Constructing argument graphs with deductive arguments: a tutorial

3.1Arguments based on simple logic

Simple logic is based on a language of literals and simple rules where each simple rule is of the form α1∧⋯∧αk→β, where α₁ to α_k and β are literals. A simple logic knowledgebase is a set of literals and a set of simple rules. The consequence relation is modus ponens (i.e. implication elimination) as defined next.

So each simple argument is minimal, but not necessarily consistent (where consistency for a simple logic knowledgebase Δ means that for no atom α does Δ⊢sα and Δ⊢s¬α hold). We do not impose the consistency constraint in the definition for simple arguments as simple logic is paraconsistent, and therefore can support a credulous view on the arguments that can be generated.

Simple logic is a practical choice as a base logic for argumentation. Having a logic with simple rules and modus ponens is useful for applications because the behaviour is quite predictable in the sense that given a knowledgebase it is relatively easy to anticipate the inferences that come from the knowledgebase. Furthermore, it is relatively easy to implement an algorithm for generating the arguments and counterarguments from a knowledgebase. The downside of simple logic as a base logic is that the proof theory is weak. It only incorporates modus ponens (i.e. implication elimination) and so many useful kinds of reasoning (e.g. contrapositive reasoning) are not supported.

3.2Arguments based on classical logic

Classical logic is appealing as the choice of base logic as it better reflects the richer deductive reasoning often seen in arguments arising in discussions and debates.

We assume the usual propositional and predicate (first-order) languages for classical logic, and the usual the classical consequence relation, denoted ⊢. A classical knowledgebase is a set of classical propositional or predicate formulae.

So a classical argument satisfies both minimality and consistency. We impose the consistency constraint because we want to avoid the useless inferences that come with inconsistency in classical logic (such as via ex falso quodlibet).

Given the central role classical logic has played in philosophy, linguistics, and computer science (software engineering, formal methods, data and knowledge engineering, artificial intelligence, computational linguistics, etc.), we should consider how it can be used in argumentation. Classical propositional logic and classical predicate logic are expressive formalisms which capture deeper insights about the world than is possible with restricted formalisms such as simple logic.

4.1Counterarguments based on simple logic

For simple logic, we consider two forms of counterargument. For this, recall that literal α is the complement of literal β if and only if α is an atom and β is ≠gα or if β is an atom and α is ≠gβ.

So in simple logic, a rebut attacks the claim of an argument, and an undercut attacks the premises of the argument (either by attacking one of the literals or by attacking the consequent of one of the rules in the premises).

Simple arguments and counterarguments can be used to model defeasible reasoning. For this, we use simple rules that are normally correct but sometimes incorrect. For instance, if Sid has the goal of going to work, Sid takes the metro. This is generally true, but sometimes Sid works at home, and so it is no longer true that Sid takes the metro, as we see in the next example.

So by having appropriate conditions in the antecedent of a simple rule we can disable the rule by generating a counterargument that attacks it. This in effect stops the usage of the simple rule. This means that we have a convention to attack an argument based on the inferences obtained by the simple logic (e.g. as in Examples 4.2 and 4.3), or on the rules used (e.g. Example 4.4).

This way to disable rules by adding appropriate conditions (as in Example 4.4) is analogous to the use abnormality predicates used in formalisms such as circumscription (see, for example, McCarthy 1980). We can use the same approach to capture defeasible reasoning in other logics such as classical logic. Note, this does not mean that we turn the base logic into a nonmonotonic logic. Both simple logic and classical logic are monotonic logics. Hence, for a simple logic knowledgebase Δ (and similarly for a classical logic knowledgebase Δ), the set of simple arguments (respectively, classical arguments) obtained from Δ is a subset of the set of simple arguments (respectively, classical arguments) obtained from Δ∪{α}, where α is a formula not in Δ. But at the level of evaluating arguments and counterarguments, we have non-monotonic defeasible behaviour. For instance in Example 4.2, with just A₁ we have the acceptable claim useMetro, but then when we have also A₂, we have to withdraw this claim. In other words, if the set of simple arguments is {A₁}, then we can construct an argument graph with just A₁, and by applying Dung's dialectical semantics, there is one extension containing A₁. However, if the set of simple arguments is {A₁, A₂}, then we can construct an argument graph with A₁ attacked by A₂, and by applying Dung's dialectical semantics, there is one extension containing A₂. This illustrates the fact that the argumentation process is non-monotonic.

4.2Counterarguments based on classical logic

Given the expressivity of classical logic (in terms of language and inferences), there are a number of natural ways that we can define counterarguments.

