Argumentation has played a significant role in understanding and unifying under a common framework di↵erent forms of defeasible reasoning in AI. Argumentation is also close to the original inception of logic as a framework for formalizing human argumentation and debate. In this context, the purpose of this paper is twofold: to draw a formal connection between argumentation and classical reasoning (in the form of Propositional Logic) and link this to support defeasible, Non-Monotonic Reasoning in AI. To this e↵ect, we propose Argumentation Logic and show properties and extensions thereof.
Introduction
Over the past two decades argumentation has played a significant role in
understanding and unifying under a common framework defeasible Non-Monotonic
Reasoning (NMR) in AI [
This paper reexamines the foundations of classical logical reasoning from an argumentation perspective, by formulating a new logic of arguments, called Argumentation Logic (AL), and showing how this relates to Propositional Logic. AL is formulated by bringing together argumentation theory from AI and the syllogistic view of logic in Natural Deduction (ND). Its definition rests on a reinterpretation of Reductio ad Absurdum (RA) through a suitable argumentation semantics. One consequence of this is that in AL the implication connective behaves like a default rule that still allows a form of contrapositive reasoning. The reasoning in AL is such that the ex-falso rule where everything can be derived from an inconsistent theory does not apply and hence an inconsistent part of a theory does not trivialize the whole reasoning with that theory.
The main motivation for studying this argumentation perspective on logical reasoning is to examine its use to bring together classical reasoning and nonmonotonic commonsense reasoning into a single unified framework. The paper presents a preliminary investigation into building such a NMR framework based on AL that integrates into the single representation framework of AL classical reasoning, as in Propositional Logic including forms of Reductio ad Absurdum, with defeasible reasoning. In particular, we study, in the context of examples, the possible use of preferences over sentences of an AL theory to capture NMR defeasible reasoning and naturally combine this with the classical reasoning of AL. Our vision is for all forms of reasoning to be captured in the argumentation reasoning of AL and its extensions with preferences.
This paper is an extract of [13]. 2
Preliminaries on Natural Deduction Let L be a Propositional Logic language and ` denote the provability relation of Natural Deduction (ND) in Propositional Logic.4 Throughout the paper, theories and sentences will always refer to theories and sentences wrt L. We assume that ? stands for ^ ¬ , for any 2 L .
Definition 1. Let T be a theory and a sentence. A direct derivation for (from T ) is a ND derivation of (from T ) that does not contain any application of RA. If there is a direct derivation for (from T ) we say that is directly derived (or derived modulo RA) from T , denoted T`MRA .
!
)}. The following is a direct derivation for Example 1. Let T1 = {↵ ! ( ↵ ^ ! (from T1): d↵ ^ hypothesis ↵ ^ E ↵ ! ( ! ) from T ! ! E
^ E c ! E ↵ ^ ! ! I Thus, T1 `MRA ↵ ^ ! (and, trivially, T1 ` ↵ ^ ! ). Consider now T2 = {¬(↵ _ )}. The following d↵ hypothesis ↵ _ _ I ¬(↵ _ ) from T ?c ^ I ¬↵ RA is a ND derivation of ¬↵ that is not direct. Since there is no direct derivation of ¬↵ , T2 ` ¬↵ but T2 6`MRA ¬↵ . 4 See the appendix for a review of the ND rules that we use. Definition 2. A theory T is classically inconsistent i↵ T ` ? . A theory T is directly inconsistent i↵ T `MRA ? . A theory T is (classically or directly) consistent i↵ it is not (classically or directly, respectively) inconsistent. Trivially, if a theory is classically consistent then it is directly consistent. However, a directly consistent theory may be classically inconsistent, as the following example shows.
! ? , ¬↵ ! ?} . T1 is classically inconsistent, ↵ RA ? ^ I However, T1 is directly consistent, since T1 6`MRA ? . Consider instead T2 = {↵, ¬↵ }. T2 is classically and directly inconsistent, since T2 `MRA ? , as follows: ↵ from T ¬↵ from T ? ^ I We will use a special kind of ND derivations, that we call Reduction ad Absurdum Natural Deduction (RAND). These are ND derivations with an outermost application of RA: Definition 3. A RAND derivation of ¬ 2 L from T is a ND derivation of ¬ from T of the form d hypothesis ... ...
