Block Argumentation

01/18/2019 ∙ by Ryuta Arisaka, et al. ∙ 0

We contemplate a higher-level bipolar abstract argumentation for non-elementary arguments such as: X argues against Ys sincerity with the fact that Y has presented his argument to draw a conclusion C, by omitting other facts which would not have validated C. Argumentation involving such arguments requires us to potentially consider an argument as a coherent block of argumentation, i.e. an argument may itself be an argumentation. In this work, we formulate block argumentation as a specific instance of Dung-style bipolar abstract argumentation with the dual nature of arguments. We consider internal consistency of an argument(ation) under a set of constraints, of graphical (syntactic) and of semantic nature, and formulate acceptability semantics in relation to them. We discover that classical acceptability semantics do not in general hold good with the constraints. In particular, acceptability of unattacked arguments is not always warranted. Further, there may not be a unique minimal member in complete semantics, thus sceptic (grounded) semantics may not be its subset. To retain set-theoretically minimal semantics as a subset of complete semantics, we define semi-grounded semantics. Through comparisons, we show how the concept of block argumentation may further generalise structured argumentation.

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1 Introduction

In higher-level argumentation [12], or temporal/modal argumentation networks [3], an argumentation may be substituted into an argument of a given argumentation . In the case of higher-level argumentation, the key is to find out which arguments in may be attacking or attacked by other arguments in . In meta-level argumentation [15, 22], properties of argumentation at object level - such as whether an object-level argument attacks another object-level argument, or the trustworthiness of an arguer - may be discussed in an argumentation about the object-level argumentation.

In a realistic argumentation, it can of course happen that some agent argues about some attack of an argument on an argument . However, that may not be at meta-level; the agent’s argument could be in the same argumentation with and , interacting with any arguments in the argumentation. Thus, the clear-cut distinction between object-level and meta-level argumentations does not always apply. Further, it may be that some argumentation itself as an argument attacks an argument, in which case it is not the case that we must seek the origin of the attack in arguments in , which does not conform to the principle of higher-level argumentation.

Seeing the gap, in this work, we formulate block argumentation, where an argumentation may be an argument and vice-versa, for non-elementary arguments such as: X argues against Y’s sincerity with the fact that Y has presented his argument to draw a conclusion C, by omitting other facts which would not have validated C. This argument is in the form: “The argument:[the argument:[other facts attack C] attacks the argument:[some facts support C]] attacks Y’s sincerity.”, indicating an argument may be a block of argumentation.
Having such an argument blurs the boundary between an argumentation and an argument, leaving the differentiation only up to the perspective one employs. While block argumentation as we show is a specific instance of Dung-style bipolar abstract argumentation (see practical motivation for bipolar argumentation in, e.g. [9, 1, 20]) with the dual nature in arguments, it propels us to contemplate internal consistency of an argument. For example, one may (or may not) find “The conclusion by X in the past that Y was a terrorist corroborates X’s sinister personality” factually inconsistent if there was no such conclusion by X.111A possible-world-semantic observation that some arguments may be unusable was made [3]. Ergo, we posit a set of constraints, of graphical (syntactic) and semantic nature, which one may choose to impose on a block argumentation, and we characterise its acceptability semantics in accordance with them. As we are to show, classical Dung acceptability semantics do not in general hold good once these constraints are taken into consideration. In particular, unattacked arguments may not be outright acceptable. Further, there may be more than one minimal member in complete semantics, thus the sceptic (grounded) semantics may not form its subset; such situation has already arisen in the context of weighted argumentation [5, 4]. To retain set-theoretically minimal semantics as a subset of complete semantics, we define semi-grounded semantics.

The paper has the following structure: in Section 2 we motivate our approach with a real legal example from a popular case. Section 3 reports the necessary technical preliminaries. Section 4 presents block (bipolar) argumentation with formal results (e.g. on the existence of semantics). Finally in Section 5, we wrap up the paper with related work, where we discuss ABA-style structured argumentation [11], noting how the concept of block argumentation may further generalise the formalism.

