The aim of this essay is to investigate the so-called discursive dilemma, sometimes referred to in the literature as the doctrinal paradox, which is a related but less general concept. This impossibility theorem generalizes Condorcet’s classic voting paradox from the eighteenth century. Our focus is to highlight certain properties of the dilemma within a very general framework, emphasizing the role of the notion of (in)consistency.

We will initially examine the use of an underlying supra-classical paraconsistent logic that can controllably re-encode classical logic. This approach allows for a more fine-grained analysis of the concept of consistency, specifically through the use of Logics of Formal Inconsistency (LFIs).

keywords: Knowledge Representation and Reasoning (AI); Social Choice; Judgment Aggregation; Collective Rationality; Non-classical Logics; Paraconsistent Logics.

State-of-the-art

How can a group of individuals make consistent collective judgments on a set of propositions based on the members’ individual judgments? This is the central question that underlies Judgment Aggregation (JA), a relatively recent theory that combines aggregation problems previously studied by Social Choice theory with the tools and methods of Mathematical Logic.

As introduced by Grossi and Pigozzi (2014), upon whose book the state-of-the-art section of this essay is based, the study of social decisions has a long history. Amartya Sen (1999) recalls that as early as the fourth century B.C., Aristotle in Greece and Kautilya in India explored how different individuals could make social decisions. However, the systematic and formal study of voting and committee decisions began only during the French Revolution, thanks to French mathematicians like Borda, Condorcet, and Laplace.

These issues have traditionally been of interest to disciplines like Economics, Political Science, and Philosophy. Recently, given the general and unifying paradigm for the formalization and understanding of aggregation problems, JA has also captured the attention of disciplines in Computer Science, especially in the areas of Artificial Intelligence, Knowledge Representation and Reasoning, and Multi-agent Systems. JA is a multidisciplinary theory that provides tools for defining the ascription of propositional contents to collective agents and explores logical methods to aggregate conflicting information from sensors, heterogeneous opinions of experts, and databases.

Discursive dilemma

Much of the literature has claimed that Judgment Aggregation raises serious concerns, particularly some well known impossibility results. The very formal work on JA had risen from one of them, the so-called doctrinal paradox advanced by Kornhauser and Sager (1986) from the area of jurisprudence, to be here introduced following List (2002).

Doctrinal paradox. Suppose a collegial court consisting of three judges has to reach a verdict in a breach-of-contract case. There are three propositions on which the court is required to make judgments:

p : The defendant was contractually obliged not to do a particular action.

q : The defendant did that action.

r : The defendant is liable for breach of contract.

According to legal doctrine, propositions p and q are jointly necessary and sufficient for proposition r – being so, r↔p∧q. Suppose now that the three judges disagree about the case, as shown in Table 1.

	p	q	r
Judge 1	True	True	True
Judge 2	True	False	False
Judge 3	False	True	False
Majority	True	True	False

If the three judges take a majority vote on the conclusion r, the outcome is a ’not liable’ verdict. But if the majority is taken on the premises, then the background legal doctrine dictates a ’liable’ verdict. The court’s decision thus depends on the aggregation rule used.

A similar problem by defining a collective outcome by majority rule (the paradigmatic democratic aggregation rule) was already stated by Condorcet (1785) – given individual preferences the pairwise majority rule can lead to an inconsistent outcome. As later proved by Arrow (1963), that issue is a more general one: there exists no social welfare function (a function that maps any n-tuple of individual preference orders to a collective preference order) that satisfies a specific small number of desirable conditions intended to cope with some notion of fairness. Informally, we have the following.

Arrow’s impossibility theorem. No ranked preferences of individuals can be converted into a community-wide (complete and transitive) ranking satisfying the following criteria:

• Nondictatorship: No individual possesses the power to always determine the group’s preference.
• Pareto Efficiency: Unanimous individual preferences must be respected.
• Independence of Irrelevant Alternatives: If a choice is removed, then the others’ orders should not change.
• Unrestricted Domain: Voting must account for all individual preferences.
• Social Ordering: Each individual should be able to order the choices in any way and indicate ties.

Just as Arrow’s theorem shows that Condorcet’s paradox is a particular case of a wider result, List and Pettit (2002) re-interpreted the Doctrinal Paradox by adding the very legal doctrine (i.e.  the fact that r ↔p∧q  is true) to the set of issues on which the judges have to vote. The result is that the group reaches an inconsistent decision like {p, q, r↔p∧q, ¬r} – that’s the so-called discursive dilemma. All those results show that when individual choices are combined into a collective one, some expected rationality constraints that were satisfied at the individual level are lost.  Just as majority voting can yield cyclical collective preferences, logical consistency can be lost in collective judgments on any set of binary issues with certain logical interconnections. The formal approach to those issues are the core of Judgment Aggregation.