To illustrate these different notions of classical counterargument, we consider the following examples, and we relate these definitions in Figure 3 where we show that classical defeaters are the most general of these definitions.

Figure 3.

We can represent the containment between the classical attack relations as above where an arrow from R₁ to R₂ indicates that R₁ ⊆ R₂. For proofs, see Besnard & Hunter (2001) and Gorogiannis & Hunter (2011).

Using simple logic, the definitions for counterarguments against the support of another argument are limited to attacking just one of the items in the support. In contrast, using classical logic, a counterargument can be against more than one item in the support. For example, in Example 4.7, the undercut is not attacking an individual premise but rather saying that two of the premises are incompatible (in this case that the premises lowCostFly and luxuryFly are incompatible).

Trivially, undercuts are defeaters but it is also quite simple to establish that rebuttals are defeaters. Furthermore, if an argument has defeaters then it has undercuts. It may happen that an argument has defeaters but no rebuttals as illustrated next.

There are some important differences between rebuttals and undercuts that can be seen in the following examples.

So an undercut for an argument need not be a rebuttal for that argument, and a rebuttal for an argument need not be an undercut for that argument.

Arguments are not necessarily independent. In a sense, some encompass others (possibly up to some form of equivalence), which is the topic we now turn to.

Roughly speaking, a more conservative argument is more general: it is, so to speak, less demanding on the support and less specific about the claim.

We can use the notion of “more conservative” to help us identify the most useful counterarguments amongst the potentially large number of counterarguments.

So choosing classical logic as the base logic gives us a wider range of choices for defining attacks. This has advantages if we want to better capture argumentation as arising in natural language, or to more precisely capture counterarguments generated from certain kinds of knowledge. However, it does also mean that we need to be aware of the consequences of our choice of definition for attacks when using a generative approach to instantiating argument graphs (as we will discuss in the next section).

5.1Descriptive graphs

For the descriptive approach to argument graphs, we assume that we have some abstract argument graph as the input, together with some informal description of each argument. For instance, when we listen to a debate on the radio, we may identify a number of arguments and counterarguments, and for each of these we may be able to write a brief text summary. So if we then want to understand this argumentation in more detail, we may choose to instantiate each argument with a deductive argument. So for this task we choose the appropriate logical formulae for the premises and claim for each argument (compatible with the choice of base logic). Examples of descriptive graphs are given in Figure 4 using simple logic, and in Example 5.1 and Figure 5 using classical logic.

Figure 4.

A descriptive graph representation of the abstract argument graph in Figure 1 using simple logic. The atom bp(high) denotes that the patient has high blood pressure. Each attack is a simple undercut by one argument on another. For the first argument, the premises include the assumptions ok(diuretic) and≠g give(betablocker) in order to apply its simple rule. Similarly, for the second argument, the premises include the assumptions ok(betablocker) and≠g give(diuretic) in order to apply its simple rule.

Figure 5.

A descriptive graph representation of the abstract argument graph in Figure 1 using classical logic. The atom bp(high) denotes that the patient has high blood pressure. The top two arguments rebut each other (i.e. the attack is classical defeating rebut). For this, each argument has an integrity constraint in the premises that says that it is not ok to give both beta blocker and diuretic. So the top argument is attacked on the premise ok(diuretic) and the middle argument is attacked on the premise ok(betablocker). So we are using the ok predicate as an normality condition for the rule to be applied (as suggested in Section 4.1).

So for the approach of descriptive graphs, we do not assume that there is an automated process that constructs the graphs. Rather the emphasis is on having a formalisation that is a good representation of the argumentation. This is so that we can formally analyse the descriptive graph, perhaps as part of some sense-making or decision-making process. Nonetheless, we can envisage that in the medium term natural language processing technology will be able to parse the text or speech (for instance in a discussion paper or in a debate) in order to automatically identify the premises and claim of each argument and counterargument.

Since we are primarily interested in representational and analytical issues when we use descriptive graphs, a richer logic such as classical logic is a more appealing formalism than a weaker base logic such as simple logic. Given a set of real-world arguments, it is often easier to model them using deductive arguments with classical logic as the base logic than a “rule-based logic” like simple logic as the base logic. For instance, in Example 5.1, the undercut does not claim that the flight is not low cost, and it does not claim that it is not luxury. It only claims that the flight cannot be both low cost and luxury. It is natural to represent this exclusion using disjunction. As another example of the utility of classical logic as base logic, consider the importance of quantifiers in knowledge which require a richer language such as classical logic for reasoning with them. Moreover, if we consider that many arguments are presented in natural language (spoken or written), and that formalising natural language often calls for a richer formalism such as classical logic (or even richer), it is valuable to harness classical logic in formalisations of deductive argumentation.