.
.
?c .
¬ RA A sub-derivation (of some 2 L ) of a derivation d is a RAND sub-derivation of d i↵ it is a RAND derivation.
¬↵ ↵ of the derivation (d) of ? are RAND sub-derivations (of d).
The ND derivation of ¬↵ given T2 in example 1 is a RAND derivation. Also, given T1 in example 2, the sub-derivations5 d↵ d¬↵ ↵ ! ? ¬ ↵ ! ? ?c ?c 5 If clear from the context, we omit to give the ND rules used. 3
Argumentation Logic Frameworks Given a propositional theory we will define a corresponding argumentation framework as follows.
Definition 4. The argumentation logic (AL) framework corresponding to a theory T is the triple hArgsT , AttT , Def T i defined as follows: – ArgsT = {T [ ⌃ |⌃ ✓ L}
in L; – given a, b 2 ArgsT , with a = T [
AttT i↵ a [ b `MRA ? ; – given a, d 2 ArgsT , with a = T [ 1. d = T [ {¬ } (d = T [ {
¬ 2 ), or 2. d = T [ {} and a `MRA ? .
is the set of all extensions of T by sets of sentences , b = T [ , such that
6= {}, (b, a) 2 , (d, a) 2 Def T i↵ }) for some sentence 2 (respectively In the remainder, b attacks a (wrt T ) stands for (b, a) 2 AttT and d defends or is a defence against a (wrt T ) stands for (d, a) 2 Def T .
Note also that, since T is fixed, we will often equate arguments T [ ⌃ to sets of sentences ⌃ . So, for example, we will refer to T [ {} = T as the empty argument. Similarly, we will often equate a defence to a set of sentences. In particular, when d = T [ defends/is a defence against a = T [ we will say that defends/is a defence against (wrt T ).
The attack relation between arguments is defined in terms of a direct derivation of inconsistency. Note that, trivially, for a = T [ , b = T [ , (b, a) 2 AttT i↵ T [ [ `MRA ? . The following example illustrates our notion of attack: Example 3. Given T1 = {↵ ! ( ! )} in example 1, {↵, } attacks {¬ } (and vice-versa), {↵, ¬ } attacks { } (and vice-versa), {↵, ¬↵ } attacks { } (and viceversa) as well as any non-empty set of sentences (and vice-versa). Note that the attack relationship is symmetric except for the case of the empty argument. Indeed, for a, b both non-empty, it is always the case that a attacks b i↵ b attacks a. However, the empty argument cannot be attacked by any argument (as the attacked argument is required to be non-empty), but the empty argument can attack an argument. For example, given T2 = {↵, ¬↵ } in example 2, {} attacks {↵ } and {} attacks { } (for any 2 L ), since T `MRA ? . Finally, note that our notion of attack includes the special case of attack between a sentence and its negation, since, for any theory T , { } attacks {¬ } (and vice-versa), for any 2 L .
The notion of defence is a subset of the attack relation. In the first case of the definition we defend against an argument by adopting the complement6 of some sentence in the argument, whereas in the second case we defend against any directly inconsistent set using the empty argument. Then, in example 3, {¬↵ } 6 The complement of a sentence is ¬ and the complement of a sentence ¬ is . defends against the attack {↵, } and {} defends against the (directly inconsistent) attack {↵, ¬↵ }. Note that the empty argument cannot be defended against if T is directly consistent. Finally, note that when T is directly inconsistent the attack and defence relationships trivialise, in the sense that any two non-empty arguments attack each other, the empty argument attacks any argument, and any argument can be defended against by the empty argument. 4
Argumentation Logic In this section we assume that T is directly consistent.
