The Attack as Strong Negation, Part I

The Attack as Strong Negation, Part I

D. Gabbay
King’s College London,
Department of Informatics,
Strand, London, WC2R 2LS, UK;
Bar Ilan University, Ramat Gan, Israel
and
University of Luxembourg, Luxembourg.
dov.gabbay@kcl.ac.uk
M. Gabbay
Cambridge University, UK.
mg639@cam.ac.uk
Submitted to Logic Journal of the IGPL 1.4.15. Revised 1.6.15. Paper 540
Compiled on July 19, 2019
Abstract

We add strong negation to classical logic and interpret the attack relation of “ attacks ” in argumentation as . We write a corresponding object level (using only) classical theory for each argumentation network and show that the classical models of this theory correspond exactly to the complete extensions of the argumentation network. We show by example how this approach simplifies the study of abstract argumentation networks. We compare with other translations of abstract argumentation networks into logic, such as classical predicate logic or modal logics, or logic programming, and we also compare with Abstract Dialectical Frameworks.

1 Background from classical logic: non-logical axiomatic theories

This paper introduces a particularly intuitive and simple representation/translation of abstract argumentation networks into classical propositional logic. All we need is a simple version of strong negation. So our starting point must be to introduce this negation.

Classical propositional logic can be properly axiomatised in many ways. For simplicity, let us take the set of all tautologies as axioms and the rule of modus ponens as the deduction rule. Let us assume that the connectives used are the usual ones and the atomic sentences are the set . Classical logic is strongly complete for the classical semantics. Models are assignments giving values in to the atoms of .

We have satisfaction defined as follows for wffs and theories .

• iff , for

• iff

• iff and

• iff or

• iff implies .

• iff for all .

The notion of proof can be defined in classical logic in many ways and completeness holds for any set of wffs and any :

• iff for all models we have implies .

We now introduce the notion of a set of sentences being a specific non-logical axiomatic theory .

Consider again the set of atomic wff

 P={p1,p2,p3,…}.

Suppose we insist, for our own reasons, that we want to consider only those models satisfying the restriction below:

• For all even index atoms we have implies

There is a theory of classical propositional logic whose models are exactly all the models satisfying . This theory is

 Θn={p2→¬p1,p4→¬p3,p6→¬p5,…}

Let us now for the sake of clarity, rename the atoms of with the help of a new symbol . We rename as follows:

 q1=def. p1Nq1=def. p2q2=def. p3Nq3=def. p4    ⋮qi=def. p2i−1Nqi=def. p2i   ⋮

We can thus write the set of atoms as as the set

 Q={q1,Nq1,q2,Nq2,…}.

The theory becomes the theory

 Θn={Nq→¬q|q∈{q1,q2,…}}

is considered a non-logical set of axioms on the symbol .

Note that cannot be iterated and can be applied only to atoms taken from .

We said that we regard as a non-logical axiomatic theory on the symbol . This is common practice in logic and model theory. Consider, for example, the classical theory of Abelian groups formalised in classical logic for the multiplication symbol and the constant . We add to the logical axioms of predicate logic the non-logical group axioms below

In our case our non-logical axioms are .

Thus our logic for is a theory (of ) within classical propositional logic, much in the same way as the theory of Abelian groups is a theory within the classical predicate calculus. The next section defines the logic CN formally.

2 The logic Cn

We add to classical propositional logic the strong negation symbol . We can thus form atomic sentences like as well as atoms of the form . We do this as explained in Section 1.

At this point we do not allow iterations of . We have the usual connectives (negation), . We shall discuss iterated use of in a later section. We require the only additional non-logical axioms . We can thus view as strong negation. For example might be true or might be false (i.e.  is true) or might be strongly false ( true) or might be false but not strongly false ( and true).

Let us define our logic directly.

Definition 2.1 (The Cn classical propositional logic with atomic strong negation)
1. {540-D1}

2. Let be a language with a set of atoms and the connectives . are the usual classical connectives and is a unary connective applied once to atoms only.

3. An atomic formula has the form or where .

4. A wff has the form atomic formula or where and are wffs.

5. We regard as axioms for our logic CN the nonlogical set of axioms as discussed in Section 1. The proof theory CN is relying on the proof theory of classical logic C, as follows:

• iff (def) .

6. A model for the logic CN is an assignment giving each atomic wff of the form or a value in such that if then .

7. Satisfaction is defined in the usual way.

Theorem 2.2

CN is complete for the proposed semantics.

• Proof. Our discussion in Section 1 presented CN as a non-logical theory of the classical propositional calculus of item 4 of Definition 2.1 defined the consequence for CN, via the classical consequence. Since we have strong completeness for C we also have it for CN.

Remark 2.3

Note that what we are calling our ‘logic’ CN is actually a theory in a two sorted classical propositional logic with two sorts of atoms of the form and . In Appendix Appendix: Formal definition of the modal logic we will turn our logic into a proper modal logic, which we will call .

We now have enough machinery to faithfully represent abstract argumentation networks in the logic CN. This is the job of the next section.