5.2Generative graphs based on simple logic

Given a knowledgebase Δ, we can generate an argument graph G=(A,R), where A is the set of simple arguments obtained from Δ as follows and R⊆A×A is simple undercut.

This is an exhaustive approach to constructing an argument graph from a knowledgebase since all the simple arguments and all the simple undercuts are in the argument graph. We give an example of such an argument graph in Figure 6.

Figure 6.

An exhaustive simple logic argument graph where Δ={a, b, c, a ∧ c→≠g a, b→≠g c, a ∧ c→≠g b}. Note that this exhaustive graph contains a self-cycle and an odd length cycle.

Simple logic has the property that for any argument graph, there is a knowledgebase that can be used to generate it: let (N, E) be a directed graph (i.e. N is a set of nodes, and E is a set of edges between nodes), then there is a simple logic knowledgebase Δ such that (Argumentss(Δ),Attackss(Δ)) is isomorphic to (N, E). So simple exhaustive graphs are said to be constructively complete for graphs.

To show that simple exhaustive graphs are constructively complete for graphs, we can use a coding scheme for the premises so that each argument is based on a single simple rule where the antecedent is a conjunction of one or more positive literals, and each consequent is a negative literal unique to that simple rule (i.e. it is an identifier for that rule and therefore for that argument). If we want one argument to attack another, and the attacking argument has the consequent ≠gα, then the attacked argument needs to have the positive literal α in the antecedent of its simple rule. The restriction of each rule to only have positive literals as conditions in the antecedent, and a negative literal as its consequent, means that the rules cannot be chained. This ensures that the premises of each argument has only one simple rule. We illustrate this in Figure 6.

Simple exhaustive graphs provide a direct and useful way to instantiate argument graphs. There are various ways the definitions can be adapted, such as defining the attacks to be the union of the simple undercuts and the simple rebuts.

5.3Generative graphs based on classical logic

In this section, we consider generative graphs for classical logic. We start with the classical exhaustive graphs which are the same as the simple exhaustive graphs except we use classical arguments and attacks. We show that whilst this provides a comprehensive presentation of the information, its utility is limited for various reasons. We then show that by introducing further information, we can address these shortcomings. To illustrate this, we consider a version of classical exhaustive graphs augmented with preference information. This is just one possibility for introducing extra information into the construction process.

5.3.1Classical exhaustive graphs

Given a knowledgebase Δ, we can generate an argument graph G=(A,R), where A is the set of arguments obtained from Δ as follows and R⊆A×A is one of the definitions for classical attack.

Definition 5.3

Let Δ be a classical logic knowledgebase. A classical exhaustive graph is an argument graph G=(A,R), where A is Argumentsc(Δ) and R is AttackscX(Δ)) defined as follows where X is one of the attacks given in Definition 4.5 such as defeater, direct undercut, or rebuttal.

Argumentsc(Δ)={⟨Φ,α⟩∣Φ⊆Δand⟨Φ,α⟩is a classical argument}AttackscX(Δ)={(A,B)∣A,B∈Argumentsc(Δ)×Argumentsc(Δ)∣AisXofB}

This is a straightforward approach to constructing an argument graph from a knowledgebase since all the classical arguments and all the attacks (according to the chosen definition of attack) are in the argument graph as illustrated in Figure 7. However, if we use this exhaustive definition, we obtain infinite graphs, even if we use a knowledgebase with few formulae. For instance, if we have an argument ⟨{a},a⟩, we also have arguments such as ⟨{a},a∨a⟩, ⟨{a},a∨a∨a⟩, etc., as well as ⟨{a},¬¬a⟩, etc.

Figure 7.

An exhaustive classical logic argument graph where Δ={a, b,≠g a ∨≠g b} and the attack is direct undercut. Note argument A₄ represents all arguments with a claim that is implied by b, argument A₅ represents all arguments with a claim that is implied by a, argument A₆ represents all arguments with a claim that is implied by≠g a ∨≠g b, argument A₇ represents all arguments with a claim that is implied by a ∧≠g b except≠g b or any claim implied by a or any claim implied by≠g a ∨≠g b, argument A₈ represents all arguments with a claim that is implied by≠g a ∧ b except≠g a or any claim implied by b or any claim implied by≠g a ∨≠g b, and argument A₉ represents all arguments with a claim that is implied by a ∧ b except≠g (≠g a ∨≠g b) or any claim implied by a or any claim implied by b.