As conventional in argumentation, we define a notion of acceptability of sets of arguments to determine which conclusions can be dialectically justified (or not) from a given theory. Our definition of acceptability and non-acceptability is formalised in terms of the least fix point of (monotonic) operators on the cartesian product of the set of arguments/sentences in L, as follows: Definition 5. Let hArgsT , AttT , Def T i be the AL framework corresponding to a directly consistent theory T , and R the set of binary relations over ArgsT . is defined as follows: for any acc 2 R – The acceptability operator AT :R!R and a, a0 2 ArgsT : (a, a0) 2 A T (acc) i↵ • a ✓ a0, or • for any b 2 ArgsT such that b attacks a wrt T , ⇤ b 6✓ a0 [ a, and ⇤ there exists d 2 ArgsT that defends against b wrt T such that (d, a0 [ a) 2 acc. – The non-acceptability operator NT : R ! R nacc 2 R and a, a0 2 ArgsT : (a, a0) 2 N T (nacc) i↵ is defined as follows: for any • a 6✓ a0, and • there exists b 2 ArgsT such that b attacks a wrt T and ⇤ b ✓ a0 [ a, or ⇤ for any d 2 ArgsT that defends against b wrt T , (d, a0 [ a) 2 nacc. These AT and NT operators are monotonic wrt set inclusion and hence their repeated application starting from the empty binary relation will have a least fixed point.
Definition 6. ACCT and N ACCT denote the least fixed points of AT and NT respectively. We say that a is acceptable wrt a0 in T i↵ ACCT (a, a0), and a is not acceptable wrt a0 in T i↵ N ACCT (a, a0).
Thus, given the AL framework hArgsT , AttT , Def T i (for T directly consistent) and a, a0 2 ArgsT : – ACCT (a, a0), i↵ • a ✓ a0, or • for all b 2 ArgsT such that b attacks a: ⇤ b 6✓ a0 [ a, and ⇤ there exists d 2 ArgsT such that d defends against b and
ACCT (d, a0 [ a); – N ACCT (a, a0), i↵ • a 6✓ a0 and • there exists b 2 ArgsT such that b attacks a and ⇤ b ✓ a0 [ a, or ⇤ for all d 2 ArgsT such that d defends against b it holds that
N ACCT (d, a0 [ a).
We will often equate the (non-)acceptability of an argument T [ wrt an argument T [ 0 to the (non-)acceptability of the set of sentences wrt the set of sentences 0.
Note that non-acceptability, N ACCT (a, a0), is the same as the classical negation of ACCT (a, a0), i.e. N ACCT (a, a0) = ¬ACCT (a, a0). We choose to give the definition of non-acceptability explicitly to aid readibility. We will use these two versions of non-acceptability interchangeably.
Note that the empty argument is always acceptable, wrt any other argument. Note also that the “canonical” attack of a sentence on its complement (i.e. of T [ { } on T [ {¬ } and vice-versa) does not a↵ect the acceptability relation as it can always be defended against by this complement itself.
The following examples illustrate non-acceptability.
{¬O } (since T [{¬ }`MRA? ) {↵ O }
(since T [{ ↵ }[{ }`MRA? ) {} { KS } {¬O }
(since T [{¬ }`MRA? ) {}
Example 4. Let T = {↵ ^ ! ? , ¬ ! ?} . T is classically and directly consistent, T [ {¬ } is classically and directly inconsistent, and T [ { ↵ } is classically inconsistent but directly consistent. It is easy to see that N ACCT ({¬ }, {}) holds, as illustrated in figure 1 (left)7, since {¬ } 6✓ {} , b = {} attacks {¬ } 7 Here and throughout the paper we adopt the following graphical convention: " denotes an attack and * denotes a defence. and {} ✓ {¬ Indeed:
}. Also, N ACCT ({↵ }, {}) holds, as illustrated in figure 1 (right). – since {↵ } 6✓ {} , b = { } attacks {↵ } and {¬ } is the only defence against b, to prove that N ACCT ({↵ }, {}) it suces to prove that N ACCT ({¬ }, {↵ }); – since {¬ } 6✓ { ↵ }, b = {} attacks {¬ } and {} ✓ { ↵, ¬ }, it follows that
N ACCT ({¬ }, {↵ }) holds as required.