Judgement aggregation

JA’s framework was first proposed by List and Pettit (2002) and further developed by Dietrich and List (2007) and other subsequent papers. The underlying logic considered is PCL. An aggregation is defined simply as a set of agent’s decisions upon a given set of issues.

Definition 2.1 (Judgment aggregation problem). Let L be a propositional language on a given set of atoms At. A judgment aggregation problem for L is a tuple J=⟨N, A⟩ where:

• N is a finite non-empty set;
• A ⊆ L such that A = {ϕ|ϕ ∈ I} ∪ {¬ϕ|ϕ ∈ I} for some finite I ⊆ L which contains only positive contingent formulae.

N is the set of agents (individuals or voters). A is the agenda and I is the pre-agenda (set of issues) of A. ±I denotes an agenda based on I. [A] denotes a pre-agenda of A.

Definition 2.2 (Judgment set). Let J = ⟨N, A⟩ be a judgment aggregation problem. A judgment set for J is a set of formulas J ⊆ A such that:

• J is consistent;
• J is complete, i.e., for all ϕ ∈ A, either ϕ ∈ J or ¬ϕ ∈ J.

Instead of ϕ ∈ J, it is used the notation J |= ϕ to indicate that ϕ belongs to a judgment set J.

Definition 2.3 (Profile). Let J ⊆ ℘(A) be the set of all judgment sets J ⊆ A. A

judgment profile P = ⟨J_i⟩_i∈N ∈ J^|N| is a |N |-tuple of judgment sets.

Notation 2.4. P_i denotes the i^th entry of P , namely the judgment set of agent i in P . For ϕ ∈ A we use P_ϕ to denote the set of individuals accepting ϕ in P : {i ∈ N |P_i |= ϕ}. P denotes the set of all judgment profiles.

J_i ∈ P denotes (as a notation abuse) that a judgment set J_i belongs to P .

Definition 2.5 (Aggregation function). Let J = ⟨N, A⟩ be a judgment aggregation problem. An aggregation function for J is a function f : P→ ℘(A). The output set f (P ), where P = ⟨J_i⟩_i∈N , is sometimes denoted by J, to be called collective set. A collective set J which is a judgment set is called a collective judgment set.

There are in the literature several aggregation rules for defining aggregation func- tions. One example is majority rule, which asserts that ϕ is collectively accepted if and only if there is a majority of individuals accepting it.

Example 2.6 (Majority rule).

Where, for x ∈ Q, [x| is the smallest integer greater or equal to x.

Aggregation conditions are properties of aggregation functions, and the major issues in JA is how to justify the application of those conditions over aggregation functions, and vice-versa. There are basically two sets of conditions: output conditions, expressing the properties of an aggregation function’s output, and mapping conditions, expressing properties of how the input is mapped.

Definition 2.7 (Output conditions). Let J = ⟨N, A⟩ be a judgment aggregation problem and X ⊆ A. An aggregation function f for J is:

• Consistent iff for all P ∈ P, f (P ) is a consistent set of formulas.
• Complete iff for all ϕ ∈ A, for all P ∈ P, ϕ ∈ f (P ) or ¬ϕ ∈ f (P ).
• Closed iff for all ϕ ∈ A, for all P ∈ P, if f (P ) |= ϕ then ϕ ∈ f (P ).
• Collectively rational (RATIO) iff f (P ) is consistent and complete.

Some conditions that state constrains on how the aggregation maps the input to the output, appealing to some intuition of what is reasonable (or fair) are, among many others, the following.

Definition 2.8 (Mapping conditions). Let J = ⟨N, A⟩ be a judgment aggregation problem. An aggregation function f for J is:

• Anonymous (AN) iff for all P, P‘∈ P such that P‘ is a permutation of P: f (P )=f(P^‘); where the permutation µ(P ) of P = ⟨J_i⟩_1≤i≤|N| is the profile ⟨J_µ(i)⟩_1≤i≤|N|.
• Unanimous (U) iff for all ϕ ∈ A, for all P ∈ P : if [for all i ∈ N : P_i |=ϕ]  then ϕ ∈ f (P ).
• Responsive (RES) iff for all ϕ ∈ A, exists P, P^‘ ∈ P such that ϕ ∈ f (P ) and ¬ϕ ∈ f (P^‘).
• Dictatorial (D) iff exists i ∈ N such that for all P ∈ P : f (P ) = P_i.
• Oligarchic (O) iff exists O ⊆  N such that O≠∅, for  all  ϕ  ∈  A,  for  all  P ∈ P :[∩_i∈O P_i =_ϕ f (P ) ].
• Monotonic (MON) iff for all ϕ∈ A, for all i∈ N, for all P, P^‘∈ P: if [P =_-i P and P_i /|=ϕ and P‘_i |= ϕ] then [if ϕ ∈ f (P ) then ϕ ∈ f (P^‘)].
• Independent (IND) iff for all ϕ ∈ A, for all P, P ‘∈ P: if [for all i ∈ N : P_i =_ϕ P’_i] then f (P ) =_ϕ f (P^‘).
• Neutral (NEU) iff for all ϕ, ψ ∈ A, for all P ∈ P : if [for all i ∈ N : P_i |= ϕ iff P_i |= ψ] then  [ϕ ∈ f (P ) iff ψ ∈ f (P )].
• Systematic (SYS) iff for all ϕ, ψ  ∈A, for all P, P^‘ ∈ P: if [for all i ∈ N : P_i |= ϕ iff P‘_i |= ψ] then [ϕ ∈ f (P ) iff ψ ∈ f (P^‘)] .
• Unbiasedness (UNB) iff for all ϕ ∈ A, for all P, P^‘ ∈ P: if  [for all i  ∈   N : P_i |= ϕ iff P’_i |= ¬ϕ] then [ϕ ∈ f (P ) iff ¬ϕ ∈ f (P^‘)].

Observation 2.9 (Some properties of f_maj). Let J = ⟨N, A⟩ be a judgment aggregation problem:

1. f_maj does not satisfy D;
2. f_maj satisfies U, RES, AN, MON, IND, NEU, SYS, UNB;
3. If A is simple, then f_maj satisfies RATIO iff |N | is odd.

Where simple agenda is an agenda that does not contain one set X minimally inconsistent with 3 or more issues.

Corolary 2.10 (Discursive dilemma). f_maj does not satisfy RATIO in general.

The discursive dilemma is only one of several impossibility results that have driven the literature on JA. In a nutshell, what all of them have in common is the fact that seemingly desirable constraints on the agendas and aggregation functions may lead to undesirable properties or even to degenerative forms of aggregation – for instance, to functions satisfying D or O. Just to name one example, Nehring and Puppe (2010) have shown that, if an agenda is non-simple, every aggregation function satisfying RATIO, SYS and MON is a dictatorship.

As advanced by several authors, possibility results can be found by relaxing agenda and aggregation conditions: relaxing output conditions, mapping conditions (mostly IND, for example when choosing between premise/conclusion-based functions in the doctrinal paradox), or even relaxing the universal domain in the very definition of an aggregation function (cf. List (2012)).

Paraconsisten Jugment Aggregation?

List and Pettit’s (2002) seminal model was the starting point of several advancements in JA to distinct directions. In fact, some works have shown the fact that the so-called JA paradoxes are not a peculiarity of PCL: as it is highlighted by Grossi and Pigozzi (2014), aggregation is potentially difficult whenever the to-be-aggregated issues are related to some notion of (not necessarily classical) inconsistency, and hence of logical consequence.

(Para)consistency in general JA framework

An important advancement in this direction is presented by Dietrich (2007). By abstracting away from the specificity of PCL, and focusing on logical consequence from a general structural standpoint, Dietrich advances an unified (general) JA model, despite of the diferences between particular logics.  Several previous results (suitably restated to cope with the general framework) are shown to hold in general logics. Specifically only two conditions are necessary for formalizing aggregation issues in terms of JA, namely:

1. Decision should be able to be logically constructed as the acceptance or rejection of propositions.
2. The informational basis of a decision can be construed as the individuals’ acceptances/rejections of these propositions, according to some relevant notion of individual acceptance/rejection.

Let a logic be a pair (L, |=) consisting of a non-empty set L of propositions, such that p ∈ L implies ¬p ∈ L; and a binary relation |= (⊆ ℘(L) × L) between sets A ⊆ L and propositions p ∈ L. Dietrich (2007) shows that (im)possibility results can be reassessed based on constrains entailed by rather general conditions on the underlying logic, namely:

• Self-entailment (L1) for any p ∈ L, p |= p
• Monotonicity (L2) for any p ∈ L, A ⊆ B ⊆ L, if A |= p then B |= p.
• Completability (L3) ∅ is consistent, and each consistent set A ⊆ L has a consistent superset B ⊆ L containing a member of each pair p, ¬p ∈ L.
• Non-paraconsistency (L4) For any A ⊆ L and p ∈ L, if A ∪ ¬p is inconsistent then A |= p
• Compactness (L5) For any p ∈ L and A ⊆ L, if A |= p then B |= p for some finite subset B ⊆ A.