Even though the graph is infinite, we can present a finite representation of it, by just presenting a representative of each class of structurally equivalent arguments (as considered in Amgoud,Besnard, & Vesic 2011), where we say that two arguments A_i and A_j are structurally equivalent in G=(A,R) when the following conditions are satisfied: (1) if A_k attacks A_i, then A_k attacks A_j; (2) if A_k attacks A_j, then A_k attacks A_i; (3) if A_i attacks A_k, then A_j attacks A_k; and (4) if A_i attacks A_k, then A_j attacks A_k. For example, in Figure 7, the argument A₄ is a representative for ⟨{b},b⟩, ⟨{b},a∨b⟩, ⟨{b},¬a∨b⟩, etc.

We can also ameliorate the complexity of classical exhaustive graphs by presenting a focal graph (as discussed in Section 2). We illustrate this in the following example.

Example 5.4

Consider the knowledgebase Δ={a,b,¬a∨¬b} as used in Figure 7. Suppose we have the focus Π={A1,A2} (i.e. the arguments of interest), then the focal graph is as follows.

Various postulates have been proposed for classical exhaustive graphs (e.g. Gorogiannis &Hunter 2011). Some of these are concerned with consistency of the set of premises (or set of claims) obtained from the arguments in an extension according to one of Dung's dialectical semantics. We give a useful example of one of these consistency postulates (taken from Gorogiannis & Hunter2011): for a classical exhaustive graph G, the consistent extension property is defined as follows where ExtensionS(G) is the set of extensions obtained from the argument graph G according to the semantics S (e.g. grounded, preferred, etc.).

⋃A∈ΓSupport(A)⊭⊥,for allΓ∈ExtensionS(G)

What has been found is that for some choices of attack relation (e.g. rebuttal), the consistent extension property is not guaranteed (as in Example 5.5), whereas for other choices of attack relation (e.g. direct undercut), the consistent extension property is guaranteed. We illustrate a consistent set of premises obtained from arguments in a preferred extension in Example 5.6.

Example 5.5

Let Δ={a∧b,¬a∧c}. For the reviewed semantics for rebut, the following are arguments in any extension: A1=⟨{a∧b},b⟩ and A2=⟨{¬a∧c},c⟩. Clearly {a∧b,¬a∧c}⊢⊥. Hence, the consistent extension property fails for rebut.

Example 5.6

Consider the argument graph given in Figure 7. There are three preferred extensions {A1,A5,A6,A7}, {A2,A4,A6,A8}, and {A3,A4,A5,A9}. In each case, the union of the premises is consistent. For instance, for the first extension,

Support(A1)∪Support(A5)∪Support(A6)∪Support(A7)⊬⊥

The failure of the consistent extension property with some attack relations is an issue that may be interpreted as a weakness of the attack relation or of the specific semantics, and perhaps raises the need for alternatives to be identified. Another response is that it is not the attack relation and dialectical semantics that should be responsible for ensuring that all the premises used in the winning arguments are consistent together. Rather, it could be argued that checking that the premises used are consistent together should be the responsibility of something external to the defeat relation and dialectical semantics, and so knowing whether the consistent extension property holds or not influences what external mechanisms are required for checking. Furthermore, checking consistency of premises of sets of arguments may be part of the graph construction process. For instance, in García and Simari (2004) proposal for dialectical trees, there are constraints on what arguments can be added to the tree based on consistency with the premises of other arguments in the tree.

In the previous section, we saw that the definition for simple exhaustive graphs (i.e. simple logic as the base logic, with simple undercuts) is constructively complete for graphs. In contrast, the definition for classical exhaustive graphs (i.e. classical logic, with any of the definitions for counterarguments) is not constructively complete for graphs. Since the premises of a classical argument are consistent, by definition, it is not possible for a classical argument to attack itself using the definitions for attack given earlier. But, there are many other graphs for which there is no classical logic knowledgebase that can be used to generate a classical exhaustive graph that is isomorphic to it. To illustrate this failure, we consider in Example 5.7 the problem of constructing a component with two arguments attacking each other. Note this is not a pathological example as there are many graphs that contain a small number of nodes and that cannot be generated as a classical exhaustive graph.

Example 5.7

Let Δ={a,¬a} be a classical logic knowledgebase. Hence, there are two classical arguments A₁ and A₂ that are direct undercuts of each other. Plus, there is the representative A₃ for arguments with a claim that is strictly weaker than a (i.e. the claim b is such that {a}⊢b and {b}⊬{a}), and there is the representative A₄ for arguments with a claim that is strictly weaker than ≠g a (i.e. the claim b is such that {¬a}⊢b and {b}⊬{¬a}).

