In computational argumentation, values adjudicate between conflicting arguments, where values hold of an argument as a whole rather than of any constituent parts; preference rankings between values determine the winning argument. We propose a novel formal framework towards an account for motivated reasoning, which is the widespread, natural observation that an agent constructs (instantiated) arguments from those propositions which are a selective subset of the set of all available propositions; and more specifically, an agent selects those propositions which accord with their values; as such, the propositions and arguments indirectly reflect an agent's values. Conflicts between arguments are grounded in conflicts between the values associated with the constituent propositions rather than with the arguments per se.
Values are widely acknowledged as a way to select amongst alternative arguments and resolve
conflict in reasoning and decision-making. Kahneman discusses how values play a role in
everyday motivated reasoning [
In computational argumentation, values adjudicate conflicts between arguments [
While formally clear and well-developed, the relationship between an argument and associated
value is opaque. In addition, a key aim of our work is account for motivated reasoning, where from
a common pool of accessible information, agents use their values to select how to construct their
arguments; e.g., in justifying what item to purchase such as a camera, each purchaser associates
an item’s features with values relative to them, thus coming to a highly subjective justification
[
To formalise motivated reasoning, we assume: each agent has a value profile that expresses what is abstractly important to the agent and to what degree, e.g., family, freedom, security, and others; each agent assesses the relation between propositions and values. This is grounded in the observation that, apart from logical contradictions (i.e., p vs ¬p) and commonsense incompatibilities (i.e., an agent cannot be in two places simultaneously) which are “objective relations”, there are conflicts which are subjective and related to values. An agent filters from the available propositions those in an agent’s knowledge base, which is a subjective knowledge base; from these available propositions, the agent constructs their arguments. As agents differ in their value profiles, then so too do their knowledge bases and arguments. As values profiles and relations of propositions to values can conflict, conflicts then arise between arguments.
In this paper, we initially develop a formal, novel approach to motivated argumentation as
valuebased instantiated argumentation (VIA), which represents how values intervene in argumentation.
The present work is focused on the development of subjective knowledge bases, which can be
used in instantiated theories of argumentation. In this regard, VIA is scoped and agnostic with
respect to theories of abstract or instantiated argumentation theories (e.g., [
The structure of the paper follows the layers of analysis. In Section 2, we provide the vocabulary, basic predicates, and filters which are built relations amongst agents, propositions, values, and weights. In Section 3, subjective knowledge bases and reasoning chains are detailed, where reasoning chains are an argumentation-theoretic agnostic approach to chaining together of rules. Section 4 presents a core, novel concept of conflict between arguments which is rooted in the conflict between values. A worked example illustrates the formalisations. We conclude with a discussion of related work in Section 5 and discussion in Section 6.
In this section, we develop a language for knowledge bases and reasoning. We assume the following denotations.
• a finite set of agents Agent = {agent0, . . ., agentn} of entities. • a finite set AtomicProposition of atomic propositions (aka prop). • a finite set incompProp containing pairs of elements of AtomicProposition which cannot co-occur. The relation is symmetric. The props of such a pair are called objectively incompatible, otherwise, they are objectively compatible. • a finite set of values Value = {value0, . . ., valuen} of abstract objects. • a totally ordered, finite set of scalar elements Scale = { weight0, . . ., weightn}. • a designated unordered element ?, used where a weight is indeterminate. • a set Weight = Scale ∪ {?}. There may be alternative interpretations of ‘weights’. Here, they reflect the relative ‘importance’ to an agent, e.g., family might be a very important value and personal status very unimportant. • as ? is unordered with respect to other elements of Scale, any prop comparing ? to the other entity is false, e.g., ? > weight1 is false.
To quantify over any type in the basic vocabulary, we have: variables for each type, represented by Greek subscripts, e.g., agentα , agentβ , ....; and constants of each type represented by Latin subscripts agenta, agentb, .... In addition, we can quantify over variables and constants over tuples defined with respect to the basic vocabulary.
Basic Predicates With the following predicates, we construct the predicate PropBaseCleanagenth for some agenth, which contains all and only those props that are ifltered according to an agent’s value profile. We show how Agents may “hold” different sets of props relative to their values such that values can be taken as the root of conflict.
AgentValue represents all the values an agent might consider.
AgentValue = { < agentα , valueβ >| ∀agentα ∀valueβ (agentα ∈ Agent ∧ valueβ ∈ Value)} From this, we construct an agent’s value profile , AgentValueToWeight, which indicates the degree of importance that the agent ascribes to the value, where the higher the weight, the more important and the lower the weight the less important. Given an agent and Value, the value can only have one weight in order to avoid conflicts.
