In a previous work we introduced a vector-based extension of value-based argumentation for public interest communication aimed to provide an articulated model of the impact of a communication campaign on a set of target audiences. The proposed model was monotonic, intuitively meaning that adding arguments to a campaign and enlarging the set of the values they cover cannot decrease the efectiveness of the campaign itself. As this property does not always hold in practice, in this paper we extend the model in order to encompass nonmonotonic efects both at the level of quantitative measures of campaign impact and of the acceptability of the campaign arguments with respect to a standard argumentation semantics. In both cases, we identify some suficient conditions for monotonicity and provide a preliminary discussion about their relevance and applicability in practice.
— Increasing fruit and vegetable consumption supports animal welfare. — A plant-based diet has a lower environmental footprint. — Choosing locally sourced produce benefits the local economy.
Despite their importance, public interest campaigns often face significant challenges. Inefectiveness
or backfire efects, caused by poorly targeted communication, are common issues, as evidenced by
numerous unsuccessful and costly campaigns [
Since PIC campaigns are essentially made of arguments, it is natural to leverage on tools from formal
argumentation [
These assumptions imply somehow that, in terms of argument impact, the more the better, an efect
we will refer to, in what follows, as monotonicity. While monotonicity may hold in some contexts, there
are also situations where the addition of arguments and an extended coverage of values can be harmful
and lead to a less efective communication. In fact, theories of basic human values as [
Capturing this kind of situations requires a nonmonotonic behavior to be represented by the adopted
model. In this work we address this requirement by investigating modifications of the model introduced
in [
The paper is organised as follows. Section 2 recalls the necessary background notions, while Section
3 reviews the monotonic model of argument impact proposed in [
In this section we provide the necessary background on Dung’s theory of abstract argumentation (with focus on the notion of acceptability) hinting at how to expand it into value-based argumentation so that it fits the purposes of our conceptual model.
Dung’s theory of abstract argumentation treats arguments as abstract entities, whose internal structure and properties are abstracted away, and focuses only on conflicts between them. The key notion is that of an argumentation framework, defined as follows: Definition 2.1. An argumentation framework (AF) is a pair = ⟨, ℛ⟩, where is a set of arguments and ℛ ⊆ ( × ) is a binary relation on .
When (, ) ∈ ℛ (also denoted as ℛ) we say that attacks . For a set ⊆ and an argument ∈ we write ℛ if ∃ ∈ : ℛ and ℛ if ∃ ∈ : ℛ, and we denote the arguments attacking as − ≜ { ∈ | ℛ} and the arguments attacked by as + ≜ { ∈ | ℛ}.
The relation of attack is the basis for the evaluation of the acceptability of arguments, given that the conflict among them prevent to accept them all together. Acceptability is determined by argumentation semantics. Formally, an argumentation semantics specifies the criteria for identifying, for a generic AF, a set of extensions, where each extension is a set of arguments considered to be acceptable together. Given a generic argumentation semantics , the set of extensions prescribed by for a given framework is denoted as ℰ ( ).
Typical (minimal) criteria for a set of arguments constituting an extension are conflict-freeness , the
absence of conflict between its members, and self-defense the capacity of attacking every external
attacker. Definition 2.2 recalls these notions and the definition of the grounded semantics, which is the
only one we use in this paper. For more details, the reader is referred to [
Definition 2.2. Let = ⟨, ℛ⟩ be an argumentation framework, ∈ and ⊆ . is conflictfree, denoted as ∈ ℰCF( ), if ∩ − = ∅. is acceptable with respect to (or is defended by ) if {}− ⊆ +. The function : 2 → 2 which, given a set ⊆ , returns the set of the acceptable arguments with respect to , is called the characteristic function of . is admissible (denoted as ∈ ℰAD( )) if ∈ ℰCF( ) and ⊆ (). is the grounded extension (denoted as = GR( ) or ∈ ℰGR( )) if is the least fixed point of .
Value-based argumentation frameworks have been introduced in [
ℛ and () ⊀ () In words, an argument defeats for audience only when attacks in the ordinary sense and the audience does not rank the value of higher than that of . The acceptability of arguments according to audience can then be derived by applying an argumentation semantics to .