To conclude our discussion of classical exhaustive graphs, the definitions ensure that all the ways that the knowledge can be used to generate classical arguments and classical counterarguments (modulo the choice of attack relation) are laid out. However, for some choices of attack relation, there is a question of consistency (which may be an issue if no further consistency checking is undertaken). Also, the definition for classical exhaustive graphs is not structurally complete for graphs (which means that many argument graphs cannot be generated as classical exhaustive graphs). Perhaps more problematical is that even for small knowledgebases, the classical exhaustive graphs that are generated are complex. Because of the richness of classical logic, the knowledge can be in different combinations to create many arguments. Whilst we can ameliorate this problem by presenting argument graphs using a representative for a class of structurally equivalent arguments, and by using focal graphs, the graphs can still be large. What is evident from this is that there needs to be more selectivity in the process of generating argument graphs. The generation process needs to discriminate between the arguments (and/or the attacks) based on extra information about the arguments and/or information about the audience. There are many ways that this can be done. In the next section, we consider a simple proposal for augmenting the generation process with preferences over arguments.

5.3.2Preferential exhaustive graphs

One of the first proposals for capturing the idea of preferences in constructing argument graphs was preference-based argumentation frameworks (PAF) by Amgoud & Cayrol (2002). This generalises Dung's definition for an argument graph by introducing a preference relation over arguments that in effect causes an attack to be ignored when the attacked argument is preferred over the attacker. So in PAF, we assume a preference relation over arguments, denoted ⪯, as well as a set of arguments A and an attack relation R. From this, we need to define a defeat relation D as follows, and then (A,D) is used as the argument graph, instead of (A,R), with Dung's usual definitions for extensions.

D={(Ai,Aj)∈R∣(Aj,Ai)∉⪯}

So with this definition for defeat, extensions for a preference-based argument graph (A,R,⪯) can be obtained as follows: for S denoting complete, preferred, stable, or grounded semantics, Γ⊆A, Γ is an extension of (A,R,⪯) w.r.t. semantics S iff Γ is an extension of (A,D) w.r.t. semantics S.

We now revise the definition for classical exhaustive graphs to give the following definition for preferential exhaustive graphs.

Definition 5.8

Let Δ be a classical logic knowledgebase. A preferential exhaustive graph is an argument graph (Argumentsc(Δ),Attacksc,⪯X(Δ)) defined as follows where X is one of the attacks given in Definition 4.5 such as defeater, direct undercut, or rebuttal.

Argumentsc(Δ)={⟨Φ,α⟩∣Φ⊆Δ&⟨Φ,α⟩is a classical argument}Attacksc,⪯X(Δ)={(A,B)∣A,B∈Argumentsc(Δ)&AisXofB&(B,A)∉⪯}

We give an illustration of a preferential exhaustive graph in Figure 8, and we give an illustration of a focal graph obtained from a preferential exhaustive graph in Example 5.9.

Figure 8.

The preferential exhaustive graph where the knowledgebase is Δ={a, b,≠g a ∨≠g b} and the attack is direct undercut. This is the same knowledgebase and attack relation as in Figure 7. For the preference relation, A₁ is preferred to all other arguments, A₂ is preferred to all other arguments apart from A₁, and the remaining arguments are equally preferred. So for all i such that i≠1, A₁ ≺ A_i, and for all i such that i≠1 and i≠2, A₂ ≺ A_i.

Example 5.9

This example concerns two possible treatments for glaucoma caused by raised pressure in the eye. The first is a prostaglandin analogue (PGA) and the second is a beta blocker (BB). Let Δ contain the following six formulae. The atom p₁ is the fact that the patient has glaucoma, the atom p₂ is the assumption that it is ok to give PGA, and the atom p₃ is the assumption that it is ok to give BB. Each implicational formula (i.e. p₄ and p₅) captures the knowledge that if a patient has glaucoma, and it is ok to give a particular drug, then give that drug. Formula p₆ is an integrity constraint that ensures that only one treatment is given for the condition.

p1=glaucomap4=glaucoma∧ok(PGA)→give(PGA)p2=ok(PGA)p5=glaucoma∧ok(BB)→give(BB)p3=ok(BB)p6=¬ok(PGA)∨¬ok(BB)

There are numerous arguments that can be constructed from this set of formulae such as the following.