Note that if an argument a is attacked by the empty argument, then it is acceptable wrt a0 i↵ a ✓ a0, since there is no defence against the empty argument. This observation is used in the following example.
Example 5. Given T = T1 = {↵ ! ? , ¬↵ ! ?} in example 2, N ACCT ({↵ }, {}) and N ACCT ({¬↵ }, {}) both hold. Indeed, for N ACCT ({↵ }, {}), {↵ } is attacked by {}.
The following example shows a case of non-acceptability making use of a nonempty attack for the base case.
Example 6. Let T = {↵ ^ ¬ ! ? , ^ ! ? , ↵ ^ ^ ¬ ! ?} . T is classically (and directly) consistent, and T [ { ↵ } is classically inconsistent but directly consistent. N ACCT ({↵ }, {}) holds, as illustrated in figure 2. Indeed: {↵ O }
(since T [{ ↵ }[{¬ }`MRA? ) {¬KS } { O }
(since T [{ }[{ }`MRA? ) – since {↵ } 6✓ {} , b = {¬ } attacks {↵ } and { } is the only defence against b, to prove that N ACCT ({↵ }, {}) it suces to prove that N ACCT ({ }, {↵ }); – since { } 6✓ { ↵ }, b0 = { } attacks { } and {¬ } is the only defence against b0, to prove N ACCT ({ }, {↵ }) it suces to prove N ACCT ({¬ }, {↵, }); – since {¬ }✓ {6 ↵, }, b00 = {↵, } attacks {¬ } and b00 ✓ { ↵, , ¬ },
N ACCT ({¬ }, {↵, }) and so N ACCT ({ }, {↵ }) and N ACCT ({↵ }, {}) hold. The following example illustrates non-acceptability in the case of an empty (and thus classically consistent) theory. Example 7. For T = {}, N ACCT ({¬( ¬_ )}, {}) holds, as illustrated in figure 3. Also, trivially, N ACCT ({ ^ ¬ }, {}) holds, since it is attacked by the empty argument.
{¬( _ ¬O
)}
A novel, alternative notion of entailment can be defined for theories that are directly consistent in terms of the (non-) acceptability semantics for AL frameworks, as follows: Definition 7. Let T be a directly consistent theory and 2 L . Then is ALentailed by T (denoted T |=AL ) i↵ ACCT ({ }, {}) and N ACCT ({¬ }, {}). This is motivated by the argumentation perspective, where an argument is held if it can be successfully defended and it cannot be successfully objected against.
In the remainder of the paper we will study properties of |=AL and discuss extensions thereof to support NMR. 5
Basic Properties The following result gives a core property of the notion of AL-entailment wrt the notion of direct derivation in Propositional Logic, for directly consistent theories. Proposition 1. Let T be a directly consistent theory and T `MRA . Then T |=AL . 2 L such that Proof: Let a = T [ be any attack against { }, i.e. T [ { } [ `MRA ? . Since T `MRA then T [ `MRA ? . Since T is directly consistent, 6= {}. Hence any such a can be defended against by the empty argument. Since ACCT ({}, ⌃ ), for any ⌃ ✓ L , then ACCT ({ }, {}) holds. Moreover, since T `MRA , necessarily T [ {¬ } `MRA ? . Hence the empty argument attacks {¬ } and thus N ACCT ({¬ }, {}) holds. QED
The following theorem shows (one half of) the link of AL with Propositional Logic by showing how the RA rule, deleted from the ND proof system within `MRA, can be recovered back through the notion of non-acceptability.8 Theorem 1. Let T be a directly consistent theory and 2 L . If N ACCT ({ }, {}) holds then there exists a RAND derivation of ¬ from T .9 ¬↵ For example, the RAND derivation corresponding to the proof of N ACCT ({↵ },{}) in figure 1 is d1 in figure 4.10 Here, the inner RAND derivation in d1 corresponds to the non-acceptability of the defence {¬ } against the attack { } against {↵ }. Derivation d2 in figure 1 is an alternative RAND of ¬↵ , but this cannot be obtained from any proof of N ACCT ({↵ }, {}), because there is a defence against the attack { } given by the empty set (in other words, d2 does not identify a useful attack, that cannot be defenced against, for proving non-acceptability). 6
AL for Propositional Logic The following result gives a core property of the notion of non-acceptability for classically consistent theories.
