Senses of ‘argument’ in instantiated argumentation frameworks^†

2.1.Abstract argumentation (Dung, 1995)

An AF is defined as follows (Dung, 1995).

The relevant auxiliary definitions are as follows, where S⊆N:

There are a variety of other semantics, for example, grounded, stable, and others (see Baroni, Caminada, and Giacomin, 2011 for an introduction), but for our purposes, we only consider preferred extensions (Dung, 1995).

As it is our intention to clarify the notion of argument itself, we want to use content neutral terminology in the expression of the syntax rather than relying on terminology such as argument and attacks, which introduce semantic interpretations. Thus, we usually refer to the so-called arguments of AFs defined above as graph-theoretic nodes (denoted by N) and their attack relations as arcs (denoted by O).

2.2.Instantiated argumentation (Caminada & Amgoud, 2007)

In this section, we briefly review the key components of the benchmark argument instantiation method ASPIC (Caminada & Amgoud, 2007), which we later exemplify and compare to our proposal.

In constructing arguments, several functions are introduced: Prem is the set of premises of the argument, Conc returns the last conclusion of an argument, Sub returns all the subarguments of an argument, DefRules returns all the defeasible rules used in an argument, and TopRule returns the last inference rule used in the argument. Theory Bases T comprised strict and defeasible implications. Arguments have a deductive form and are constructed recursively from the rules of the Theory Base. To distinguish strict or defeasible rules from the deductive form of arguments, we use short arrows, → and ⇒, for the former and long arrows, → and ⇒, for the latter. For brevity, we only provide the clauses for the construction of strict arguments as the clauses for the construction of defeasible arguments are analogous (including among the DefRules the TopRule(A) that is defeasible) (Caminada & Amgoud, 2007).

Such a system determines the acceptability status of propositions generally in three steps as in Figure 1 (from Caminada & Wu, 2011). Starting with an inconsistent KB comprised of facts and rules, we construct arguments (nodes) and attacks (arcs) from this KB, resulting in an AF (Step 1); evaluate the AF according to a variety of semantics, resulting in extensions (sets) of arguments (Step 2); and extract the conclusions from the arguments, resulting in extensions of conclusions (Step 3). Thus, from a KB that is initially inconsistent (or derives inconsistency), we can nonetheless identify consistent sets of propositions.

Figure 1.

Three steps of argumentation.

In Section 5.1, we provide an example of a KB and arguments as generated using ASPIC.

3.1.Theory Base T

We refer to the literals in bd(r) as premises and the literal in hd(r) as the claim.

For easy reference to the ‘content’ of the rule, we assume each rule has an associated definite description as follows. For r∈Rstr, the definite description of r has the form r:bd(r)→hd(r), where hd(r)∈L and bd(r)⊆L. Similarly, the definite description for r∈Rdfs has the form r:bd(r)⇒hd(r). Where a rule has an empty body, bd(r)=∅, we have r:→hd(r) or r:⇒hd(r), which are strict and defeasible assertions, respectively. To refer distinctly to the set of rules with non-empty bodies and those with empty bodies (assertions), we have R=PRules∪ARules, where PRules={r∣r∈R∧bd(r)≠∅} and ARules={r∣r∈R∧bd(r)=∅}.

We constrain a Theory Base, which we refer to as a Well-formed Theory, with the following Constraints 1–4. The purpose of the constraints is to reduce the expressivity of the language that appears in the Theory Base, guiding the knowledge engineer to produce a sound formalism. The first two constrain the relation of literals and rules; the third constrain reuse of literals between strict and defeasible rules; while the fourth prohibits strictly asserted contradiction.

First, every literal appears in some rule.

In addition, every rule has a claim.

Furthermore, the relationship between literals of strict and defeasible rules is constrained such that a strict rule is not, in effect, ‘contained within’ a defeasible rule:

Finally, no literal and its negation can both be strictly asserted.