2 Motivation for Block Argumentation

During the still ongoing trial over the death of Kim Jong-Nam222https://en.wikipedia.org/wiki/Assassination_of_Kim_Jong-nam., a certain argumentation was deployed by a suspect’s defence lawyer as he cast a blame on Malaysian authorities for having released only portions of CCTV footage of the fatal attack. Broadly:

Prosecutor:

the CCTV footage released by Malaysian Police shows a suspect walking quickly to an airport restroom to wash hands after attacking the victim with VX, which produces an impression that the suspect, contrary to her own statement that she thought she was acting for a prank video, knew what was on her hands.

Defence Lawyer:

however, the CCTV footage in its entirety shows the suspect adjusting her glasses after the attack, with VX on her hands, which counter-evidences her knowledge of the substance. Since Malaysian authorities know of the omitted footage, they are clearly biased against the suspect, intentionally tampering with evidence.

Assume the following arguments (with attacks and supports):

:

After the victim was attacked with VX, the suspect walked quickly to a restroom for washing hands.

:

The suspect knew VX was on her hands.

:

The suspect was acting for a prank video.

:

The suspect adjusted her glasses with VX on her hands before walking to restroom.

:

Malaysian authorities are biased against the suspect, tampering with evidence by intentional omission of relevant CCTV footage.

:

supports .  :   attacks .  :   attacks .

Then we can model the example argumentation as in A. Malaysian Police uses for ( supports ) to dismiss ( attacks ). All these three arguments are made available to the audience. The defence lawyer uses to counter . is also available to the audience as attacking . He then uses , which is itself an argumentation, to attack Malaysian Police’ argumentation . This is also presented to the audience. Finally, he uses , an argumentation, for .

Arguments of the kinds of , and are themselves argumentations, so “ attacks ” could be detailed as in B, and “ supports ” as in C.

2.0.1 Constraints.

These non-elementary arguments occur in natural language constructs such as “The fact: [something criticises (supports) something] is at odds with (in support of) something”, “The sentiment: [something is being defeated by something] enforces something”, and so on, which thus appear rather commonly in practice. By recognising these arguments, however, we seem to face some challenges to the semantic account given in Dung classic abstract argumentation theory

[10]. Assume, for instance, that argumentation as in B: “ attacks ”, i.e. “The fact: [the suspect adjusted her glasses with VX on her hands before walking to restroom counter-evidences her knowledge of the substance on her hands] attacks the argument: [she knew VX was on her hands because, after attacking the victim with VX, she walked quickly to a restroom for washing hands].”, is given without any prior mention of arguments , and . Then, we do not even know what relations may hold between them. Regardless, any argument not attacked is always justified in classic theory, according to which , i.e. that attacks , is an acceptable argument, while in the first place nothing is known about and that would enable us to see ’s attack on [21]. Essentially, the problem boils down to one’s interpretation of acceptance of an argument. When we accept an argument, are we only accepting the fact that it has resisted refutations, or are we also accepting the information in the argument?

On many occasions, acceptance of an argument means the latter, for which the classic prediction could appear hasty and possibly unsafe, more so in block argumentation, for an argument can be explicitly seen appearing multiple times, and one may like to impose certain constraints to enforce argument’s dependency on the same argument that occurs in a given argumentation. For B, the constraint to be imposed could be the presence of the argumentation involving and outside the blocks, as in A. With it, it is immediate whether the argumentation in the argument or actually refers to a part of an already known argumentation. We envision both graphical (syntactic) and semantic constraints in this paper (more details are provided in Section 4).

3 Technical Preliminaries

Let be a class of abstract entities we understand as arguments. We refer to a member of by with or without a subscript and/or a superscript, and a finite subset of by with or without a subscript. A bipolar argumentation framework (e.g. see [9]) is a tuple with two binary relations and over . For any , is said to attack, or support, if and only if, or iff, there exist and such that (for attack), or (for support).
An extension-based acceptability semantics of is a family of (i.e. a subset of power set of ). When is not taken into account, a semantics of is effectively that of , a Dung abstract argumentation framework [10]. In Dung’s, is said to defend iff each attacking is attacked by at least one member of , and said to be conflict-free iff does not attack . is said to be: admissible iff it is conflict-free and defended; complete iff it is admissible and includes all arguments it defends; preferred iff it is a maximal complete set; and grounded iff it is the set intersection of all complete sets. Complete / preferred / grounded semantics is the set of all complete / preferred / grounded sets. The grounded set defined as the set intersection may not be a complete set [5] in non-classic setting; it, however, reflects the attitude of sceptic acceptance, the whole point of the grounded semantics, which would not be necessarily fulfilled if the grounded set were defined the least complete set.