The approach summarized above, where each argument relates exactly to a single value and values are ordered diferently by diferent recipients, is arguably the most immediate one for representing how a value dimension may be attached to argumentative discourse. Yet, there are at least three aspects that are worth rediscussing in the context of our analysis of PIC campaigns. First, arguments may refer to more than a single value, as soon as they pertain to an articulated conceptual framework, while there can also be arguments which are not associated to any value. Second, as stressed by diferent theories of human values [14, 15], an argument may be anchored to values with diferent degrees of intensity and this matters for the recipient. Third, also relating to these empirical findings in psychology, some values may be mutually incompatible: a strong appeal to one value may negatively impact the persuasive force of an argument for certain audiences.
Regarding the first aspect, a generalizing approach by [ 16] already allows arguments to refer to multiple values (or none), where the ordering relation is lifted accordingly to sets of arguments. However, besides other shortcomings of this approach, the second and third aspect are hard to deal with a simple ordering of elements, but rather require a numerical treatment. Bringing these desiderata together, it is natural to use vectors with weighted coordinates to handle reference to multiple values and with diferent intensity. We instead replace the simple ordering among values, by a measure of their impact on an audience, as resulting from the distribution of weights to the vector’s coordinates. The formal details of this approach are recapitulated in Section 3.
In this section we recall, with some minor formal adjustments, the model introduced in [
We assume that given a domain2 of interest (e.g., the promotion of a greener diet) there is a reference
universe of potential arguments, denoted as and a set of relevant values . Both sets are assumed
to be finite. The key point of our approach is that each argument ∈ is characterized by the set of
values it promotes, where an argument can promote several values to diferent extents.
Definition 3.1 (Space of values). We assume that there is a finite set of values with cardinality
|| = . Each value is identified by a number in 1 . . . . The space of values is = [
Arguments in the universe may attack each other, giving rise to a universal argumentation framework for the considered domain.
Definition 3.2 (Universal argumentation framework). Given a universe of potential arguments for a given domain, we assume the existence of a binary attack relation ℛ ⊆ × . The universal argumentation framework, denoted as ℱ , for the domain is defined as ℱ = ⟨, ℛ⟩. The attack-relation ℛ is assumed to be subject-independent. When (, ) ∈ ℛ we say that is a potential attacker of .
In each domain, there is a set of audiences, representing the potential targets of communication campaigns. Each audience ∈ will have their own preferences among values. Based on the discussion of Section 2, we represent this associating a profile of weights (one for each value coordinate) with each audience.
Given a set of audiences and a space of values , an audience specific value function asv : → assigns to each audience ∈ a vector asv() ∈ whose -th entry represents the importance that the audience assigns to value . 2To avoid a too heavy notation, in the following we leave implicit the domain of interest , as it is unique for each application context. For instance we indicate the universe of arguments as rather than as or any other notation evidencing the connection with . 3In the following we will use this notation to identify the -th element of any kind of vector occurring in the paper.
Besides having preferences on values, each audience has an initial mindset on the considered domain, represented by the arguments in the universe which are known to the audience before a communication campaign is started.
Definition 3.4. Given a set of audiences and a a universe of potential arguments, the set of arguments initially known to each audience ∈ is denoted as 0 ⊆ .
A campaign in a given domain consists of one or more arguments which are communicated to a selected set of audiences representing the target of the campaign. We assume that each campaign has a goal and that, on the basis of the goal, the subset of the arguments which are eligible for the campaign can be identified.
Definition 3.5. A PIC campaign is characterized by a goal , whose nature is left unspecified. We assume that for each possible goal a set ⊆ of relevant arguments is identified. The set of all possible PIC campaigns for a goal , denoted as consists of all non-empty subsets of , namely = { ⊆ | ̸= ∅}. Given a campaing ∈ the corresponding set of arguments is denoted as args(). In the following, with a little abuse of notation, we will sometimes equate a campaign with its set of arguments where this is not ambiguous.
