 1 0 1 0 1 0

The collective set is not consistent in classical logic. In proof-theoretic terms, it means that , and that can be shown as follows.

L L

Theorem 12 can be extended to various classes of functions, in particular all those classes that include the majority rule (cf. List and Puppe (2009); Endriss et al. (2012)).

Thus, by playing with our definitions, we can infer that classical logic is not safe for the majority rule and for any class of axioms that define procedures that include the majority rule.

Because of Theorem 12, in JA it is important to characterize which type of agendas are safe for a certain set of axioms. In particular, Theorem 12 can be refined by saying that the agendas that are not safe are those that violates the median property.

Theorem 13 (Nehring and Puppe (2007)).

An agenda satisfies the median property iff is safe for MAJ.

For larger classes of functions, the median property is no longer sufficient. The following proposition summarizes the relationships between agenda properties and classes of functions for classical logic.

Proposition 14.

[Endriss et al. (2012)] The following facts hold:

  • satisfies the SMP iff it is safe for

  • satisfies the SMP iff it is safe for

  • satisfies the SSMP iff it is safe for

  • satisfies the kMP iff it is safe for , (for , with

For the class of uniform quota rules, note that the choice of the threshold is crucial as well to preserve consistency. The median property is the condition that guarantees that an agenda in classical logic is safe for the majority rule. Since the median property is defined in terms of minimally inconsistent sets of formulas, any logic that conservatively extends classical propositional logic shall suffer the same problems of aggregation. If is minimally inconsistent in classical logic, then is minimally inconsistent in any conservative extension of classical logic. Therefore, there is no hope to mend propositional inconsistency by enriching the language of the logic.

Corollary 15.

Any extension of classical logic is not safe for MAJ.

For instance, any modal logic and any description logic, as soon as they extend propositional reasoning, are not safe (cf. for instance Porello and Endriss (2014) for description logics, and Pauly (2007), Endriss and Grandi (2014) for general Kripke structures).

5. Judgment aggregation in Intuitionistic Logic

We start the study of judgment aggregation in non-classical logics by dealing with intuitionistic logic. For , we do not assume that individual judgments sets are complete. Recall that intuitionistic negation is defined by .141414Assuming that the agenda may contain does not entail that the agenda contains the formula for false . That is, in this case, the agenda is not closed under the atoms occurring in the formulas. Such an agenda is called non truth-functional in Nehring and Puppe (2008). For intuitionistic logic, weakening holds, hence consistency and robust consistency are equivalent. We can easily see that with respect to judgment aggregation, intuitionistic logic does not significantly differ from classical logic.

Theorem 16.

Intuitionistic logic is not safe for MAJ.

Proof.

It is sufficient to exhibit an agenda w.r.t. which an aggregation problem generates an inconsistent set. For instance, we show that any agenda that includes is not safe for the majority rule in intuitionistic logic. There exists indeed a profile of judgments J (adapt the one we encountered in Section 4) such that . This set is inconsistent in intuitionistic logic, as the following proof shows.

R L

Also in the case of intuitionistic logic, the median property is the appropriate condition that characterizes safe agendas for the majority rule. The following proof largely adapts the known result for the case of non-complete judgments sets (cf. Porello and Endriss (2014)).

Theorem 17.

An agenda is safe for MAJ iff it satisfies the median property.

Proof.

In one direction, we show that if satisfies the median property, then the majority rule is consistent. Suppose by contradiction that there is an agenda that satisfies the median property and that there is a profile J such that is inconsistent. Since the median property holds, if is inconsistent, then there is a minimally inconsistent set with cardinality at most 2. Firstly, cannot have cardinality 1, otherwise there must be a contradictory formula in the agenda.151515In case we do assume contradictions in the agenda, we can reason as follows: if there should be a majority of agents for which is in , violating the assumption of consistency of the individual sets Suppose then that . Since is accepted by majority, we have that and . This entails that there is an individual , such that both and are in , against assumption that every individual judgment set is consistent.

In the other direction, we prove the contrapositive statement: if violates the median property, then is inconsistent. Suppose that violates the median property, then there exists a subset that is minimally inconsistent of size strictly bigger than 3. We construct a profile that violates the consistency of the majority rule. Suppose and that and are distinct formulas in . The first individuals accept all formulas of but , the last individuals accept all formulas of but , and the individual accepts just and . Each individual set is consistent, however, by majority, is accepted. That is, it is contained in . Since satisfies , if is inconsistent, then is inconsistent. ∎

Note that we can use the sole consistency assumption to conclude the argument that shows that the median property is necessary because satisfies weakening.