3 Expressing argumentation networks in Cn

This section will show a simple way of translating formal argumentation networks into CN. We assume we are dealing with finite argumentation networks.111Actually, we do not need the requirement that the network is finite. All we need is that it is finitary, namely that each point is attacked by a finite number of attackers. This will allow us to write classical wffs describing the attacks.

{540-S3}

We shall use the Caminada labelling characterisation of complete extensions. See [1] for a survey. Given an argumentation network , with and , a legitimate Caminada labelling on is a function giving values “in”, “out”, “undecided” to each , satisfying the following properties.

• in iff either or out).

• out iff in)

• und iff in) and und).

Each legitimate Caminada labelling gives rise to a unique complete extension and all complete extensions can be obtained in this way. See [1].

Definition 3.1

Let be a formal argumentation network which is finitary, i.e. each point has a finite number of attackers This means that and is the attack relation. Define a theory of the logic CN as follows

1. {540-D3}

2. We can assume that (i.e. the arguments of are identified as atoms of the logic).

3. , where Attack.

Theorem 3.2 (First Correspondence Theorem)

Let . Then the models of correspond exactly to the complete extensions of .

• Proof.

1. Let be a model of . We show it defines a complete extension on (note that two different models can yield the same complete extension): We use the Caminada labelling function. Let . Define

 λh(x)=⎧⎪⎨⎪⎩in, if h(x)=1out, if h(Nx)=1und, if h(x)=h(Nx)=0

we prove the following

1. is well defined. For each we can have exactly one of the three cases mentioned in the definition of . The reason for that is that we have the axiom and so the case and does not arise.

2. The crucial points to show are

1. If is not attacked then in. This holds because .

2. If and in then . Therefore (because is in the theory) and so out.

3. Suppose for all such that we have out. We want to show the in. We have for all such that that hence and therefore (because of the theory), and so in.

4. Suppose und. Then . Let . We cannot have in because then and this implies . Thus none of the attackers of are in. Thus for all such that we have . Can we have that for all of such ? If this were the case, since is in we get i.e.  in, contradicting our assumption that und.

Therefore for some such that we have . But this means that und. We thus got that if und then none of the attackers of are in and at least one of them is undecided.

5. If out, (i.e. ), show that for some such that , we have in, (i.e. ). Otherwise for all such that , we have . We ask about any such , is ? If the answer is yes for all of them then we must have , by item (iii) above. So for some we have . Then by the axiom we get that .

Either way, the assumption that for all such that we have , leads to a contradiction.

Thus is a legitimate labelling, giving rise to a complete extension.

2. Let be a legitimate labelling giving rise to a complete extension. We show that it gives a model for .

Define as follows:

 hλ(x)=1 if λ(x)=inhλ(Nx)=1 if λ(x)=out.

We show that all axioms of hold

1. Show that otherwise . This means that out and in which is not possible.

2. Second we show that if then . Otherwise we have and . The former implies in. Therefore out and so by definition , a contradiction.

3. We now show that

 hλ⊨y↔⋀zRyNz.

Assume . Then in. So for all such that out, and so by definition of , for all such and hence .

Assume . Therefore in. Then either out or und. If out then for some such that we have in. Therefore out and so .

If und, then for some an und again out and so .

Thus for sure if then for some and and so .

4. We show that . If the antecedent holds then all attackers of are not in and one of them is undecided. Therefore is undecided and so the consequent holds.

Corollary 3.3

Let be an argumentation network. Consider . Then is CN consistent.

• Proof. Since has the complete ground extension, by the previous Theorem 3.2 this yields a model for .

Example 3.4
1. {540-E5}

2. Consider the argumentation network of Figure 1.

Its theory in CN is . Since we have the axiom we get that is provable and so is provable and so is also provable. This means we have only one model with and therefore the only extension is und.

3. Consider the additional axiom added to the theory namely

 Stable:{x∨Nx|for all x}.

The theory stable does not have models in which for any . Thus this axiom characterises all stable extensions.

Theorem 3.5

Let be an argumentation network. Consider and let . Then is the ground extension of .

• Proof. We know that is consistent. Hence is consistent. We show that is a complete extension. Since other complete extension corresponds to a model of , contains . Thus would be the smallest complete extension — namely is the ground extension. We now show that is a complete extension.

1. is conflict free. Let . If holds, then and hence , which contradicts the consistency of since .

2. Assume protects . We show . Let . then for some holds. Hence is in and so . Thus we have

 ΔA⊢CN⋀zRxNz

and hence and so .

Remark 3.6

This is a clarifying remark about the correspondence between extensions of an argumentation network and CN models of the theory .

{RMarch22}

Consider the three argumentation networks in Figure 2. They all have the same extension, in, out, in.

Their theories are different, but they have the same models.