An impact measure is meant to give an account of how much each argument is efective for a given
audience. In [
‖ · ‖ : →− − val ⊆ R →−n R
In words, given an argument , its impact on audience is determined on the basis of its value of vectors
val() which is then synthesised into a single real number, measuring the impact, through an audience
specific function n. In particular, in [
In order to model the impact of a campaign consisting of several arguments and directed to a set of audiences, some further notions have to be introduced. First we define the set of possible targets of a campaign, which need to be associated with a weighting, since diferent audiences may matter to diferent degrees with respect to the goal of the campaign (e.g. improving dietary habits can be more important for younger than for elder people).
Definition 3.6. A set of target audiences for a campaign is any non-empty subset of the set of audiences. Given a set representing a set of target audiences, we assume that each audience ∈ is assigned a weight (), representing its importance in the campaign, which satisfies the following constraints.
∀ ∈ , () ≥ 0, ∑︁ () = 1. ∈
The overall impact of a campaign for a given set of target audiences can then be defined as a weighted sum of the impact, on each audience, of the arguments it contains4.
Definition 3.7. Given a campaign for a goal consisting of a set of arguments args() ⊆ and a set of target audiences, the overall impact of with respect to , denoted as ℐ(, ) is defined as
∑︁ ∈args() ‖‖ (2)
The model of impact summarized above is monotonic in two respects: with respect to the values promoted and with respect to the arguments included in the campaign.
As to the former aspect, given two value vectors 1, 2 ∈ , assume the following relation: 1 ⪯ 2 if for every 1 ≤ ≤ 1 ≤ 2. Then, it is easy to see that assuming the use of equation (1), and indeed of any monotonic seminorm n, for any audience it holds that if for two arguments and val() ⪯ val(), then ‖‖ ≤ ‖ ‖. In words, the more an argument is able to promote all values the higher is its impact, or to say it compactly promoting more values cannot do harm.
As to the second aspect, the purely additive model encompassed by Definition 3.7 is based on the idea that each argument included in the campaign represents a further opportunity to convince the target audiences and hence, to say it compactly making more articulated campaigns cannot do harm.
Both aspects of monotonicity can be reasonable in some contexts, but fail to capture phenomena like backfire efects and would correspond to a rather simple criterion for campaign design: include as many arguments as possible promoting as many values as possible. The task of campaign design in practice is typically much more complex than this criterion would suggest. In next section, we discuss therefore alternative modeling options encompassing nonmonotonic impact measures.
The measure introduced in equation (1) captures the intuition that the contribution of each value to the impact is non-negative. The idea is that if an audience does not care about a value (e.g. the relevant weight in asv() is zero), it will be essentially indiferent to the presence of among the values promoted by an argument , but this will not afect negatively the impact determined by the other values promoted by . While this is reasonable in some contexts, there are also situations where one can imagine that the impact of an argument depends on how close the values promoted by the argument are to the values which are important for the audience. This means that the presence of a value which is not shared by the audience can afect negatively the impact of the relevant argument, possibly countering the role of other values. This may occur, for instance, in the cases where an audience has some strong biases and/or radical opinions and is inclined to reject every communication which, even partially, bears some similarity with positions perceived as “opposite” in some sense.
To represent this kind of situations, a nonmonotonic behavior of the argument impact measure is required, such that an audience can react negatively to the promotion of some values. We consider here the option of assessing impact in terms of the distance between the value vectors of the argument and of the audience. This leads to consider a combined vector whose elements result from the element-wise diference (in absolute value) rather than product of the value weights. This resorts to assessing the distance between the value vector the argument refers to and the audience specific value function. The impact of the combined vector can, in general, be measured through a norm, as seen in Section 3.
In general, a family of alternative distance measures can be considered. We assume that the impact
is the complement to 1 of the distance between the value vector of the argument and the one of the
audience. In words, if the two vectors coincide the impact is 1, otherwise it decreases with the distance
4This is a very simple instance of aggregation method, the consideration of alternative aggregation functions like min, max
and, more generally, OWA operators [17] is left to future work.
and becomes 0 if the two vectors contain the opposite extremes (i.e. one is 0 and the other is 1 or vice
versa) for each value. This can be captured by a distance-based function ‖ · ‖ : → [
⎷ =1 Definition 4.1. Given a campaign for a goal , with argument list ℒ() of cardinality , and a set of target audiences, the position-aware overall impact of with respect to , denoted as ℐ(, ) is defined as ℐ(, ) = ∑︁ () · ∈
∑︁ ‖‖ · ()pos()
Equation 3 overcomes the first kind of monotonicity discussed at the end of Section 3. Here, promoting more values, namely increasing some element of val() can be harmful since it may increase the distance term which is subtracted to 1 and hence may decrease the impact measure.
