Reasoning by Cases in Structured Argumentation.
Abstract
We extend the framework for structured argumentation so as to allow applications of the reasoning by cases inference scheme for defeasible arguments. Given an argument with conclusion ‘ or ’, an argument based on with conclusion , and an argument based on with conclusion , we allow the construction of an argument with conclusion . We show how our framework leads to different results than other approaches in nonmonotonic logic for dealing with disjunctive information, such as disjunctive default theory or approaches based on the ORrule (which allows to derive a defeasible rule ‘If ( or ) then ’, given two defeasible rules ‘If then ’ and ‘If then ’). We raise new questions regarding the subtleties of reasoning defeasibly with disjunctive information, and show that its formalization is more intricate than one would presume.
1 Introduction
When formulated in terms of the material implication connective ‘’, the pattern of reasoning by cases is valid in classical logic:
(RBC) 
Many formalisms of nonmonotonic logic likewise allow defeasible applications of reasoning by cases, where a formula ‘’ reads ‘If , then normally/usually/probably ’:
(RBC) 
In the context of formal argumentation, it is natural to include a more general argumentative version of the RBC rule. Relative to a knowledge base this rule would allow the construction of an rbc argument with conclusion given (i) an argument with the disjunctive conclusion , and (ii) for each , an argument with conclusion based on an extended knowledge base including . To the best of our knowledge, no such rule has yet been introduced and studied in the context of formal argumentation. The aim of this paper is to do exactly this, and to investigate the nature of ‘argumentation by cases’.
For defining rbcarguments and attacks on rbcarguments, we will extend the framework [8]. is a framework for instantiating abstract argumentation frameworks as conceptualized by Dung [4]. We introduce abstract argumentation frameworks in Section 2, and define our formalism in Section 3. Our approach is limited in at least two ways. First, we do not allow for nested or iterated rbcarguments. Second, we do not yet take into account priorities assigned to arguments, nor do we include undercutting attacks. The removal of these limitations is left for future work.
The present approach raises new questions regarding the nature of arguing by cases. For instance, what happens if in the rbcargument one of the arguments is rebutted by an independent argument? Is this a sufficient condition for rejecting argument ? After all, there are other defeasible ‘paths’ leading to ’s conclusion. Our formalism implements a cautious rationale according to which a successful rebut on one of its paths is indeed sufficient for rejecting an rbcargument. In Section 4 we show how this approach leads to intuitive outcomes different from those obtained by other formalisms in nonmonotonic logic.
2 Abstract argumentation
A Dungstyle abstract argumentation framework (AF) is a pair where is a set of arguments and is a binary relation of attack. Relative to an AF, Dung defines a number of extensions – subsets of – on the basis of which we can evaluate the arguments in .
Definition 1 (Defense).
A set of arguments defends an argument iff every attacker of is attacked by some .
Definition 2 (Extensions).
Let be an AF. If is conflictfree, i.e. there are no for which , then (i) is a complete extension iff whenever defends ; (ii) is a preferred extension iff it is a set inclusion maximal complete extension; and (iii) is the grounded extension iff it is the set inclusion minimal complete extension.
Dung [4] showed that for every AF there is a unique grounded extension. On Dung’s abstract approach from [4], arguments are basic units of analysis the internal structure of which is not represented. In what follows we will instantiate abstract arguments by allowing for the representation of their internal logical structure.
3 Argumentation by cases
In this section we define structured argumentation frameworks (SAFs) for reasoning by cases. Our point of departure is an instantiation of the framework (without priorities, without defeasible premises and without undercuts).
We adjust in the following ways. (i) We define a new type of argument called an rbcargument (see point 4 in Definition 4). By means of rbcarguments, we can argue by cases in an argumentation formalism (Section 3.1). (ii) We generalize the attack relation so as to include argumentation by cases, using the concept of a hypothetical argument (Section 3.2), and we define a logical consequence relation for the resulting SAFs (Section 3.3). We briefly discuss the metatheoretical properties of our framework (Section 3.4).
3.1 Arguments
We illustrate our framework using the propositional fragment of classical logic () as our core logic. We denote the consequence relation of by . To obtain the formal language of , we close a denumerable stock of propositional letters under the usual connectives . We also add the verum constant and the falsum constant to . For reasons of transparency we will sometimes use subscripted letters as names for propositional letters.
Definition 3 (Argumentation theory).
An argumentation theory (AT) is a triple where:

is our formal language defined above;

is a set of strict () and defeasible () inference rules of the form and respectively (where ); and

is a consistent knowledge base.^{1}^{1}1 is consistent iff .
We assume in addition that iff . Since we keep and fixed, we will in the remainder refer to ATs as pairs .
Definition 4 (Arguments).
Given an argumentation theory , the set of arguments contains:

where


where and


where and


where

,

, ,

and for all .
We have:

Definition 4 departs in two respects from the way arguments are usually defined in . First, there is a new class of arguments constructed by means of rule 4: these arguments are called rbcarguments. They correspond to applications of the reasoning by cases scheme outlined in Section 1.
Example 1.
Let and . The following are arguments in :
and are rbcarguments constructed on the basis of the ‘cases’ and in the disjunctive conclusion of . Argument concludes that , while concludes that .
The second novel feature of Definition 4 is that we not only keep track of an argument ’s conclusion () and its subarguments (), but also of its hypothetical subarguments (). These are pairs consisting of an argument and a formula , where is constructible on the basis of the extended AT obtained by adding to the knowledge base of the original AT. For instance, .
’s hypothetical subargument contains the argument constructible on the basis of the AT .
Remark 1.
If one is interested in reducing the size of , one may only allow for minimal disjunctions when generating arguments of type 4. More precisely, where is of the form or , only if there is no where for which .
Definition 4 allows for the construction of arguments containing subarguments the conclusions of which are jointly inconsistent, such as arguments and in the following example, both of which rely on both and in their construction.
Example 2.
Let .
When is used as the underlying logic, inconsistent arguments like and may contaminate our formalism by blocking intuitively acceptable arguments like . (This is because the conclusions of and are conflicting, causing to attack and exclude , cfr. infra.) Contamination problems of this kind have been studied and tackled in [3, 10]. We can avoid them by filtering out inconsistent arguments.
Definition 5.
We define for an argument recursively as follows:


where

where

where .
Definition 6.
An argument is inconsistent iff . Otherwise is consistent. Relative to , is without inconsistent arguments.
For arguments without occurrences of our definition of inconsistent arguments is equivalent to that of [10]. In the remainder we will focus on the set rather than , avoiding contamination problems.
3.2 Attacks
In , attacks are defined in terms of a generic contrariness operator. We define them in terms of ‘’, so that arguments the conclusions of which are classical contradictories attack each other. We have to be careful when defining argumentative attacks when rbcarguments are involved, since we must take into account the hypothetical subarguments of an rbcargument. New questions arise here. For instance, an argument’s hypothetical subargument may conflict with a nonhypothetical argument.
Example 3.
Let , with and .
In Example 3, the nonhypothetical argument is in conflict with the argument , which belongs to ’s hypothetical subargument . The desirable outcome in this example is that attacks , but not vice versa.
As a further illustration, consider the following scenario.
Example 4.
, with and .
The following argument is constructible on the basis of the extended theory :

.
contains the intermediate conclusion based on the assumption . However, concludes that on the basis of the same assumption, . The desirable outcome in this example is to let and attack each other, since these arguments were both constructed on the basis of our knowledge base plus the assumption that , and since their conclusions are contradictories.
To handle examples like these, we introduce the set of all arguments that can be constructed on the basis of some disjunct used in the construction of an rbcargument in .
Where is an argument and a formula, and . We lift the definition as usual: where , .
Where , and , we denote by .
Definition 7 (Hypothetical Arguments).
Where , is the set of all such that for some .
Definition 8 (Attacks, Rebuts).
For a given theory , we define a direct attack relation
as follows: directly attacks iff is of the form , ( or ), and

and or

and or

and for some .
We lift the definition recursively in the following way: attacks if directly attacks or it attacks some .
For an argument to directly attack an argument , the following requirements need to be fulfilled: The conclusion of conflicts with the conclusion of and (either is nonhypothetical, or and are hypothetical arguments based on the same assumption ).
3.3 Consequence relations
Definition 9.
The structured argumentation framework (in short, SAF) defined by the theory is the pair .
Given a SAF, we can use the argumentation semantics from Section 2 to define consequence relations:
Definition 10.
Let , , let , and let , , and denote the sets of ’s complete extensions, ’s preferred extensions, and ’s grounded extension respectively.

iff for every there is an with .