Proposition 2. Let T be classically consistent and 2 L . If N ACCT ({¬ }, {}) holds then ACCT ({ }, {}) holds.
Proof: By theorem 1, since N ACCT ({¬ }, {}), then T ` . Suppose, by contradiction, that ACCT ({ }, {}) does not hold. Then N ACCT ({ }, {}) holds (since N ACCT ({ }, {})=¬ACCT ({ }, {})) and by theorem 1 there is a RAND derivation of ¬ from T and thus T ` ¬ . This implies that T is classically 8 The other half of this result shows how (under some conditions) a RAND derivation of ¬ implies N ACCT ({ }, {}), proven in [12]. 9 The proof of this theorem is included in [13]. 10 Here and elsewhere in the paper, c( ), for any 2 L , indicates that is the hypothesis of an ancestor sub-derivation copied within the current sub-derivation.
from T inconsistent: contradiction. Hence ACCT ({ }, {}) holds.
QED Thus, in Propositional Logic, trivially AL-entailment reduces to the notion of non-acceptability: Corollary 1. Let T be a classically consistent theory and i↵ N ACCT ({¬ }, {}). 2 L . Then T |=AL The following property sanctions that AL-entailment implies classical derivability: Corollary 2. Let T be a classically consistent theory and then T ` . 2 L . If T |=AL Proof: If N ACCT ({¬ }, {}), then, by theorem 1, there is a RAND derivation of ¬¬ from T and thus T ` . QED This corollary gives that consequences of a classically consistent theory under |=AL are classical consequences too. Proposition 1 sanctions that direct consequences are not lost by |=AL. However, in general not all classical consequences are retrieved by |=AL, namely the converse of corollary 2 does not hold, as the following example shows.
Example 8. Let T = {¬↵ }. We show that T 6|=AL ↵ ! by showing that N ACCT ({¬(↵ ! )}, {}) does not hold. A standard ND derivation of ↵ ! from T is: ¬¬ RA
c ¬E ↵ ! ! I This does not help with determining N ACCT ({¬(↵ ! )},{}). This is related to the fact that the inconsistency in the inner RAND derivation of ¬¬ is derived without the need of the hypothesis, ¬ , of this RAND derivation. In general, any RAND derivation of ¬¬(↵ ! ) (and hence of ↵ ! ) from this theory, T , contains such a RAND sub-derivation relying on the inconsistency of the copy of ↵ from a (! I) sub-derivation, with ¬↵ from T . This means that N ACCT ({¬(↵ ! )}, {}) cannot hold, since, otherwise, by theorem 1, we would have a RAND derivation of ¬¬(↵ ! ) without such a sub-derivation. This is because by construction of the corresponding RAND derivation given by theorem 1 the existence of such a RAND sub-derivation would violate the non-acceptability of some defence in the assumed non-acceptability of ¬(↵ ! ).
This example shows, in particular, that implication is not material implication under |=AL.