A label-based acceptability semantics [8] for Dung’s makes use of the set of three elements, say , and the class of all functions from to . While, normally, it is , by we avoid direct acceptability readings off them. is said to be a complete labelling of iff:

  • iff for every that attacks .

  • iff there is some with that attacks .

is the set of all arguments that map into under complete labelling iff

is a complete set, thus a label-based semantics provides the same information as an extension-based semantics does, and more because it classifies the remaining arguments in

and ?.
In bipolar argumentation where properly matters for an acceptability semantics, the notion of support is given a particular interpretation which influences the semantics.333See [9] for a survey of three popular types: deductive support [6] (for any member of a semantics, if and if is in the member, then is also in the member); evidential support [19, 1] (for any member of a semantics, if is in the member, then it can be traced back through to some indisputable arguments in the member) and necessary support [17, 18] (for any member of a semantics, if and if is in the member, then so must also be). can be used also for the purpose of expressing premise-conclusion relation in structured argumentation [16, 11] among arguments, which will be separately discussed in Section 5.

4 Block (Bipolar) Argumentation

Let be the class of natural numbers including 0. We refer to its member by with or without a subscript. Let be a class of an uncountable number of abstract entities. We refer to a member of by with or without a subscript. It will be assumed that every member of is distinguishable from any other members. Further, every member of has no intersection with any others. Lack of these assumptions is not convenient if one wants to know equality of two arguments for graphical and semantic constraints.444Alternatively, we can assume a domain of discourse and interpretation as customary in formal logic; see [14] or any other standard text for the foundation of mathematical logic.

Definition 1 (Arguments and argumentations)

We define a (block) argument to be either for some or for some and some binary relations and over . We say is unitary iff is some .
We define a Block (Bipolar) Argumentation (BBA) to be some argument . We say that it is finite iff the number of occurrences of symbols is finite in .555 As in set theory (see [13]), this is not equivalent to finiteness in the number of members of . We denote the class of all finite BBAs by , a subclass of , and refer to its member by with or without a subscript.

Example 1 (Bba argumentation)

Denote the argumentation A in Section 2 by .

  • .

  • for .     .

  • .       .

4.1 Representation of Argument(ation)s

To refer to arguments in a specific position in , we make use of:

Definition 2 (Flat representation)

Let denote the class of all sequences of natural numbers (an empty sequence is included), whose member is referred to by with or without a subscript, or by a specific sequence of natural numbers. We use ‘.’ for sequence concatenation. Let be such that is the least set that satisfies all the following.

  1. .

  2. For any , if is not unitary and is some , then for every such that all are distinct.

For any and any , we say that is its flat representation.

Example 2 (Flat representation)

(Continued) For the same example argumentation in A, we can define its flat representation to be the set of all the following. Just for disambiguation, we demarcate the constituents in a sequence of natural numbers with ‘.’.

  • .        for .        and .

  • and .                             and .

  • and .                     and .

Definition 3 (Order in flat representation)

Let be such that , synonymously written as , iff for some . We write iff and .666and” instead of “and” is used in this paper when the context in which it appears strongly indicates truth-value comparisons. It follows the semantics of classical logic conjunction.

4.2 Characterisation of Complete Sets with No Constraints

We characterise complete sets with no constraints initially.

Definition 4 (Arguments, attacks and supports in )

Let be such that iff for some and . Let be such that:

For any and any , we say:

  • is the set of arguments in iff .

  • attacks in iff .

  • supports in iff .

While there are three typical interpretations (deductive, necessary, evidential) of support in the literature, they enforce a strong dependency between arguments and the arguments that support them concerning their acceptance. In light of the example in Section 2, our interpretation of support here is weaker, almost supplementary, as in the following definitions. Briefly, it is not necessary that an argument be in a complete set when its supporter/supportee is in the set. A supporter can, however, prevent an argument attacked by an attacker from being strongly rejected (with labels, it concerns the difference of whether the argument gets (which leads to strong rejection) or ?). We look at extension-based complete set characterisation first.