Turning to second kind of monotonicity, a further consideration concerns the possible negative aspects of multiplicity. In the model introduced in Section 3, potentially every addition of a further argument to a campaign can increase its collective impact. However, this clashes against the intuition that long and verbose messages are often less efective than concise and focused ones. Moreover, if there are multiple items in a message, it is common that they do not receive the same attention, as (some) people more easily focus on the first (and possibly the last) parts of a communication while it is more likely that they overlook the contents in the middle.
To provide a formal counterpart to this intuition we need a small modelling refinement, namely that
the arguments forming a campaign for a goal , i.e. the arguments in args(), are arranged in an
ordered list denoted as ℒ() ∈ *, where, for a set , * denotes the set of all (finite) sequences of
elements of . For ∈ args() the position of in the list ℒ() will be denoted as pos(). Then, we
assume that each audience has its own capabilities and attitudes concerning the reception of ordered
communications of a given length and that, in this respect, the efectiveness of each item is afected by
its position in the list. To represent this aspect we associate to each audience a position-weighting
function : N → [
The considerations above lead to revise Definition 3.7 as follows. (3) (4)
Definition 4.1 generalises Definition 3.7, which can be regarded as a special case where for every audience () returns a vector consisting of 1’s. We can make some comments on the nature () and how it gives rise to nonmonotonicity of the impact measure.
First, it seems rather reasonable, though the model allows also other choices, that for every audience (1) = ⟨1⟩, i.e. that being the only argument used in a campaign cannot be detrimental to the impact of the argument per se, which in this case is at the same time in the first position, in the last one and (in a degenerate sense) in all the intermediate positions. Peculiar exceptions to this assumptions can however be conceived. For instance, if one assumes an audience whose attention needs to be “warmed up”, a single argument may not be received with full weight, this being reserved to arguments occurring after a given position.
Second, it seems reasonable that the greater is the higher is the chance that () contains some zeros or anyway some very low values. This can be intuitively related to an excess of cognitive load, which can manifest itself either in forgetting the first arguments or losing attention on the last ones (or both) leading respectively to having zeros at the beginning or at the end of the vector () for a large . It is also interesting to note that diferent audiences may have diferent attention capabilities or cognitive load thresholds. These can be put in correspondence with the value of the sum of the elements of () with the increase of . For instance, if for a given and two audiences 1 and 2 it holds that ∑︀ =1 2 (), this means that the audience 2 is altogether more capable =1 1 () < ∑︀ to sustain the cognitive load of receiving arguments.
While this model is admittedly rough (for instance the cognitive load may depend not only on the number of the arguments but also on their structure or form of presentation) it allows to capture a variety of possible situations and in particular to introduce nonmonotonicity with respect to the set of arguments included in the campaign, even if each argument would in isolation have a nonnegative impact on all audiences. In particular, it may be the case that for some audience it holds that ∑︀=11 (1) < ∑︀=21 (2) for some 1, 2 such that 1 > 2. This would correspond to a situation where a too long list afects negatively the impact of all the arguments it contains and a shorter list would be more efective.
In general, it can be interesting to identify conditions on ensuring that the efectiveness of a campaign cannot decrease by adding new arguments. To draw considerations in this respect we need to impose some constraints on the order of the arguments in the initial and in the extended campaign. In particular we assume that the arguments with greater impact (i.e. arguments with higher values of ‖‖) are put in the positions with greater positional weight. This is expressed by the notion of position-optimal campaign.
Definition 4.2. A campaign for a goal , with argument list ℒ() of cardinality is position-optimal for an audience if for every pair of arguments , ∈ args() it holds that ‖‖ ≤ ‖ ‖ if and only if ()pos() ≤ ()pos(). A campaign is position-optimal for a set of audiences if and only if it is position-optimal for all the audiences in the set.
We can now establish a simple non-decreasing condition for .