iff there is a with .
Since the grounded extension is unique both definitions coincide for .
Relative to a theory , the ‘virtual’ arguments in are capable of attacking arguments in (in their hypothetical subarguments), and consequently of preventing the derivability of conclusions of arguments in . However, the conclusions of virtual arguments are never themselves derivable from AT.
3.4 Rationality postulates
In [2, 3] several postulates were proposed to evaluate formalisms for structured argumentation. In the present context these postulates read as follows. Given a SAF where :^{2}^{2}2The proofs of these properties are to be found in the technical appendix as indicated below, except for te proof of noninterference which we omit due to space restrictions. The authors in [2] distinguish between direct and indirect consistency: in our framework these definitions are equivalent since our strict rules are closed under .
4 Related work
4.1 Disjunctive Defaults
In [5] a generalization of default logic, disjunctive default logic, is proposed that is more apt to deal with disjunctions than Reiter’s original approach. For instance, given the default theory with and the defaults and , is not a default consequence in Reiter’s approach since the only extension of this theory is . In disjunctive default logic, an alternative disjunction is available: enforces that or is in any extension of the theory. So, for the default theory consisting of , and the defaults and we have two extensions, and . Now is a skeptical consequence. Default consequents can also make use of : a disjunctive default is of the form:
where is the prerequisite, are justifications, and are consequents of the default. A set of formulas is an extension of a disjunctive default theory consisting of the disjunctive defaults in (we here follow the convention in [5] according to which ’facts’ are considered as disjunctive defaults with empty prerequisite and empty justification) if it satisfies the following requirements: (i) for any , if and then for some , (ii) , and (iii) is minimal with properties (i) and (ii).^{3}^{3}3This definition in [5] is suboptimal in that it gives undesired results: e.g. for the theory also will form an extension. The problem can easily be fixed though by defining extensions analogous to Reiter.
We compare our approach to disjunctive default logic by thinking of as a default conditional: encodes the normal default . We start our comparison with the example given above. Let . has two extensions, and and so is a skeptical consequence. This outcome corresponds to our approach for the theory : the argument is in all complete extensions of .
Next, consider Poole’s broken arm example [9]. Let be “having a left broken arm”, “having a right broken arm”, “writing legibly”. On our approach, the theory gives rise to the argument , which is in all complete extensions of . In contrast, the disjunctive default theory has two extensions and . Since , is not a skeptical consequence in disjunctive default logic.
Finally, reconsider the AT from Example 3. There are various ways in which we can translate this AT into a disjunctive default theory, e.g.^{4}^{4}4Our discussion also applies if we translate by .
For we have the extensions , , and , so is not a skeptical consequence. This corresponds to our approach in which the argument from Example 3 is excluded due to the attack by . However, on our approach is in all complete extensions, so contrary to disjunctive default logic we obtain as a skeptical consequence.
4.2 The OR metarule
A different approach for dealing with disjunctive information is to allow for inference rules that produce new conditionals from given conditionals. We, for instance, find the following rule in system P [6] and in several Input/Output logics [7]:
One could define an like system where arguments are constructed as in rules 1–3 in Definition 4, and in which the defeasible rules are closed under OR. For instance, given from Section 4.1 this allows to derive from and , so that we can construct the argument and obtain as a consequence.
Adding OR is not sufficient to always get the intuitive outcome. Let . One would want to build an argument for , but we cannot put OR to much use (except for deriving ). We would have to combine OR with e.g. rightweakening (RW: ), or generalize OR to
in order to produce . Since also can be derived, we now have the means to construct the argument . In many systems of nonmonotonic logic, e.g. in system P and in many Input/Output logics, OR and RW are available (and thus gOR is a derived rule). We now contrast our approach with such (g)ORbased approaches.
A striking difference concerns the handling of Example 3. In the gORbased system we can construct:
This argument is not attacked by the argument . An alternative argument for is given by . Recall that neither in our approach nor in disjunctive default logic is a skeptical consequence. Even if we add to the knowledge base in Example 3, remains derivable in the gORbased approach since remains unchallenged. This is counterintuitive: both the argumentative path via and the path via are barred in view of the unchallenged arguments and . An advantage of using the reasoningbycases rule 4 in Definition 4 is that it provides more finegrained ways of tracking commitments in subarguments. This enables us to block the undesired consequence in this example.^{5}^{5}5An additional advantage is that argument strength can be tracked in a more finegrained way when using RbC in contrast to ORbased approaches. See also Section 5.
5 Conclusion and outlook