AgentValueToWeight = {<< agenta, valueb >, weightc >| ∀agenta∀valueb∃!weightc(< agenta, valueb >∈ AgentValue ∧ weightc ∈ Weight)} AgentValueToWeight is a total function. We indicate each agent’s value profile with a subscript, e.g., AgentValueToWeightagentk for agent agentk. Given the ? weight, the importance an agent associates with a value can be indeterminate, reflecting the view that the agent has no specific association with respect to the value. Note that any two agents may ascribe different weights to the same value, perhaps at opposite ends of the Scale; as such, this can be taken to represent antithetical assessments on the values, one agent viewing the value as more important and the other less important. Note that since differences in AgentValueToWeight of particular agents represent differences in the levels of importance the agents puts on those values, they can be seen as a subjective or personal value profiles of those agents.
To represent how an agent assesses an AtomicProposition with respect to a Value and a Weight, we first associate Values and Weights.
ValueWeight = { < valueα , weightβ >| ∀valueα ∀weightβ (valueα ∈ Value ∧ weightβ ∈ Weight)} Then, we have a functional relation that expresses an agent’s disposition towards an AtomicProposition with respect to values and weights.
AgentToAtomPropToValueWeight =
{ < agentα , < propβ , < valueγ , weightδ >>>|
∀agentα ∀propβ ∀valueγ ∃!weightδ (agentα ∈ Agent ∧
propβ ∈ AtomicProposition ∧ valueγ ∈ Value ∧ weightδ ∈ Weight)}
Conceptually, props with positive weights are “proponents” or positive instances of the value
and with negative weights as “antagonists” or contrary instances. The basic notion of associating
a prop with a particular value and weight appears in [
The predicates AgentToValueWeight and AgentToAtomPropToValueWeight are taken to be conceptually distinct. In particular, we interpret AgentToValueWeight to indicate the importance that the agent ascribes to the Value. In contrast, we interpret AgentToAtomPropToValueWeight to indicate an agent’s assessment of the Value and Weight ascribed to a particular prop from AtomicProposition. In other words, AgentToValueWeight expresses an agent’s “ideal” and “global” view on Values, while AgentToAtomPropToValueWeight expresses an agent’s assessment of particular props in terms of their association with a value. In this approach, values are directly associated with props relative to agents and their value profile.
Filters on Sets of Relations To reflect an agent’s value-based world view, all the props from AgentToPropositionToValueWeight are gathered which have a Value-Weight where the Weight on the Value is no less than the agent’s assignment of the Weight on that Value in AgentToValueWeight. Propositions can be indeterminate about the Weight on Values.
AgentMinFilterOwnCleanagenth = { < agenth, < propα , < valueβ , weightγ >>>| agenth ∈ Agent ∧ ∀propositionα ∀valueβ ∀<agenth,<propα,<valueβ ,weightε>>> ∀<<agenth,valueβ >,weightγ>(propα ∈ AtomicProposition ∧ valueβ ∈ Value ∧ < agenth, < propα , < valueβ , weightε >>>∈ AgentToAtomPropToValueWeight ∧ << agenth, valueβ >, weightγ >∈ AgentValueToWeight ∧ ¬(weightγ > weightε ))} Values and weights discriminate amongst props. For a particular agent, a lower weight on a particular value implies, with respect to AgentMinFilterOwnClean, that there is a lower discriminatory threshold on the acceptability of props, which themselves are associated with that Value and the Weight. Simply put, if an agent has a lower weight on a particular value, then more props may pass the filter, as they have higher weights on the same value. The higher the weight means that an agent has higher standards with respect to the value; there is greater discrimination such that fewer props pass the filter. An intuitive example may help to clarify the relations between the value-weights in AgentValueWeight and AgentToAtomPropValueWeight as they appear in AgentMinFilterOwn. Suppose an agent is not much bothered by the quality of coffee, so on the value taste the weight is low, and the agent acts accordingly. When they are served a coffee that on the value taste has weight that is also low, then the agent drinks the coffee; when served another coffee with value taste with weight high, then the agent also drinks the coffee. In effect, the taste makes little difference to this agent, as they don’t discriminate. On the other hand, suppose the agent has on the value taste a high weight. In the first instance, the agent rejects the coffee as not upholding their higher standards on the value; in the second instance, the agent is satisfied and drinks the coffee. AgentMinFilterOwnClean represents an idealised view of an agent’s assessment of a set of AtomicPropositions, where all AtomicPropositions must pass an agent’s highest weighting; presentation of other, less stringent variations remain for future work.