Definition 4.3. Given 1, 2 ∈ N with 1 < 2, we say that a position-weighting function of an audience is non-decreasing with respect to 1 and 2 if there is an injective function from the set {1, . . . , 1} to the set {1, . . . , 2} such that for each 1 ≤ ≤ 1 it holds that (1) ≤ (2)(). In words, for every position from 1 to 1 there is a distinct position included between 1 and 2 which has a not lesser positional weight.
A monotonicity guarantee follows from the above conditions.
Proposition 4.4. Let 1, 2 be two campaigns such that args(1) ⊊ args(2) and let be a set of target audiences. If for every audience ∈ it holds that 1 and 2 are position optimal and is non-decreasing with respect to |args(1)| and |args(2)| it follows that ℐ(1, ) ≤ ℐ (2, ). Proof. First note that all terms in the sum defined in (4) are nonnegative. Then observe that for every audience in , every argument in args(1) contributes to ℐ(1, ) with a term ‖‖ · ()pos1 () (and similarly for every argument in args(2)). Let 1, . . . , 1 be any ordering of the arguments in args(1) such that ‖1‖ ≥ ‖ 2‖ . . . ≥ ‖ 1 ‖. Similarly let 1, . . . , 2 be any ordering of the arguments in args(2) such that ‖1‖ ≥ ‖ 2‖ . . . ≥ ‖ 2 ‖. To show that ℐ(1, ) ≤ ℐ (2, ) we show that for every 1 ≤ ≤ 1 ‖‖ · ()pos1 () ≤ ‖ ‖ · ()pos2 (). Consider first 1: since there is some such that 1 = , it follows that ‖1‖ ≥ ‖ 1‖, moreover from the fact that both campaigns are position optimal we get that ()pos1 (1) ≥ ()pos1 () for every 1 ≤ ≤ 1 and similarly ()pos2 (1) ≥ ()pos2 () for every 1 ≤ ≤ 2. From the hypothesis that is non-decreasing with respect to |args(1)| and |args(2)| it follows that ()pos1 (1) ≤ ()pos2 (1) from which ‖1‖ · ()pos1 (1) ≤ ‖ 1‖ · ()pos2 (1). Moving to 2, from the fact that 1 = for some ≥ 1 and that 2 = for some > it follows that ‖2‖ ≥ ‖ 2‖. From the fact that both campaigns are position optimal we get that ()pos1 (2) ≥ ()pos1 () for every 2 ≤ ≤ 1 and similarly ()pos2 (2) ≥ ()pos2 () for every 2 ≤ ≤ 2. From the hypothesis that is non-decreasing with respect to |args(1)| and |args(2)| it follows that ()pos1 (2) ≤ ()pos2 (2) from which ‖2‖ · ()pos1 (2) ≤ ‖ 2‖ · ()pos2 (2). The same reasoning can be iterated for all arguments until 1 reaching the desired conclusion.
The study of other suficient conditions for monotonicity and of how the above identified condition can be reasonably met in practice is left to future work. As preliminary comments we can observe that if consists of a single audience , assuming that ‖ · ‖ and are known, it is rather reasonable to assume that one designs a campaign which is position optimal for . If consists of a diferent audiences it may be dificult, and sometimes provably impossible, to define a campaign which is position optimal for all audiences. As to the requirement of non-decreasing position-weighting functions, it appears to be heavily audience dependent, and, in general, to correspond to a sort of robustness of the attention and reception capabilities of the audience with respect to the increase of the cognitive load. Leaving the investigation of these aspects to further research, in the sequel of the paper we address the issue of monotonicity of campaign efects from the perspective of argument acceptability.
In [
The starting assumption is that it is possible to identify, for each audience , an argumentation framework, denoted as 0, representing the mental state of the audience before receiving the campaign. In particular, the arguments in 0 are a subset of the universe of arguments initially known to the audience (Definition 3.4), and the attacks are derived from those in the universal argumentation framework (Definition 3.2) following the idea of the value-based approach as specified in the following definition.
⟨0, ℛ⟩ where 0 ⊆ 0
Given an audience , the initial argumentation framework of is defined as 0 = and ℛ0 = {(, ) ∈ 0 × 0 | (, ) ∈ ℛ and ‖‖ ≮ ‖‖}.