Given these constraints, we have the following:

For the semantics, we assume standard notions of truth and falsity of literals. For the difference between strict and defeasible rules, we use quantification over circumstances (Lewis, 1975). Semantically, a rule r∈Rstr represents the notion that hd(r) ‘always’ holds if all of the literals in bd(r) simultaneously hold; that is, there are no circumstances where all of the literals in bd(r) hold and hd(r) does not. With respect to the rule, we say the bd(r) strictly implies the hd(r). Semantically, a rule r∈Rdfs represents the notion that hd(r) ‘usually’ holds if all of the literals in bd(r) simultaneously hold, but there are circumstances where ¬hd(r) holds though all of the literals in bd(r) simultaneously hold. With respect to the rule, we say the bd(r) defeasibly implies the hd(r). A formal discussion on the semantics for the Theory Base is provided in Appendix.

While the clauses are similar to the Horn Clauses of logic programming, the head literal can be in a positive or negative form. We only have classical negation, not negation as failure; we do not allow iterated negation – that is, ¬¬x≡x. The rationale for this choice of clauses is that it naturally supports our analysis of argument and its various senses as discussed in Section 4.

3.2.Deriving an AF from a Theory Base

A core element of our approach is the concept of the AF derived from a Theory Base. The AF uses a set of labels for the nodes in the graph: {x1,…,xn}∪{¬x1,…,¬n}∪{r1,…,rn} (or for clarity, the definite description of the rule name). Thus, we can see how elements of a Theory Base, T, correspond to but are distinct from elements of the derived AF, indexing the AF to the T. While in standard approaches to AFs, the graph is described in terms of arguments, we wish the syntax of the graph to be semantically unloaded; thus we refer to nodes. However, we maintain the standard terminology attacks as arcs between nodes. We first present the syntax of the derivation, then the semantics. Examples of derived AFs are given in Section 3.3.

3.3.Examples of the definitions

We now give some examples of the basic definitions, discuss defeasibility, provide a simple combination of strict and defeasible rules, illustrate reasoning with an assertion in a partial KB, and comment on the connection between the extensions and the classical models. In Section 5, we discuss examples from the literature, which have been discussed as problematic for an ASPIC-type approach, but are unproblematic in the approach presented above.

First, we provide a Theory Base T1 with just one strict rule, the derived AF, a graphic representation of the derived AF, and then the preferred extensions. Since it is always clear in context where we have literals and rules (in a Theory Base) and where we have labels (in an AF), we use one typographic form without confusion.

We graphically represent ⟨NT1,OT1⟩ as in Figure 2.

Figure 2.

AF of x1→x2.

In ⟨NT1,OT1⟩, the WFPEs are

We observe that the WFPEs with respect to an AF correlate with the models of the Theory Base; in this respect, the derived AF is a means to build models for the Theory Base. This observation applies to strict and defeasible rules alike (Section A.3).

The following is an example of a defeasible rule.

We graphically represent the derived AF ⟨NT2,OT2⟩ as in Figure 3.

Figure 3.

AF of x1⇒x2.

Figure 4.

AF derived from T3 with x1⇒x2 and x2→x3.

In ⟨NT2,OT2⟩, the WFPEs (which coincide with the preferred extensions of ⟨NT2,OT2⟩) are as follows:

The first three preferred extensions are similar to those for a strict rule. In the last extension, ¬x2 itself attacks the rule node r2; consequently, either x1 or ¬x1 are in a preferred extension along with ¬x2. This contrasts with the preferred extension of a derived AF with just strict rules. Indeed, if x1 is asserted by attacking ¬x1, there will be two WFPEs, namely {x1,r2,x2} and {x1,x2}. Therefore, differently from strict rules, a defeasible rule is not ‘strong enough’ to ensure that x2 will hold (sceptically) since x1 holds.

Therefore, to make use of a defeasible rule, one must provide the means to choose between extensions, for example, by selecting the extension which maximises the number of applicable defeasible rules, or which uses some notion of priority or entrenchment on the rules. Different ways of making this choice give rise to different varieties of non-monotonic logic (Prakken & Sartor, 1997; Reiter, 1980). Circumscription (McCarthy, 1980) could be used by including additional designated nodes such as ab(r1) which attack the rule r1 and attack and are attacked by notab(r1). We then choose the extension containing the most notab(r1) nodes. Using DeLP defeaters (García & Simari, 2004), we can specify circumstances where the rule is not applied.

In our third example, we show the interaction of defeasible and strict rules, which was the root of several of the problems identified in Caminada and Amgoud (2007).