Now we are prepared to abstract from consideration of Values and Weights and extract those AtomicPropositions which, in effect, represent the agent’s value-based world view. A set PropBaseCleanagenth contains only all those props which pass the value-weight in the value profile for agent h for all values. It is important to emphasise that this is a very strict, formally convenient definition, which assesses all props and creates a set of only those props that pass the agent’s value filter. Matters are more complex and less strict; in other work we develop lenient, lfexible definitions.
PropBaseCleanagenth = {propα | agenth ∈ Agent ∧ ∀<agenth,<propα ,<valueβ ,weightγ >>> (< agenth, < propα , < valueβ , weightγ >>>∈ AgentMinFilterOwnCleanagenth )} Any set of PropBaseClean may contain incompatible props. Moreover, two agents may each accept the same prop, yet for different settings of values and weights.
We assume PropBaseClean represents static (all at once) and private (inaccessible to others) associations of value-weights to props by an agent, which contrasts to the publicly reported reasoning chains, as discussed below. Note that we do not analyse whether the props in PropBaseClean are true or believed to be true; the set presents props which the agent can accept in the light of his/her value profile. The key point of creation of PropBaseClean is to distinguish props which are coherent with the agent’s value profile, i.e. props which pass AgentMinFilterOwnClean. By the same token, the complement to PropBaseCleanagenth reflects all those props which are incompatible with the Agent’s values and weights. Both notions are used in Section 4.
PropBaseCleanagenth = {propβ | ∀propβ (propβ ∈ AtomicProposition ∧ propβ ̸∈ PropBaseCleanagenth ) } Consider the various ways props may appear in intersecting or complementing sets of two Agents’ PropBaseClean. For intersecting: the Agents have the same value-weight profile and same value-weights on same prop; same value-weight profile and different value-weights on prop, but not sufficient to block; different value-weight profile and same value-weight on prop, but not sufficient to block; different both, but not sufficient to block. Where they have the same props, neither the value-weight profile nor value-weight on prop is sufficient to discriminate. For complementing: same value-weight profile and different value-weights on prop, and sufcfiient to block; different value-weight profile and same value-weight on prop, and sufficient to block; different both and sufcfiient to block. In other words, differences in sets of props arise where value-weight profiles or value-weights on props are sufficient to discriminate. Broadly speaking, where a prop appears in the intersection of the sets of PropBaseClean of two Agents, we can say the agents agree on that prop in one sense or another, while where the prop is in complementary distribution (in one set, but not the other), we say there is some sense of disagreement. Note that there may be different justifications for the agreement or disagreement as well as different extents of such justification, e.g., greater or lesser difference in weights associated with the value. Relatedly, two agents can have the same denotations for their respective PropBaseClean, yet different value-weight profiles or different value-weights on the same prop.
Example 1. [Creating PropBaseClean] To illustrate PropBaseCleans, suppose 2 agents (agentA and agentB), 3 propositions {pX , pY , pZ}, and 2 values: V = {vQ, vP}, e.g., privacy or law enforcement, where AgentValueWeightagentA = {< agentA, valueQ, 3 >, < agentA, valueP, − 2 >} AgentValueWeightagentB = {< agentB, valueQ, 2 >, < agentB, valueP, 1 >} AgentA has very high requirements concerning value Q, and very low requirements concerning value P, while agentB has a more balanced value profile.
For AgentToAtomPropToValueWeight, Table 1 is a tabular form for instances of agents, props, values, and weights. The props that pass the AgentMinFilterOwnClean are indicated in bold. In Table 2, we have PropBaseClean for each Agent (the props that pass AgentMinFilterOwnClean).
Given PropBaseClean, we construct subjective knowledge bases, from props that are most
important relative to an Agent’s values; in this sense, a knowledge base is relativised to an
Agent and their values. Given this, we then are in position to compare and relate alternative
justifications across Agents relative to each of their values. Given a subjective knowledge-based,
we can construct reasoning chains, which here are intended to be theoretically agnostic about the
construction of arguments (e.g., [
Definition 2 (Subjective Knowledge Base). Given a set of atomic propositions Pi ⊆
AtomicProposition, a set of rules R j ⊆ R, and an AgentA, a subjective knowledge base
KBAgentA is < Pi, R j > where ∀pα , s.t. Pi ∪ R j ⊢ pα , pα ∈ PropBaseCleanagentA
Informally, a subjective knowledge base represents what an Agent intuitively accepts relative to
that Agent’s values. Set-theoretic relations can hold between the Agents subjective knowledge
bases. A subjective knowledge base can be inconsistent, as is widely accepted in argumentation
theory [
Finally, we define reasoning in subjective knowledge bases.