AL for Non-Monotonic Reasoning-Discussion Here we present a first investigation on how AL can be used as a basis for NMR unifying classical and defeasible reasoning, in the context of the well known tweety example. Our examination is based on the (expected) need to extend AL with preferences and the observation that when a theory is (directly) inconsistent we have the possibility to reason with its sub-theories, considering these as arguments that support their conclusions under AL. For the illustration we use the following (abbreviations of) sentences: bf = [bird(tweety) ! f ly(tweety)] p¬f = [penguin(tweety) ! ¬f ly(tweety)] pb = [penguin(tweety) ! bird(tweety)] ¬f = [¬f ly(tweety)] p = [penguin(tweety)] ¬b¬p = [¬bird(tweety) ! ¬penguin(tweety)] Example 9. Let T = { bf , pb, ¬f } (T is classically consistent). It is easy to see that T |=AL ¬bird(tweety) as {} attacks {bird(tweety)} and therefore N ACCT ({bird(tweety)}, {}). Similarly, T |=AL ¬penguin(tweety). In absence of other information, we believe that these conclusions are legitimate/desirable. Note that AL does not distinguish default rules and facts and it supports contrapositive reasoning with the single form of implication it allows. In example 9, default logic [19] would derive the same conclusions only by labelling T as facts, but would not derive either conclusion if the first sentence were labelled as a default rule, as conventional.
Example 10. Let T = { bf , pb, ¬f , p¬f } (T classically consistent, obtained
by adding p¬f to T in example 9). As in example 9, T |=AL ¬bird(tweety) and
T |=AL ¬penguin(tweety). From a commonsense reasoning perspective, this is
counter-intuitive, as it disregards the newly added sentence and the alternative
possibility for ¬f ly(tweety) it supports, namely penguin(tweety).
By comparison, default logic with the first and last sentences in T labelled as
default rules (as conventional) would (sceptically) derive no conclusion as to
whether tweety is (or not) a bird or penguin. Arguably, this is too sceptical a
behaviour. Note that we have the same counter-intuitive behaviour of deriving
¬penguin(tweety) when the sentence ¬f ly(tweety) is deleted from the theory of
example 10. In order to accommodate within AL the intuitive kind of reasoning
pointed out for these examples, we can extend AL with priorities over sentences,
so that, in particular, exceptions may override rules, in the spirit of prioritised
default logic [
By introducing priorities we can also use preference-based argumentation, as in e.g. [14, 18], to distinguish between strengths of AL-entailment from subtheories, and, in particular, allow for stronger sub-theories to dominate, as illustrated by the following example: Example 11. Let T ={ bf , p, ¬f , ¬b¬p} (T is directly but not classically consistent). Then, correctly, in absence of other information, T 6|=AL bird(tweety) and T6|=AL¬bird(tweety). The sub-theories T1 = { bf , ¬f } and T2 = { p, ¬b¬p} AL-entail ¬bird(tweety) and bird(tweety) respectively and hence dispute each other. If we now take into account ¬b¬p > bf , then, under a preference-based argumentation approach, T2 would dominate T1 and thus T would correctly entail bird(tweety).
The core technical challenge of using priorities over sentences is to understand how these could influence the reasoning by contradiction a↵orded by RA in AL. In our illustrative setting we want the priorities (especially p¬f > bf ) to restrict the application of RA. There are other cases, however, where RA gives intuitive results and should not be restricted. For example, from the theory {bird(tweety), pb, p¬f , bf } with pb > p¬f > bf we expect that ¬penguin(tweety) is entailed since f ly(tweety) is an intuitive default conclusion of this theory and then, by RA, penguin(tweety) cannot be entailed (as otherwise through the stronger sentence of p¬f , the sentence ¬f ly(tweety) would follow). Similarly, given the theory {f ly(tweety), pb, p¬f , bf } with pb > p¬f > bf , we expect that ¬penguin(tweety) is entailed as penguin(tweety) would give ¬f ly(tweety) due to the higher strength of p¬f . To accommodate such cases it may be necessary to use the priority information more tightly within the definition of AL, i.e. within the definition of (non-)acceptability. 8
Related Work
AL is based on a notion of acceptability of arguments which is in the same
spirit as that in [
The link of argumentation to NMR has been the topic of extensive study for many years. Most of these studies either separate in the language the classical reasoning from the defeasible part of the theory (e.g. in Default Logic) or restrict the classical reasoning (e.g. in LP with NAF) or indeed as in the case of circumscription [17] the theory is that of classical logic but a complex prescription of model selection is imposed on top of the classical reasoning.