AF ⟨NT3,OT3⟩ has the following six preferred extensions:

4.1.Rebuttal, undermining, and undercutting

Given these structures we can express the various familiar notions of attack on an argument: a Undermining of an Argument is an Argument with claim that is the negation of the premise of another Argument; a Rebuttal of an Argument is an Argument with a claim that is the negation of the claim of another Argument; the Rebuttal of a Case is similar to the Rebuttal of an Argument; and an Undercutter of an Argument is an attacker of the rule node of an argument.

Given the definitions of the three senses of ‘argument’ and various notions of attack, it would be possible to define a more abstract, derivative AF in which we represent structures at levels ArgS1 or ArgS2 as nodes in that AF and which, given the requisite attack relations defined above, are in a higher level attack relation. However, we leave such proposals and the analysis of them for future research.

4.2.Assertions

So far we have only PRules in a Single-point Debate. Usually in a Theory Base there are assertions which further restrict the preferred extensions. In our framework, assertions are ARules. A Case such as ArgS2(y) is a directed tree where y is the root and the literals in the bodies of constituent arguments at the level where ρk+1(y)=ρk(y) are the leaves. If all the leaves are strictly asserted and all the rules are strict rules, then the root is sceptically accepted. However, where even just one of the leaves of ArgS2(y) is defeasibly asserted, then the root of the tree is only credulously accepted. Such a Case is vulnerable to attack. Similarly, where there are defeasible rules, the root is always credulously accepted. Where we have ArgS2(y) and ArgS2(¬y) for and against a claim and the leaves of each Case are strictly asserted, then the resultant Single-point Debate must be adjudicated according to some principle for choosing between preferred extensions. Matters are complex where we have strict and defeasible assertions of literals at different levels of the tree. Further investigation would consider how to adopt some notion of accrual in AFs to overcome some of these limitations (Modgil & Bench-Capon, 2010; Prakken, 2005). However, these matters are beyond the scope of this paper.

5.1.Comparison to ASPIC with a base-case example

In this section, we exemplify the differences between our approach and ASPIC. In Section 5.1.1, we show how the ex falso quodlibet and the tertium non datur are automatically satisfied by our approach, while they require additional rules in ASPIC. Section 5.1.1 provides an example from Caminada and Amgoud (2007), which highlights an issue in the instantiation method of ASPIC (Caminada and Amgoud, 2007) (see Section 2.2) as well as motivates the Rationality Postulates. We then show how such problems do not arise in our approach.

5.1.1.Credulous reasoning

Let us consider a Theory Base Tcr with the following rules:

rcr1:→¬x1;rcr2:⇒x1;rcr3:x1⇒x2.

ASPIC constructs the following three arguments:

Acr1:[→¬x1];Acr2:[⇒x1];Acr3:[[⇒x1]⇒x2].

Since ¬x1 is strictly asserted, then Acr1 defeats both Acr2 and Acr3. This results in having ¬x1 as the only ‘accepted’ conclusion.

Instead, using the approach described in Section 3, the resulting framework is shown in Figure 6.

Figure 6.

Credulous reasoning.

In particular, the shown framework has two WFPEs, namely {rcr1,¬x1,x2}, and {rcr1,¬x1,¬x2}. ¬x1 is sceptically accepted, but the system acknowledges the existence of x2 and forces the choice between assuming either x2 or ¬x2.

To obtain the same extensions using ASPIC, Tcr should be enlarged with the following rules:

⇒x2;⇒¬x2.

These rules give rise to two further arguments in favour of assuming either x2 or ¬x2:

Acr4:[⇒x2];Acr5:[⇒¬x2].

Thus, as pointed out in the introduction, ASPIC requires auxiliary rules that our approach does not.

5.1.2.Caminada and Amgoud Example 1

Consider a Theory Base with strict and defeasible rules from which we construct arguments according to the definitions of ASPIC (Caminada & Amgoud, 2007) outlined in Section 2.2. We examine Example 5 of Caminada and Amgoud (2007).

Example 5

Let T4 be a Theory Base with the following rules:

r21:→x1;r22:→x2;r23:→x3;r24:x4,x5→¬x3;r25:x1⇒x4;r26:x2⇒x5.