By enlarging the class of aggregation procedures beyond the majority rule, that is by focusing on uniform quota rules, we can show that the situation is similar w.r.t. the classical case.

Proposition 18.

satisfies the kMP iff it is safe for (A, I, N,M), for with .

The proof largely adapt the case for classical logic (cf Dietrich and List (2007), Corollary 2), so we omit it.

6. Judgment Aggregation in the Lambek Calculus

We start discussing substructural logics by studying the Lambek calculus . In this case, we do not assume that individual judgments are complete.161616In fact, the Lambek calculus is an intuitionistic logic, Abrusci (1990). Individual and collective judgments are (finite) lists of formulas that are sublists of . Since the aggregation procedures for return lists of judgments, we have to be explicit in defining how the aggregation computes the outcome. We present the case of the majority rule, other aggregation procedures can be handled in a similar way. We have to keep track of the positions of the formulas in the s with respect to the order of formulas in . Denote by the -th element of the list . We write a judgment set by means of , where is a designated symbol. is defined by if and is the position of in , otherwise . For example, the sublist of can be written by .

Given a profile J, let , that is, denotes the set of individuals that place at the -th position of their list of judgements.

Define the list , where is the length of the list and each is either a formula of the agenda of the designated symbol : if and , if . is then the sublist of that is obtained by removing all the symbols.

It is easy to see that collective inconsistencies may emerge also for .

Theorem 19.

The Lambek calculus is not safe for MAJ.

Proof.

Take any agenda that includes and a profile of judgments J as follows.

1 1 1 0
1 0 0 1
0 1 0 1
maj. 1 1 0 1

The outcome of the majority rule on J is the list , which is not consistent in , as the following proof shows.

L L

In consistency and robust consistency are not equivalent. For instance, is consistent, since it does not entail , whereas is inconsistent. The following example shows that the majority rule for may return an inconsistent list even in an extremely simple agenda.

Example.

Take the agenda and the following profile of (incomplete) lists of judgments.

1 1 1
1 0 0
0 0 1
maj. 1 0 1

Each individual judgment list is consistent, however the majority returns the inconsistent list .

Therefore, in case we assume just the consistency of the individual judgments, the median property is not sufficient to guarantee the consistency of every collective outcomes. The condition of robust consistency is required, as the following proof shows.

Theorem 20.

An agenda satisfies the median property iff is safe for MAJ.

Proof.

From left to right, suppose by contradiction that satisfies the median property and is inconsistent. Then, contains a minimally inconsistent list of size smaller or equal than 2. cannot be 1, since we excluded contradictions in the agenda. Thus , and suppose that and take the -th and -th positions (respectively) in the list . Therefore, there exist and , with , such that and , which entails that there is an individual such that is a sublist of . Note that and may be contiguous in or not, however, in both cases, this contradicts the assumption of robust consistency.

From right to left, we show that if violates the median property, then the majority rule is inconsistent. Assume that contains a list that is minimally inconsistent of size bigger than 3. Define a profile J that coincides with the one in the proof of Theorem 17 for the formulas in and, for the other formulas of the agenda, the individuals reject all of them. Note the judgment sets are now incomplete. By construction, we get , which is inconsistent in . ∎

For the larger class of uniform quota rules, we can adapt the known results, by noticing that in this case, again, the safety result applies to the condition of robust consistency.

Proposition 21.

satisfies the kMP iff it is safe for , for with .

7. Judgment Aggregation in Linear Logic

We show that for linear logic analogous impossibility results can be stated. We assume that the judgments form multisets of formulas and that they are complete. In , due to the lack of weakening, consistency and robust consistency are not equivalent.

Theorem 22.

(Multiplicative additive) linear logic is not safe for MAJ.

Proof.

Take the agenda and the following profile:

1 1 1 0 0 0
1 0 0 0 1 1
0 0 1 1 1 0
maj. 1 0 1 0 1 0

The outcome of the majority rule on J is the multiset which is inconsistent w.r.t. :


Again, the median property is not sufficient to preserve consistency, in case we assume that individual judgments are just consistent.

Example.

Take the agenda closed under complements . The agenda satisfies the median property, since there is no inconsistent multiset of size strictly bigger than 2. Take the following profile.