A1=⟨{p1,p2,p4,p6},give(PGA)∧¬ok(BB)⟩A5=⟨{p1,p2,p4},give(PGA)⟩A2=⟨{p1,p3,p5,p6},give(BB)∧¬ok(PGA)⟩A6=⟨{p1,p3,p5},give(BB)⟩A3=⟨{p2,p3},ok(PGA)∧ok(BB)⟩A7=⟨{p2,p6},¬ok(BB)⟩A4=⟨{p6},¬ok(PGA)∨¬ok(BB)⟩A8=⟨{p3,p6},¬ok(PGA)⟩

Let Argumentsc(Δ) be the set of all classical arguments that can be constructed from Δ, and let the preference relation ⪯ be such that Ai⪯A1 and Ai⪯A2 for all i such that i≠1 and i≠2. Furthermore, let Π={A1,A2} be the focus (i.e. the arguments of interest). In other words, we know that each of these two arguments in the focus contains all the information we are interested in (i.e. we want to determine the options for treatment taking into account the integrity constraint). This would give us the following focal graph.

By taking this focal graph, we have ignored arguments such as A₃ to A₈ which do not affect the dialectical status of A₁ or A₂ given this preference relation.

Using preferences is a general approach. There is no restriction on what preference relation we use over arguments, and there are various natural interpretations for this ranking such as capturing belief for arguments (where the belief in the argument can be based on the belief for the premises and/or claim), capturing the relative number of votes for arguments (where a group of voters will vote for or against each argument), etc.

To conclude, by introducing preferences over arguments, we can reduce the number of attacks that occur. Using preferences over arguments is a form of meta-information, and with the definition for preference-based argumentation (as defined by Amgoud & Cayrol 2002), it supports selectivity in generating argument graphs that discriminates between arguments and thereby between attacks. With this definition more practical argument graphs can be constructed than with the definition for classical exhaustive graphs. Furthermore, using an appropriate definition for the preference relation, the definition for preference exhaustive graphs is structurally complete for graphs, and for some choices of preference relation and dialectical semantics, the consistent extension property holds (see, for example, the use of probability theory for obtaining preferences over arguments by Hunter 2013).

6.1Assumption-based argumentation

This is a general framework for defining logic-based argumentation systems. As with deductive argumentation, a language needs to be specified for representing the information. Rule-based languages and classical logic are possibilities. In addition, a set of inference rules needs to be specified. This corresponds to a base logic consequence relation augmented with a set of domain-specific inference rules. A subset of the language is specified as the set of assumptions, and each argument is a subset of these assumptions. The assumptions in an argument can be viewed as the premises of the argument, with the claim being implicit. The approach incorporates a generalised notion of negation that specifies which individual formulae contradict an assumption, and a form of undercut is defined using this notion. Given a knowledgebase, all the arguments and counterarguments can be generated, and the attacks relation identified, thereby producing an “exhaustive graph". However, a significant feature of the approach is a set of proof procedures for determining whether there exists an argument with a given claim in a preferred, grounded, or ideal extension. Many of the examples used to illustrate and develop assumption-based argumentation (ABA) are a simple rule-based language (analogous to simple logic for deductive argumentation). However, the approach is general and can use a wide variety of underlying logics including classical logic (though the same issues as raised in Section 5.3.1 would arise for ABA).

6.2ASPIC+

This is a general framework for defining structured argumentation systems. As with deductive argumentation and ABA, a language needs to be specified for representing the information. Rule-based languages and classical logic are possibilities. As with ABA, a set of inference rules needs to be specified. Inference rules can be domain specific. This contrasts with deductive argumentation where inference rules are specified by the base logic. An argument in ASPIC+ contains the structure of the proof used to obtain the claim from the premises, and so the same pair of premises and claim can potentially result in many arguments. The defeasibility of an argument based on a defeasible rule can come from a counterargument that attacks the use of defeasible rule. This is done via a naming mechanism for the rules, so an attacker of a defeasible rule has a claim that is the contrary (negation) of the name of that rule. The defeasibility of a defeasible rule can also come from counterarguments that either rebut or undercut (called undermine in ASPIC+) an argument. ASPIC+ incorporates a preference relation over arguments which allows for some attacks to be ignored. Given a knowledgebase, all the arguments and counterarguments can be generated, and the attacks relation identified, thereby producing an “exhaustive graph". A main objective in the development of ASPIC+ is to identify conditions under which instantiations of ASPIC+ satisfy logical consistency and closure properties.