Definition 5 (Extension-based complete set when no constraints)

For any and any , we say: defends in iff and and every attacking in is: attacked by at least some ; and not supported by any .
We say that is conflict-free in iff and for any .
We say that is standard complete in iff and is conflict-free and includes all arguments it defends in .

Example 3 (Complete set with no constraints)

(Continued) For in A, we have only one standard complete set in 0 with this characterisation, namely . To explain the role of a supporter to prevent strong rejection of an argument, notice that , which is in the standard complete set in 0, supports , which is attacked by in the same standard complete set in 0. If it were not for the supporter, would be in the standard complete set in 0.

We can also have a label-based characterisation with .

Definition 6 (Complete labelling when no constraints)

Let be the class of all such that for some . For any and any , we say that is a standard complete labelling of iff every satisfies all the following.

  • [leftmargin=0.3cm]

  • , , iff every , , attacking in satisfies .

  • , , iff there exists some such that and that attacks in and there is no such that and that supports in .

Theorem 4.1 (Correspondence between standard complete sets and standard complete labellings)

For any and any , is standard complete in only if there is some standard complete labelling of such that is equivalent to for any . Conversely, is a standard complete labelling of only if, for every , is a standard complete set in .

On the basis of this correspondence, we will work mainly with labels, as they simplify referrals of arguments that do not get .

4.3 Graphical (Syntactic) Constraints

We now characterise constraints, which can be graphical (syntactic) or semantic. The former enforces that any argumentation within an argument(ation) has already occurred, while the latter enforces that acceptability statuses of arguments in any argumentation respect in a certain way those of the same arguments that have already occurred. We begin with a graphical one, for which we need to be able to tell equality of two arguments.

Definition 7 (Argument equality)

Let be such that iff one of the following conditions is satisfied.

  • [leftmargin=0.3cm]

  • for some .

  • If is some , then is some such that:

    • ;

    • iff ;

    • and iff

    for .

Definition 8 (Sub-argumentation)

Let be such that , written synonymously as iff and are such that:

  • [leftmargin=0.3cm]

  • .

  • For each , , there exists some , , such that .

  • For each two , , if , then there exist some , such that , that , and that .

  • For each two , , if , then there exist some , such that , that , and that .

We say that is a sub-argumentation of iff .

Definition 9 (Graphical (syntactic) constraints)

For any , we say that satisfies G iff both of the conditions below hold.

  1. If and if is not unitary, then there exists some such that and that .

  2. Every satisfies G.

Example 4 (Graphical (syntactic) constraints)

(Continued) As has been the case so far, let be the argumentation in A. and let its flat representation be as given in Example 2. Then clearly satisfies G for any , since:

  • : there is nothing to show, as the sequence is 0.

  • , : there is nothing to show, as are unitary.

  • : .

  • , : there is nothing to show, as and are unitary.

  • : .

  • : there is nothing to show, as and are unitary.

  • : .

  • : see above for .

  • : see above for .

  • : there is nothing to show.

  • and : there is nothing to show.

For comparison, however, suppose that is the argumentation in B, i.e. , , and . Assume , then neither of them satisfies G, because none of are in .

Since the graph structure of a given never changes, violation of the graphical constraint monotonically propagates up to (0 excluded) from longer sequences. Thus, it is rather straightforward to handle graphical constraint satisfaction.

4.4 Semantic constraints

By contrast, semantic constraint satisfaction depends on what labels are assigned to arguments, which adds to technical subtlety. We define a partial order on , and characterise semantic constraints based on them.

Definition 10 (Order in labels)

Let be . We write alternatively as .

Definition 11 (Semantic constraints)

For any and for any , we say that and satisfy:

S

iff for every , if and , then .

iff, for any , if there exist some , and such that and that , and if , then .

To speak of the use of , there should be no oddity from a semantic coherency perspective when for every such that . That just means that same arguments in are assigned the same label. However, consider our example A. There, in which is assigned by a standard complete labelling is given in by the Malaysian authorities and in by the defence lawyer. assigned to an argument in , therefore, can be interpreted flexibly in argumentation in , that it can be any of , , depending on which part of the argumentation in is selected to be included in the argumentation in (for some non-empty ). This is the intuition for and its use in S.
The symbol denotes a semantic constraint among the members of for some , to prevent the same arguments from being assigned a different label.

Example 5 (Semantic constraints)

Denote the argumentation in D by . Assume that as shown in E, and assume that are all unitary such that for no distinct . Assume . The following are all the standard complete labelling distinct for .