After receiving a campaign each audience updates its argumentation framework. In particular, we assume the existence of an audience-specific attention-trigger map which specifies, given a set of arguments , which arguments are brought to the attention of the audience after being exposed to the arguments in . As a minimal requirement, we assume that at least the arguments included in the campaign are brought to the attention of the audience.
Definition 5.2. Given an audience , an attention-trigger map for , denoted as , is a function : 2 → 2, satisfying the requirement that for every ⊆ it holds that ⊆ ().
The argumentation framework of the audience after receiving a campaign includes also the arguments triggered by the campaign and the relevant attacks, again following the value-based perspective. Definition 5.3. Given an audience and a campaign , the argumentation framework of after receiving is defined as = ⟨, ℛ⟩, where = 0 ∪ (args()) and ℛ = {(, ) ∈ × | (, ) ∈ ℛ and ‖‖ ≮ ‖‖}
For simplicity, we assume here the use of a single-status semantics, namely the grounded semantics5, for the assessment of acceptance of the arguments in an argumentation framework. An audience meets the goal of a campaign if at least one of the arguments in is accepted by in the updated argumentation framework. To formalize this we resort to the following definition. 5The consideration of other semantics is left to future work.
Definition 5.4. Given a campaign with goal and an argumentation framework , we say that the goal of the campaign is met with respect to , denoted as OK (, ), if ∩ GR( ) ̸= ∅.
Note that of course it can happen that an audience meets the goal of the campaign even before receiving it, namely it can happen that OK (, 0). It follows that with respect to the efects of a campaign we can identify three cases.
Given a campaign and an audience we say that — is positively afected by , denoted as E +(, ) if OK (, ) while not OK (, 0); — is negatively afected by , denoted as E − (, ) if OK (, 0) while not OK (, ); — is unafected by otherwise.
The assessment of a campaign can then consist of the sum of the weights of the audiences on which it has a positive efect from which the weights of the audiences on which it has a negative efect should be subtracted.
Definition 5.6. Let be a campaign and be a set of target audiences. The conviction value of with respect to is defined as
CV (, ) =
∑︁ ∈ |E+(,) () −
∑︁ ∈ |E− (,) () (5)
Some remarks on Definition 5.6 are worth making. First, diferently from the assessment based on impact measures, it allows in principle also to derive a negative evaluation for a campaign, corresponding to situations where it has prevailing unintended efects. This is a significant improvement of the expressiveness of the model, which comes at the price of requiring the (non trivial in practice) capability of estimating not only the attitudes of the audiences with respect to values but also the arguments they initially hold and those which are triggered by the campaign. Second, it adopts a binary notion of acceptance and hence of success (and unsuccess) thus allowing a simpler analysis and discussion at this preliminary level of investigation. A finer representation could encompass some gradual notion, e.g. counting the number of accepted arguments in as a non-binary measure of success or distinguishing the status of the arguments which are attacked by the grounded extension from the status of those which are not included in the grounded extension but are not attacked by it, i.e. are in a sort of undecided, rather than definitely rejected, situation. Leaving these developments to future work, let us now discuss monotonicity issues concerning the conviction values of campaigns.
Concerning the first kind of monotonicity discussed in Section 4, namely the one referring to the role of values, we observe that in the context of the assessment introduced in this section, values play an indirect role. In fact, while ‖‖ was a term directly afecting the outcome of the assessment both in Definition 3.7 and Definition 4.1, here its role consists in determining which attacks are efective. In particular, if one assumes that a campaign only consists of sensible arguments which should be accepted by the audiences, it is generally preferable that for every argument in the campaign ‖‖ is as high as possible. Then, concerning the way ‖‖ is calculated, no additional considerations with respect to those drawn in previous sections are needed.
Concerning, the second aspect of monotonicity, namely the inclusion of more arguments in a campaign, the discussion is more articulated and partly tricky. A first key point concerns the attentiontrigger map. Besides the basic assumption that a set of argument triggers at least all its elements, there are no other hypotheses on the function , which in principle might “activate” for some audience also arguments which are fully in contrast with the goal of the campaign and which might be in line with the values of the audience, thus possibly resulting in getting to be negatively afected by and then in a decrease of CV (, ). It follows that any consideration on monotonicity must refer to some additional hypothesis on . In particular, we resort to the assumption that the campaign is goal-coherent, namely that it does not trigger any argument which is in possibly indirect conflict with any argument in .