6.3Defeasible logic programming

This is an approach to argumentation based on a logic programming language. Each argument is generated from the knowledgebase, which contains defeasible rules and strict rules, and these are used to form a tree of arguments and counterarguments: an argument for a claim of interest is used for the root, and then counterarguments to this root are children to this root, and then by recursion for each counterargument, the counterarguments to it are given as its children. The construction process imposes some consistency constraints and ensures that no branch is infinite. This tree provides an exploration of the possible reasons for and against the root of the tree being a warranted argument. To compare the approach with that of deductive argumentation, each argument in defeasible logic programming (DeLP) can be viewed as a deductive argument where the base logic is a form of logic programming consequence relation, and each counterargument can be seen as a form of undercut, but the approach does not involve instantiating abstract argument graphs. A number of variants of DeLP have been proposed incorporating features such as variable strength arguments (Martinez, Garcia, & Simari 2008) and possibility theory (Alsinet, Chesñevar,Godo, & Simari 2008).

7.1Deductive arguments and counterarguments

There have been a number of proposals for deductive arguments using classical propositional logic (Amgoud & Cayrol 2002, Besnard & Hunter 2001, Cayrol 1995, Gorogiannis & Hunter 2011), classical predicate logic (Besnard & Hunter 2005), description logic (Black, Hunter, & Pan 2009; Moguillansky, Wassermann, & Falappa 2010; Zhang & Lin 2013; Zhang, Zhang, Xu, & Lin 2010), temporal logic (Mann & Hunter 2008), simple (defeasible) logic (Governatori, Maher, Antoniou, &Billington 2004; Hunter 2010), conditional logic (Besnard, Gregoire, & Raddaoui 2013), and probabilistic logic (Haenni 1998, Haenni 2001, Hunter 2013).

There has also been progress in understanding the nature of classical logic in computational models of argument. Various types of counterarguments have been proposed including rebuttals (Pollock 1987, Pollock 1992), direct undercuts (Cayrol 1995; Elvang-Gøransson &Hunter 1995; Elvang-Gøransson, Krause, & Fox 1993), and undercuts and canonical undercuts (Besnard & Hunter 2001). In most proposals for deductive argumentation, an argument A is a counterargument to an argument B when the claim of A is inconsistent with the support of B. It is possible to generalise this with alternative notions of counterargument. For instance, with some common description logics, there is not an explicit negation symbol. In the proposal for argumentation with description logics, Black et al. (2009) used the description logic notion of incoherence to define the notion of counterargument: a set of formulae in a description logic is incoherent when there is no set of assertions (i.e. ground literals) that would be consistent with the formulae. Using this, an argument A is a counterargument to an argument B when the claim of A together with the support of B is incoherent.

Meta-arguments for deductive argumentation was first proposed by Wooldridge, McBurney, &Parsons (2005), and the investigation of the representation of argument schemes in deductive argumentation was first proposed by Hunter (2008).

7.2Properties of exhaustive argument graphs

In order to investigate how Dung's notion of abstract argumentation can be instantiated with classical logic, Cayrol (1995) presents results concerning stable extensions of argument graphs where the nodes are classical logic arguments, and the attacks are direct undercuts. As well as being the first paper to propose instantiating abstract argument graphs with classical arguments, it also showed how the premises in the arguments in the stable extension correspond to maximal consistent subsets of the knowledgebase, when the attack relation is direct undercut.

Insights into the options for instantiating abstract argumentation with classical logic can be based on postulates. Amgoud & Besnard (2009) have proposed a consistency condition and they examine special cases of knowledgebases and symmetric attack relations and whether consistency is satisfied in this context. Then Amgoud & Besnard (2010) extend this analysis by showing correspondences between the maximal consistent subsets of a knowledgebase and the maximal conflict-free sets of arguments.

Given the wide range of options for attack in classical logic, Gorogiannis & Hunter (2011) propose a series of desirable properties of attack relations to classify and characterise attack relations for classical logic. Furthermore, they present postulates regarding the logical content of extensions of argument graphs that may be constructed with classical logic, and a systematic study is presented of the status of these postulates in the context of the various combinations of attack relations and extension semantics.

Use of the notion of generative graphs then raises the question of whether for a specific logical argument system S, and for any graph G, there is a knowledgebase such that S generates G. If it holds, then it can be described as a kind of “structural" property of the system (Hunter & Woltran2013). If it fails then, it means that there are situations that cannot be captured by the system. The approach of simple exhaustive graphs is constructively complete for graphs, whereas the approach of classical exhaustive graphs is not.

Preferences have been introduced into classical logic argumentation and used to instantiate abstract argumentation with preferences by Amgoud & Cayrol (2002). Amgoud and Vesic have shown how preferences can be defined so as to equate inconsistency handling in argumentation with inconsistency handling using Brewka's preferred sub-theories (Amgoud & Vesic 2010).