  • for , for ,
    , .

  • for , for .
    , .

  • , , for .
    , .

and do not satisfy S. and do. Now, while and , S is respected, since .
Denote the argumentation in F by . Assume for . Assume and . There is a standard complete labelling of such that , . Then it could be understood that the same argument ( and ) in 0 is assigned both + and simultaneously, which on some occasions outside of rhetoric argumentation is not convenient. is the condition against such labelling.

4.5 Generalisation of standard complete labelling

Let us refine our earlier definition of standard complete labelling with those constraints. We define to be , and refer to its subset by with or without a subscript.

Definition 12 (Complete labelling)

For any , any and any , we say that is a complete labelling of under iff every satisfies all the following conditions.

  • [leftmargin=0.31cm]

  • , , iff both:

    • every , , attacking in satisfies .

    • if , then assigning + to does not lead to non-satisfaction of a semantic constraint for any .

  • , , iff both:

    • there exists some such that and that attacks in and there is no such that and that supports in .

    • if , then assigning to does not lead to non-satisfaction of a semantic constraint for any .

  • , , if and does not satisfy G.

Any argument that violates the graphic constraint will not be assigned + if . For both + and , the first condition matches exactly the condition given for a standard complete labelling. The second condition ensures conditions for a standard complete labelling to be maximally respected, in the sense that they normally apply unless by applying them there will be a non-satisfaction of a semantic constraint .

Broader intuition is as follows. If a standard labelling of satisfies all , then the standard complete labelling should itself be a complete labelling under . Thus, when a standard complete labelling is not a complete labelling under , it is because the labelling either induces violation of some semantic constraint for some arguments occurring in or + assignment to an argument that violates the graphic constraint if . In such cases, therefore, it will be required to make minimal change to the standard complete labelling so the resulting labelling satisfies semantic constraints (if they are in ) and does not assign to an argument that violates G (if it is in ). The second conditions for and ensure that the change be indeed minimal with a complete labelling.

Example 6 (Complete labelling)

(Continued) For the argumentation in D with the same assumptions and as in Example 5, is not a complete labelling under if : is assigned +, however, it does not satisfy S, so it should be assigned ?. and are a complete labelling under any . In addition, such that , , is a complete labelling. is not a standard complete labelling.

Theorem 4.2 (Conservation)

For any and any , if , then being a complete labelling of under is equivalent to being a standard complete labelling of .

Theorem 4.3 (No inclusion)

For any , there exists such that some standard complete labelling of is not a complete labelling under and that some complete labelling of under is not a standard complete labelling.

This result holds even for that does not involve any support (see Example 6).

4.6 Acceptability Semantics

Definition 13 (Types of complete sets and acceptability semantics)

For any , we say that is: complete under iff there exists a complete labelling of under such that ; grounded under iff it is the set intersection of all complete sets under ; preferred under iff it is a maximal complete set under ; and semi-grounded iff it is a minimal complete set.

We call the set of all complete / grounded / semi-grounded / preferred sets under some complete / grounded / semi-grounded / preferred semantics under .

Note that ultimately we need to tell which subsets of are acceptable: this explains why we only look at for the semantics.

Example 7 (Acceptability semantics)

(Continued) For the argumentation in D with the same assumptions as in Example 5, if , then we have the following semantics.

  • complete: .             grounded: .

  • semi-grounded: .                                 preferred: .

If , then we have the following semantics.

  • complete: .                 grounded: .

  • semi-grounded:                           preferred: .

For the argumentation in F with the same assumptions as in Example 5, if , then we have (complete), (grounded), (semi-grounded) and (preferred).

Theorem 4.4 (Existence)

For any and any , we have all the following:

  1. There exists at least one complete set under .

  2. The grounded set may not be a complete set under .

  3. A semi-grounded set is a complete set under .

  4. If there is only one semi-grounded set, then it is the grounded set under .

Additionally, if , but not necessarily otherwise, there is only one semi-grounded set under .

5 Related Work and Conclusion

Structured Argumentation. Structured argumentation [11, 16] lets arguments associated in premise-conclusion relation. Consider ABA-style structured argumentation [11]: for: some finite set of entities; ; ; and . is said to support iff . For any , if , then