Definition 3 (Subjective Reasoning chain). Given an subjective knowledge base KBAgentA = < PAgentA , RAgentA > and a reasoning chain RCAgentAj as <KBAgentAj , pAgentAj >, where KBAgentAj = <PAgentAj , RAgentAj >, PAgentAj ⊆ PAgentA , RAgentAj ⊆ RAgentA : • Every proposition derived from RCAgentAj should be in PropBaseCleanAgentA : ∀pα s.t.RCAgentAj ⊢pα (pα ∈ PropBaseCleanAgentA ) • KBAgentAj is a minimal set of atomic propositions and rules necessary to derive pi; • KBAgentAj does not contain cycles; • p j (conclusion) is single atomic proposition such that KBAgentAj ⊢ p j; • ¬(∃pQ,pT s.t. KBAgentAj ⊢ pQ and KBAgentAj ⊢ pT and < pQ, pT >∈ incompProp). In contrast to a subjective knowledge base, a reasoning chain must be internally consistent. While Def. 3 outlines a generic construction of a reasoning chain based on a subjective knowledge base, it says nothing about selection amongst the available alternatives. Nonetheless, given that the reasoning chains are constructed from a subjective knowledge base, they represent some principled selection from amongst reasoning chains otherwise constructed.
In our approach, the roots of conflict in argumentation can be found in the underlying differences between Agents’ value profiles, which surface in the props found in the knowledge bases and reasoning chains. We identify several notions of incompatibility amongst reasoning chains, which may be taken as “attacks” in the argumentation-theoretic sense, which might not otherwise be apparent. Thus, in addition to attacks based in a logical notion of incompatibility, the attacks can be grounded in the Agents’ different value-weight settings.
Definition 4 (Relations between Reasoning Chains). and their subjective knowledge bases KBagentA = < PA, RA > of a agentA and KBagentB = < PB, RB > of a agentB, we can provide reasoning chains relative to KBs.
Suppose RCagentAa and RCagentBb , where RCagentAa is < KBagentAa , ph >, where KBagentAa = < Ph ⊆ PA, Rh ⊆ RA > and RCagentBb is < KBagentBb , pl >, where KBagentBb = < PagentBb ⊆ PB, RagentBb ⊆ RB >, and RCagentAa is subjectively consistent to agentA and RCagentBb is subjectively consistent to agentB. Given two agents, agentA and agentB, if ∃pα ,pβ : (KBagentAa ⊢ pα ) ∧ (KBagentBb ⊢ pβ )∧ (pα ∈ PropBaseCleanagentB ) ∧ (pβ ∈ PropBaseCleanagentA ) 3. RCagentAa and RCagentBb are subjectively asymmetrical incompatible if 1. RCagentAa and RCagentBb are objectively incompatible iff:
∃pα ,pβ : (KBagentAa ⊢ pα ) ∧ (KBagentBb ⊢ pβ )∧ < pα , pβ >∈ incompProps 2. RCagentAa and RCagentBb are subjectively symmetrical incompatible w.r.t. agentA and agentB ∃pα s.t. : KBAa ⊢ pα , pα ∈ PropBaseCleanagentB , and ∀pγ : (KBBb ⊢ pγ → pγ ∈ PropBaseCleanagentA ) Subjective incompatibility reflects some difference between the value-weight settings of each Agent, respectively, though not specifically which setting, though we could do so. Def. 4(1) represents one way that reasoning chains may be construed to attack one or the other; it is consistent with the context independence property introduced by [18] in which, for two arguments and two knowledge bases, if argument A attacks argument B in one KB, then it also attacks in the second KB. Def. 4(2), where props are reciprocally absent in the PropBaseClean of each agent, imply argument attack between two reasoning chains in virtue of the underlying disagreement about the value preferences of the agents, even if “objectively” or logically the props are not incompatible; here, the presence or absence of props are proxies of an agent’s value profile. Def. 4(3) is the asymmetrical version of Def. 4(2), where the PropBaseClean of Agent B is a superset of Agent A. As such, Agent A finds some arguments of Agent B to be incompatible with Agent B’s value profile, but not vice versa. In such a case, Agent A would appear to be attacked in that Agent B can present reasoning chains that are not consistent with Agent A’s value profile. VIA does not uphold context independence, as conflicts between reasoning chains are Agent dependent.