7.3Importance of selectivity in deductive argumentation

Some of the issues raised with classical exhaustive graphs (i.e. the lack of structural completeness, the failure of consistent extension property for some choices of attack relation, and the correspondences with maximally consistent subsets of the knowledgebase) suggest that often we need a more sophisticated way of constructing argument graphs. In other words, to reflect any abstract argument graph in a logical argument system based on a richer logic, we need to be selective in the choice of arguments and counterarguments from those that can be generated from the knowledgebase. Furthermore, this is not just for theoretical interest. Practical argumentation often seems to use richer logics such as classical logic, and often the arguments and counterarguments considered are not exhaustive. Therefore, we need to better understand how the arguments are selected. For example, suppose agent 1 posits A1=⟨{b,b→a},a⟩, and agent 2 then posits A2=⟨{c,c→¬b},¬b⟩. It would be reasonable for this dialogue to stop at this point even though there are further arguments that can be constructed from the public knowledge such as A3=⟨{b,c→¬b},¬c⟩. So in terms of constructing the constellation of arguments and counterarguments from the knowledge, we need to know what are the underlying principles for selecting arguments.

Selectivity in argumentation is an important and as yet under-developed topic (Besnard & Hunter 2008). Two key dimensions are selectivity based on object-level information and selectivity based on meta-level information.

Various kinds of meta-level information can be considered for argumentation including preferences over arguments, weights on arguments, weights on attacks, a probability distribution over models of the language of the deductive argumentation, etc. The need for meta-level information also calls for better modelling of the audience, of what they believe, of what they regard as important for their own goals, etc., is an important feature of selectivity (see, for example, (Hunter2004a, 2004b), Hunter 2004b). Consider a journalist writing a magazine article on current affairs. There are many arguments and counterarguments that could be included, but the writer is selective. Selectivity may be based on what the likely reader already believes and what they may find interesting. Or, consider a lawyer in court, again there may be many arguments and counterarguments that could be used, but only some will be used. Selection will in part be based on what could be believed by the jury and convince them to take the side of that lawyer. Or, consider a politician giving a speech to an audience of potential voters. Here, the politician will select arguments based on what will be of more interest to the audience. For instance, if the audience is composed of older citizens, there may be more arguments concerning healthcare, whereas if the audience is composed of younger citizens, there may be more arguments concerning job opportunities. So whilst selectivity is clearly important in real-world argumentation, we need principled ways of bring selectivity into structured argumentation such as that based on deductive argumentation.

7.4Automated reasoning for deductive argumentation

For argumentation, it is computationally challenging to generate arguments from a knowledgebase with the minimality constraints using classical logic. If we consider the problem as an abduction problem, where we seek the existence of a minimal subset of a set of formulae that implies the consequent, then the problem is in the second level of the polynomial hierarchy (Eiter &Gottlob 1995). The difficult nature of argumentation has been underlined by studies concerning the complexity of finding individual arguments Parsons, Wooldridge, & Amgoud (2003), the complexity of some decision problems concerning the instantiation of argument graphs with classical logic arguments and the direct undercut attack relation Wooldridge, Dunne, & Parsons(2006), and the complexity of finding argument trees Hirsch & Gorogiannis (2009). Encodation of these tasks as quantified Boolean formulae also indicates that development of algorithms is a difficult challenge (Besnard, Hunter, & Woltran 2009), and Post's framework has been used to give a breakdown of where complexity lies in logic-based argumentation (Creignou, Schmidt, Thomas,& Woltran 2011).

Despite the computational complexity results, there has been progress in developing algorithms for constructing arguments and counterarguments. One approach has been to adapt the idea of connection graphs to enable us to find arguments. A connection graph (Kowalski 1975), (Kowalski1979) is a graph where a clause is represented by a node and an arc (φ, ψ) denotes that there is a disjunct in φ with its complement being a disjunct in ψ. Essentially this graph is manipulated to obtain a proof by contradiction. Furthermore, finding this set of formulae can substantially reduce the number of formulae that need to be considered for finding proofs for a claim, and therefore for finding arguments and canonical undercuts. Versions for full propositional logic, and for a subset of first-order logic, have been developed and implemented (Efstathiou & Hunter 2011).

Another approach for algorithms for generating arguments and counterarguments (canonical undercuts) have been given in a proposal that is based on an SAT solver (Besnard, Gregoire, Piette,& Raddaoui 2010). This approach is based on standard SAT technology and it is also based on finding proofs by contradiction.