Definition 5.7. Given a campaign with goal we say that is goal-coherent with respect to an audience if for every ∈ (args()) and every ∈ it is not the case that is an indirect attacker of .
If a campaign is goal-coherent it cannot have negative efects.
Proposition 5.8. Let be an audience and a goal-coherent campaign with goal , then cannot be negatively afected by .
Proof. To prove the statement we show that if there is some argument ∈ such that ∈ GR(0) then it also holds that ∈ GR().
Given the well-known properties of grounded semantics (see [18] and the notion of strong defense in [19]), an argument in an argumentation framework = ⟨, ℛ⟩ belongs to the grounded extension GR( ) if is strongly defended in namely (i) is unattacked in or6 (ii) for every attacker of there is an attacker of such that ̸= and is in turn strongly defended in .
By hypothesis, the above conditions hold for some argument ∈ in the argumentation framework 0. We prove then that they hold also in . Note that any additional argument in with respect to 0 is included in (args()) and that any additional attack must necessarily involve an element of (args()). Now, if ∈ is unattacked in 0, it is also unattacked in : any new attacker should belong to (args()), which would contradict the property of goal coherence.
If instead has a non-empty set of attackers in 0, we observe that, for the same reason, the set of attackers of in is the same. Considering any attacker of : we have to show that the condition (ii) still holds in . Given that ℛ0 ⊆ ℛ it still holds in that ̸= attacks . Suppose by contradiction that is no more strongly defended in . For this to happen, must have some additional indirect attacker in , but then would be also an indirect attacker of and should be a member of (args()), which would contradict the property of goal coherence.
We have thus provided a suficient condition to ensure that adding arguments does not afect negatively a campaign. The condition reflects a quite reasonable requirement of absence of conflicts (even indirects) with the arguments related to the campaign goal. The tricky point is that this requirement does not refer only to the arguments included explicitly in the campaign but also to those possibly triggered by the campaign itself for all the audiences included in the target set. In turn, this means that it is preferable that the audiences in the target set are somehow similar as far as the triggered arguments are concerned. In general, the problem is to achieve a suficiently accurate estimation of the triggered arguments: in this respect, investigating triggering mechanisms (e.g. by analogy, by opposition, by inference, . . . ) appears a very relevant direction of future work.
Defining formal models able to support the design and evaluation of public interest communication campaigns is a challenging research goal, whose complexity suggests to follow an incremental approach, by addressing step by step its many facets. Building on a previous work investigating the definition of impact measures of argumentative communication campaigns based on the values they promote, in this paper we have made a further step by considering nonmonotonic campaign efects, i.e. situations where increasing the set of promoted values and/or extending a campaign with additional arguments can turn out to be detrimental, rather than beneficial, to the campaign itself.
We addressed nonmonotonicity both in quantitative impact measures directly assessing the efectiveness of a campaign and in evaluations of audience convictions based on argumentation frameworks. In both cases, we provided suficient conditions to ensure monotonicity. This preliminary study provides the basis for further research on this topic. In addition to the future work directions indicated along the 6Condition (i) is actually a special case of condition (ii) but we keep it distinct for the sake of readability. paper we mention the consideration of bipolar argumentation frameworks [20] to represent audience opinions and the use of gradual argumentation semantics [21], encompassing a quantitative notion of degree of acceptance.
Moreover, it would be interesting to investigate relationships between the proposed model and cognitive theories of persuasion [22] in order to justify the choice of alternative impact measures with some theoretical background or experimental evidence concerning the attitudes of diferent social categories. The model will then need to be validated using data concerning past PIC campaigns, in order to check the ability of our proposal to provide a model-based explanation of their successes or failures.
This work was supported by MUR project PRIN 2022 EPICA “Enhancing Public Interest Communication with Argumentation” (CUP D53D23008860006) funded by the European Union - Next Generation EU, mission 4, component 2, investment 1.1.
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