Subjective incompatibility indirectly represents the subjective, value-based opinion of an agent and explains the root of a common real life situation in which: there is indirect conflict; and for one Agent, reasoning chains are in conflict, but not for the other one. Note that, subjective incompatibility indirectly represents conflict between the Agents’ value systems. Example 2 (subjectively asymmetrical). Suppose a set of rules R = {px → py, py → pz}. The subjective knowledge bases of agents: KBagentA =< {px, py}, {px → py} > KBagentB =< {px, py, pz}, {px → py, py → pz} > On the basis of the above the agents can create following (incomplete list of) reasoning chains: RCagentA2 =<< {px}, {px → py} >, py}, RCagentB2 =<< {px}, {px → py, py → pz} >, pz} Reasoning chains RCagentA1 and RCagentB1 are subjectively asymmetrical conflicting reasoning chains, because pz ∈ PropBaseCleanagentB but px ∈ PropBaseCleanagentA .
While there is related work bearing on legal theory [
VIA is agnostic with respect to theories of instantiated or abstract argumentation (e.g., [
Researchers [
Preferences have been widely considered in AI for decision-making and choice [19]. Broadly speaking, elements in a set are subject to choice, where preference is a comparative ordering relation between elements. The specific type of elements and the ordering relations over them can be defined in a variety of ways. In argumentation, preferences are used to adjudicate “winning” arguments which would otherwise “tie”. In our proposal, there is no direct comparison between elements and no ordering over them. Rather, values filter props according to the value profile of the agent. In future work, we aim to account for how arguments “inherit” the values of the component props and derivative concepts of argument attack.
In Value-base Argumentation Frameworks (VAF) of abstract argumentation [
Action-based Alternating Transition Systems with values (AATS+V) represent arguments
about actions in multi-agent systems [
Case-based reasoning associates factors, which are essentially props, and values [
The main aim of our paper was to take a step towards VIA, a formal model for “motivated
reasoning” in terms of value-based conflicts between reasoning chains. We observe (following [
VIA is motivated and guided by several intuitions about the role of values in reasoning. We observe that individuals are often asked to assign a value to some statement as in surveys or in social media assessments. Just how individuals make such assignments is a matter for social science and psychology. In this regard, a statement can be taken as an instance of the abstract value. The approach does not preclude being asked for a value assignment to a whole, e.g., marking argumentative essays. Indeed, different sorts of value may be more or less directly associated with different types of props, which remains for future exploration. In addition, we observe that from a common pool of accessible information, agents select what is used to construct their arguments according to their values; agents may argue for their side without considering the full spectrum of information, alternative arguments, or attacking and defeating them. As well, argumentative discussions do not necessarily proceed in the manner of explicit attack and counter-attack; arguments (as in differences of opinion or analysis) appear in a variety of dialogue types [24], where defeat may not be key. There is also voting, where distinct, antagonistic arguments can put forward, sometimes without explicit rebuttal.
VIA provides a basis for a range of future explorations, which we briefly remark on here. Broadly, we can see how VIA relates to settings of argumentation. The current report does not explore the compositional relation of values associated with a “whole”, e.g., argument with values of the “parts”, e.g., props, and how they are combined. We believe there is a connection, but leave it for future work. In relation to voting, issues such as consortia, where agents gather around shared arguments, and confirmation bias, where agents do not challenge their positions, would need to be explored further. However, the approach taken would appear to offer a fruitful line of analysis. It would be worthwhile to examine how VIA does (or does not) impact on the semantics of argumentation, i.e., extensions, as well as relevant formal properties. As it is, VIA is static, so some development could be done with respect to deliberation and persuasion dialogues. While we have abstracted over context, clearly this should be articulated; that is, how shouting “fire” in one context but not another is an instance of protected speech. In our view, this may be tied to implications that follow from one context or another. Relatedly, we believe it worth exploring how values relate in terms of implication and hierarchy. VIA has some suggestive bearing on abduction [25]; while an agent makes some value-based selection of props from the “universe” of discourse, there are still alternatives presented with respect to an Agent’s knowledge base. More concretely, we can explore how VIA might be used to represent and reason with legal decisions and the case base as well as actions, particularly with reference to the concepts of value promotion/demotion. explanations of judicial behaviour.