We construct the following arguments:

A1:[[→x1]⇒x4];A2:[[→x2]⇒x5];A3:[→x3];A4:[→x1];A5:[→x2];A6:[[[→x1]⇒x4],[[→x2]⇒x5]→¬x3].

We see clearly that arguments can have subarguments: A6 has a subargument A1, A1 has a subargument A4, and A6 has a subargument A4.

Several additional elements are needed to define justified conclusions. An argument is strict if it has no defeasible subargument, otherwise it is defeasible (non-strict). An argument Ai rebuts an argument Aj where the conclusion of some subargument of Ai is the negation of the conclusion of some non-strict subargument of Aj; rebuttal is one way an argument defeats another argument. Admissible argument orderings specify that a strict argument (containing premises that are axioms and rules that are strict) can defeat a defeasible argument, but not vice versa. Moreover, one argument can defeat another argument with respect to subarguments; in effect, defeat of a part is inherited as defeat of a whole. With respect to our example, the undefeated arguments are A1, A2, A3, A4, and A5. A3, which is a strict argument, defeats A6 but not vice versa since A6 is a non-strict argument in virtue of having a defeasible subargument. Given the arguments and defeat relation between them, we can provide an AF and the different extensions. The Output of an AF, understood as the justified conclusions of theAF, is given as the sceptically accepted conclusions of the arguments of theAF.

With respect to the example, Caminada and Amgoud (2007) claim that the justified conclusions are x1, x2, x3, x4, and x5 since these are all conclusions of arguments which are not attacked (it appears not to be an example analysed in Prakken, 2010). However, ¬x3 is not a justified conclusion, even though it is the conclusion of a strict rule in which all the premises are justified conclusions. This is so since the argument A6 of which ¬x3 is the conclusion is defeated by but does not defeat A3 because A6 has a subargument which is a non-strict argument (namely A1 or A2), so making A6 a non-strict argument, while A3 is a strict argument. Yet, given the antecedents of the strict rule are justified conclusions, it would seem intuitive that the claim of a strict rule should also be a justified conclusion. This, they claim, shows that justified conclusions are not closed under strict rules or could even be inconsistent.

In our view, these notions of argument and defeat are problematic departures from Dung (1995), which has no notion of subargument or of defeat in terms of subarguments. In addition, they give rise to the problems with justified conclusions: what is a strict rule in the Theory Base can appear in the AF as a non-strict argument in virtue of subarguments; what cannot be false in the Theory Base without contradiction is defeated in the AF; thus, what ‘ought’ to have been a justified conclusion is not. In addition, the notion of justified conclusion leads to some confusion: on the one hand, it only holds for sceptically accepted arguments, which presumably implies that the propositions which constitute them are sceptically accepted; on the other hand, there is no reason to expect that ¬x3 is sceptically accepted, given that it only follows from defeasible antecedents. Clearly the anomaly arises because of the way that arguments can have defeasible subarguments, that the defeat of the whole can be determined by the defeat of a part, and that justified conclusions depend on these notions.

In our approach, the results are straightforward and without anomaly; we do not make use of arguments with subarguments, inheritance of defeasiblity, or problematic notions of justified conclusions. We consider a key example from Caminada and Amgoud (2007) as the two other problematic examples cited in Caminada and Amgoud (2007) follow suit. The Theory Base of Example 5 appears as in Figure 7, for which all the preferred extensions for the AF are given. For clarity and discussion, we include undefeated strict and defeasible rules.

1.{x1,r21,r25,x2,r22,x3,x4,r25,¬x5}2.{x1,r21,r23,x2,r22,x3,¬x4,x5,r26}3.{x1,r21,r23,x2,r22,x3,¬x4,¬x5}4.{x1,r21,r23,r25,x2,r22,r24,r26,x4,x5}

Figure 7.

Graph of problem example.