Recently, [
Other works that aim for a tighter link between classical and defeasible
reasoning include the work of Amgoud and Vesic [
Conclusion and Future Work We have presented Argumentation Logic (AL) and shown how it allows us to understand classical reasoning in Propositional Logic in terms of argumentation. Its definition rests on capturing semantically the Reductio ad Absurdum rule through a suitable notion of acceptability of arguments. One property of the ensuing AL is that the interpretation of implication is di↵erent from that of material implication. Further results on the relationship between AL and Propositional Logic including how AL can completely capture the entailment of PL are given in [12].
Given the significant role that argumentation has played in understanding under a common framework NMR in AI we have examined the problem of how we could unify classical reasoning and NMR within the framework of AL. In this context, we have considered the following questions: How could we use AL as the underlying logic to build a NMR framework? Can AL with its propositional language provide a single representation framework for classical and defeasible reasoning without any distinctions on the type of sentences allowed in a given theory? In particular, can we understand AL as a NMR framework with sentences that would behave as default rules but also as classical rules, with a form of contrapositive reasoning with these rules allowed? In this paper we have identified this problem and the challenges it poses, and studied these questions in the context of examples.
Our preliminary investigation suggests the need for an extension of AL to
accommodate preferences amongst sentences. Many existing frameworks for NMR
use, either explicitly or implicitly, preferences to capture defeasible reasoning,
e.g. [
A
Appendix: Natural Deduction We use the following rules, for any , , ^ I : , ^ E : ^ ^ E : ^ 2 L : _ I : ^ _ _ I : _ E : _ , d . . . ! I : d . . .
! c, d . . . c c ¬E : ¬¬
_ ¬I : d . . . ?c
¬ , ! E : ! where d⇣, . . . c is a (sub-)derivation with ⇣ referred to as the hypothesis. ¬I is also called Reduction ad Absurdum (RA). 10. Dung, P.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77, 321–357 (1995) 11. Kakas, A.C., Mancarella, P., Dung, P.M.: The acceptability semantics for logic programs. In: ICLP. pp. 504–519 (1994) 12. Kakas, A., Toni, F., Mancarella, P.: Argumentation logic. Tech. rep., Department of Computer Science, University of Cyprus, Cyprus (April 2012) 13. Kakas, A., Toni, F., Mancarella, P.: Argumentation for propositional logic and nonmonotonic reasoning. In: Working notes of the 11th International Symposium on Logical Formalizations of Commonsense Reasoning (2013) 14. Kakas, A.C., Moraitis, P.: Argumentation based decision making for autonomous agents. In: The Second International Joint Conference on Autonomous Agents & Multiagent Systems, AAMAS 2003, July 14-18, 2003, Melbourne, Victoria, Australia, Proceedings. pp. 883–890. ACM (2003) 15. Kowalski, R.A., Toni, F.: Abstract argumentation. Artificial Intelligence and Law 4(3–4), 275–296 (1996) 16. Lin, F., Shoham, Y.: Argument systems: A uniform basis for nonmonotonic reasoning. In: Brachman, R.J., Levesque, H.J., Reiter, R. (eds.) Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR’89). Toronto, Canada, May 15-18 1989. pp. 245–255. Morgan Kaufmann (1989) 17. McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artificial Intelligence 13(1-2), 27–39 (1980) 18. Modgil, S., Prakken, H.: A general account of argumentation with preferences.
Artificial Intelligence (2012), in Press 19. Reiter, R.: A logic for default reasoning. Artificial Intelligence 13(1-2), 81–132 (1980) 20. Wu, Y., Caminada, M.: A labelling-based justification status of arguments. Studies in Logic 3(4), 12–29 (2010) 21. Zhang, X., Zhang, Z., Xu, D., Lin, Z.: Argumentation-based reasoning with inconsistent knowledge bases. In: Canadian Conference on AI. pp. 87–99 (2010)