Dietrich demonstrates that L1-L3 (plus L5 in some cases) is often appropriate when consistency and completeness are the only rationality conditions under consideration. L4 or weaker versions of it may become additionally useful when deductive closure is taken into account in defining a suitable form of RATIO.

JA in substructural logics: relaxing consistency

Alongside this general approach, some specific non-classical logics for JA have been investigated in the literature – most of them extensions of PCL and coping with L1- L5. However relatively little work on substructrual logics have been made, as it is outlined by Porello (2017). As emphasized by the latter, the motivations for studying JA in substructural non-classical logics are essentially three: (i) the theoretical interest in extending JA into mathematically different logics; (ii) questioning the adequacy of classical connectives to model reasoning; and (iii) investigating whether by weakening the logic standard impossibility results can be circumvented. Thus, alongside the search for possibility results by relaxing agenda and aggregation conditions, it is also possible to relax some logical constrains.

Indeed, by studying JA in weaker logics than PCL a fine-grained analysis of the inferences that are responsible of collective inconsistency can be provided. A representative example is advanced by List and Pettit (2011): even if individuals reason by means of the full power of classical logic it is still possible to reassess collective rationality with respect to weaker logics thus preserving (some notion of) consistency. A similar approach was advanced by Carnielli and Lima-Marques (1999) respect to a society semantic understanding of paraconsistent logics.

JA and Formal (In)consistency?

Reassessing RATIO with respect to JA in weaker logics provides a way to mitigate the discursive dilemma, at least in the sense that it is no longer strictly constructed as a problem of classical logical consistency. However paraconsistent logics were never discussed in details because, as emphasized by Porello (2017), the interest is studying the preservation of consistency via aggregation procedures – and the problem is that, according to the latter, this objective cannot be accomplished in paraconsistent logics where the focus is precisely the inconsistency-tolerance.

Summing up, according to the literature paraconsistent logics are weak enough to circumvent the discursive dilemma; however, too weak to cope with desirable possibility results – including the very definition of aggregation. Our focus is to originally suggest the use of a family of supraclassical paraconsistent logics – the Logics of Formal Inconsistency (LFIs) advanced by Carnielli and Marcos (2002) and further studied by Marcos (2005) and Carnielli, Coniglio and Marcos (2007). In those logics, PCL can be faithfully re-encoded where requested – preserving the aforementioned desired substructural properties and still being strong enough to cope with aggregation procedures.

LFIs

The LFIs internalize in the object language the very notions of consistency and inconsistency by means of specific connectives (primitives or not), generalizing the strategy adopted by da Costa (1974). Formally:

Definition 3.1  (LFI, Carnielli and Marcos (2000)).  Let L be a Tarskian logic with a negation ¬. The logic L is a LFI if there is a non-empty set ◯(p) of formulas in the language of L which depends only on the propositional variable p, satisfying the following:

a. ∃α∃β(¬α, α /|- β)

b. ∃α∃β(◯(α), α /|- β)

c. ∃α∃β(◯(α), ¬α /|- β)

d. ∀α∀β(◯(α), α, ¬α |- β)

For any formula α, the set ◯(α) is intended to express, in a specific sense, the consistency of α relative to the logic L. When this set is a singleton, it is denoted by

• α the sole element of ◯(α), and in this case ◦ defines a consistency operator. Roughly speaking ◦α means “α is consistent”  so that in every LFI:

                                                                                                              As aforementioned ◦ is not necessarily a primitive connective of the signature of L. It is worth mentioning that the dual connective can also be defined, that is, • where

• α means that “α is inconsistent”.

The distinctions given by the LFIs not only separates the notion of contradiction from deductive triviality (like every paraconsistent logic), but also (in)consistency from (non-)contradiction. So there is a clear distinction between contradictions that can be accepted from those that cannot. According to Carnielli and Marcos (2000), the idea to be captured is that no matter the nature of the contradictions a reasoner is willing to accept, there still are contradictions that cannot be accepted at all.

Those distinctions and relations are in the core of our future projects, serving as a basis for reassessing RATIO and, accordingly, the discursive dilemma – the LFIs are the best candidate to axiomatize List and Pettit’s intuition of collective agency – namely the fact that even if individuals reason by means of the full power of classical logic it is still possible to reassess collective rationality with respect to weaker logics.

Acknowledgements: This essay is a version of a research project developed with the assistance of Prof. Dr. Richard Booth (Cardiff University).

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