Notice that WFPEs (1)–(3) are unproblematic with respect to consistency and closure. They also satisfy Definition 7, so are the relevant extensions to consider. In contrast, extension (4) is problematic in an argumentation theory without Definition 7 since the conclusions of strict rules are missing, thus violating closure. Yet, (4) does not satisfy Constraint 5: the premises and rule nodes of strict rules are present, but the conclusions are not. With respect to those extensions that satisfy Definition 7, x1, x2, x3 are all sceptically accepted, while x4 and x5 are credulously accepted. ¬x3 is not credulously accepted given that x3 is strictly asserted. Note that every literal which is strictly asserted is sceptically acceptable. Therefore, the rule node r24 must be defeated where one or both of ¬x4 and ¬x5 hold. There is, in our view, no reason to expect ¬x3 to hold in any extension since we have no preferred extension in which both x4 and x5 are justified conclusions. Given WFPEs, we satisfy the consistency rationality postulate; closure, which is relevant only if strict rules where all the body literals hold, is not relevant to this problem. Moreover, we can provide machinery to meet Definition 7 in that we can examine the extensions relative to the rules of the AF to determine if Constraint 5 is satisfied. The analysis also corresponds well with model-building for classical logic.

5.2.Other approaches

We have considered a widely adopted approach to instantiating Theory Bases in AFs (Caminada & Amgoud, 2007) along with the problems that arise. There are other approaches to instantiating a KB in an AF that may avoid problems with the Rationality Postulates such as Assumption-based (Bondarenko et al., 1997) or Logic-based (Besnard & Hunter, 2008) argumentation. However, these approaches, like the ASPIC approach, follow the three-step structure of Figure 1; we leave further comparison and contrast to future work.

Here we comment briefly on a closely related proposal (Strass, 2013) set in the different approach of Abstract Dialectical Frameworks (ADF) (Brewka & Woltran, 2010), which is presented as a generalisation of Dungian AFs but also as a means to represent instantiated arguments, for example, logic programs (Brewka and Woltran, 2010). Broadly, we may distinguish between approaches based on Dung (1995) that make use of nodes (arguments) and arcs (attacks) alone to determine extensions and those which use auxiliary conditions to specify extensions with respect to successful attacks such as preferences (Prakken, 2010) or values (Bench-Capon, 2003). The approach of Brewka and Woltran (2010) is a generalisation of the latter approach: in addition to nodes (which can be statements or literals) and links (generalised from arcs as attacks), there are acceptance conditions, which are functions for each statement from its parents (those nodes in a single link) to {in,out}. Given this generic approach to acceptance conditions, many complex aspects of argumentation can be accommodated.

Strass (2013) translates Wyner et al. (2009, 2013) into an ADF and applies it to the same problem presented in Figure 7. The approach has a version of Constraint 4, adding support along with attack and a form of negation on rules (interpreted as inapplicability). The system is proven to abide by the Rationality Constraints, though this seems to hinge on the constraint. The main issue, in our view, is that ADF requires the generation of and reasoning with (presumably correct) acceptance conditions in addition to the AF rather than on the graph per se, which was one of the main advantages of the Dungian abstraction.

5.3.Additional advantages

Our approach has three additional strengths, which we discuss briefly here, leaving for future work further comparison and contrast.

5.3.1.The two step

In Section 2.2, we outlined the argument construction method of ASPIC, which follows the three-step structure of Figure 1 that generates arguments from a KB: arguments are complex ‘objects’ constructed from the KB, then abstracted over in theAF; once arguments and attacks are determined, we have an AF, from which we determine extensions and consequently the justified conclusions. Yet, this can be construed to overgenerate arguments: for instance in Besnard and Hunter (2008), significant effort is devoted to analysis of relations between arguments and the specification of canonical arguments (we are not aware of similar analysis in the ASPIC framework). One might regard the arguments of ASPIC or a Logic-based approach as intermediate concepts (Wyner, 2008) or as catalysts which, having served their purpose in determining justified conclusions, are no longer essential to reasoning.

By contrast, while arguments (in their various senses) can be similarly constructed as in Section 4, this is not necessary in our framework in order to calculate the justified conclusions (our WFPEs); so far as the AF is concerned, the nodes in the graph are atomic, that is, literal or rule nodes, in specified attack relations; WFPEs are determined only with respect to such nodes and attacks. In other words, arguments such as in ASPIC or Logic-based argumentation are not essential, and there is no overgeneration of arguments as complex ‘objects’. This straightforwardly simplifies analysis as well as the calculation of justified conclusions from the three steps of Figure 1 to the two steps of Figure 8.

Figure 8.

Two steps of argumentation.