The ideas developed in this paper offer many interesting avenues for further work, partially consisting of the removal of the limitations assumed in this paper. For example, we did not consider nested rbcarguments or various components that can be modelled in the ASPICframework such as defeasible premises, undermining or undercut attacks. We also plan to present a less cautious variation of this framework where an attack on an rbcargument succeeds only if each of the ’s is attacked. In addition we will investigate prioritized default rules in this framework. Here too, new questions arise. Consider, for instance, an argument . If in addition there is an argument which is preferred over , then it seems intuitive to let the former attack the latter argument. But what if the hypothetical argument has a higher degree of priority than the nonhypothetical ? Should we decide in favor of the highest priority assigned, or should we never let a hypothetical argument attack a nonhypothetical one? More generally, how do we lift the priorities assigned to the (hypothetical and nonhypothetical) constituents of an rbcargument? Difficult questions like these will have to be answered in order to resolve conflicts between prioritized rbcarguments.
In future work we also plan to investigate the use of an ordered disjunction (see, e.g., [1]). For instance, a rule can be read as: ‘If then plausibly or , where is more plausible than ’. This is especially interesting when measures of argument strength are considered and when thinking about defeat of rbcarguments based on ordered disjunctions such as .
References
 [1] Gerhard Brewka. Logic programming with ordered disjunction. In In Proceedings of AAAI02, pages 100–105. AAAI Press, 2002.
 [2] M. Caminada and L. Amgoud. On the evaluation of argumentation formalisms. Artificial Intelligence, 171:286 – 310, 2007.
 [3] M. Caminada, W. Carnielli, and P. Dunne. Semistable semantics. Journal of Logic and Computation, 22(5):1207–1254, 2012.
 [4] P.M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and nperson games. Artificial Intelligence, 77(2):321–357, 1995.
 [5] M. Gelfond, V. Lifschitz, H. Przymusinska, and M. Truszczynski. Disjunctive defaults. In Proceedings of International Conference on Principles of Knowledge Representation and Reasoning (KR), pages 230–237, 1991.
 [6] S. Kraus, D. Lehmann, and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44:167–207, 1990.
 [7] D. Makinson and L. van der Torre. Input/output logics. Journal of Philosophical Logic, 29:383–408, 2000.
 [8] S. Modgil and H. Prakken. A general account of argumentation with preferences. Artificial Intelligence, 195:361–397, 2013.
 [9] D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the First International Conference on Principles of Knowledge Representation and Reasoning, pages 333–340, San Francisco, CA, USA, 1989. Morgan Kaufmann Publishers Inc.
 [10] Y. Wu. Between Argument and ConclusionArgumentbased Approaches to Discussion, Inference and Uncertainty (PhD dissertation). University of Luxembourg, 2012.
Appendix
For the metaproofs below it will be sometimes useful to speak about subarguments in the following stronger sense. consists of and for every where . For other types of arguments (see Def. 3, items 13), the definition of is the same as the definition of . Note that for any argument , . In the following results, if not specified further, is a structured argumentation framework based on the arbitrary argumentation theory . The next theorem is a strong form of subargument closure:
Theorem 1.
If and , .
Proof.
Let . It is easy to see that every attacker of attacks . Since is complete, it defends and hence also . Again, since is complete, . ∎
Definition 11.
Where , let .
Lemma 1.
(1) and have the same attackers in ; (2) ; (3) Where , (3.1) if , ; (3.2) if and , .
Proof.
(1) follows due to the fact that the set of defeasible conclusions of is the same as the set of defeasible conclusions of . (2) follows by simple manipulations in view of Definition 5. For (3.1) note that, by (1), if some attacks then it also attacks . Since is complete, is defended and thus in . The proof of (3.2) is similar to (3.1) and left to the reader. ∎
Theorem 2 (Closure under strict rules).
Where and : if .
Proof.
Let . We show the statement by an induction over where is the number of rules from and applied in . We have to show two things: (i) and (ii) .
() Then has the form . Thus, and . Since we assume our knowledge base to be consistent, is consistent and hence . Since has no attackers and is complete, .
() We first show (i). Assume for a contradiction that . This means that . Since this implies by transitivity that . Let be minimal such that and . By the induction hypothesis, which implies . Let be a list of ordered in such a way that for all we have and . Take a minimal such that and . Then,
(1)  
(2)  
(3) 
(3) follows by simple manipulations in view of (1) and since . We distinguish the following cases concerning the form of :

where and , or

where for each , …, each s.t. ,
Consider case 1 with (the case is left to the reader). By the inductive hypothesis, (3), Theorem 1 and Lemma 1, . However, since also and attacks , this is a contradiction to the conflictfreeness of .
We now sketch the proof for case 2. By some basic manipulations it can be shown that for all
(4) 
We distinguish two cases: (a) for all , is of the form and (b) there is a for which is of the form . In case (a), , which with (4) contradicts (2). In case (b), the argument attacks in its hypothetical subargument . By the inductive hypothesis, (3), Theorem 1, and Lemma 1, . This contradicts the conflictfreeness of and completes the proof of claim (i).
Suppose that some attacks . Clearly, attacks some (where ). Since, by Theorem 1, , some attacks . Altogether this shows that defends and since is complete, . This is claim (ii). ∎
Theorem 3 (Consistency).
Where and ,