 ΔA1={x,x→Ny,y↔Nx,z,¬x∧¬Nx→¬y∧¬Ny}ΔA2={x,x→Ny,y↔Nx,y→Nz,z↔Ny,¬x∧¬Nx→¬y∧¬Ny,¬y∧¬Ny→¬z∧¬Nz}ΔA3={x,x→Ny,y↔Nx∧Nz,x∨z→Ny,¬x∧¬z∧(¬Nx∨¬Nz)→¬y∧¬Ny,y→Nz,z→Ny,¬y∧¬Ny→¬z∧¬Nz,¬z∧¬Nz→¬y∧¬Ny,y↔Nz,z↔Ny}

All three theories have only one model

 x=1,Nx=0,y=0,Ny=1,z=1,Nz=0.

The moral of the example is that the theories describe the extensions and not the geometry of the networks. In fact, are all equivalent to .

Compare the CN approach with the truly meta-level approach of [1, Section 5], discussed below in the beginning of Section 5.

We describe in predicate logic with unary predicates is in, is out, and is und, and a relation for attack. Thus using the networks of Figure 2 can be distinguished in the semantics. Note, however, that we can read the geometry of the networks from the syntax of the theories , but not from their models! This highlights the importance of proof theory.

Remark 3.7

The previous correspondence theorem reduces the idea of attack in argumentation networks to the idea of strong negation in classical propositional logic. This reduction simplifies every move we make in the argumentation area and gives us a tremendous advantage in making available to us all the machinery of classical logic. This includes

1. {540-R5}

2. extensions for formal argumentations such as joint attacks, support, higher level attacks, probabilistic argumentation, predicate/modal logic argumentation and more, all can be done more simply and easily using strong negation, see the following sections;

3. applications of argumentation become application of classical logic;

4. new ideas can be more readily imported from classical logic into argumentation;

5. the Caminada labels, is in, is out and is undecided are available in the object language as is true, is true and is true, respectively.

6. we now need to ask ourselves: what is the added value of argumentation over classical logic? We need a clear and detailed answer to this question.

4 Intermediate critical evaluation

This section pauses our formal development to evaluate what we have so far and to explain to the reader where we are going. We shall do this by a list of critical comments.

CC1. Basing argumentation on the unary notion of “being attacked”

We read as “ is being attacked”, we are not saying how and from where this attack comes. This makes a kind of strong negation (with axiom ), see [13]. This single simple idea allows us to have Theorem 3.2. It also allows us to go in the direction of turning the system CN into a paraconsisent logic of negation (see Wikipedia) by adding axioms on iterating (e.g. the axiom ). We can do this safely for as long as Theorem 3.2 is retained. We shall address this direction in later sections. We believe we can achieve similar results for logic programming by reading as “ fails”. This direction, and the connection with answer set programming, is left for a subsequent paper.

There is another direction we can go in. We can use another meaningful logic such as intuitionistic logic, linear logic or relevance logic or fuzzy logic to replace classical logic and thus get argumentation theory in those contexts. Again we shall look at this in a subsequent paper.

CC2. Simple way of defining joint attacks

The theory written for an argumentation network is comprised of several components.

1. The logic CN (the use of )

2. Formulas defining when an argument is “in”.

This is the part

 x↔⋀zRxNz

also includes the part relating to “ is not attacked”, since the empty conjunction is .

3. Formulas defining when an argument is “out”. This is the part , for all , or if we write it as a single formula, it is .

 ⋁zRxz→Nx.
4. Formulas defining when is undecided

 (⋀z∈ Attack(x)¬z)∧(⋁z∈ Attack(x)¬Nz)→¬x∧¬Nx

Once we put the above in we get the correspondence theorem, Theorem 3.2, generating complete extensions by models of .

Now we can see how easy it is to generalise argumentation to joint attacks. Joint attacks, introduced in [3, 4] can be described by the configuration in Figure 3 (using Gabbay’s notation from [3]).

The meaning is that we have is out if all of are in. This we can write in our logic CN simply as where is the joint attack relation of the form .

So we can write the formula for joint attacks as

 (⋁GR0x⋀z∈Gz)→Nx

where .

The formula for for a point will be

 x↔(⋀GR0x⋁z∈GNz)
 ⋀GRox(⋁z∈G¬z)∧(⋁GR0x⋀z∈G(z∨¬Nz))→¬x∧¬Nx

If we use these three wffs and and to define , we can study the semantics for joint attacks, in the object level, in classical logic as models of , provided we prove a correspondence theorem similar to Theorem 3.2.

The reader can compare the simplicity of this approach to what is done in the papers [3, 4].

CC3. Simple way of defining higher level attacks

Higher level attacks on attacks. These were studied in many papers [2, 5, 6, 7, 9]. They were first introduced in general in Gabbay’s paper [8]. Figure 4 illustrates the basic configuration for higher level attacks.

attacks and attacks the attack . We write it as .

We need to write the and and of this type of attack. In our set up with it is easy to write this! The is and the involves . Let us write the wff’s in detail for a network with one level of higher attacks. So our networks have the form , where are the attacks and are the attacks on the attacks. We write:

 (⋁zRx⋀yR1(z,x)z∧Ny)→Nx
 ⋀zRx⋁yR1(z,x)(Nz∨y))↔x
 (¬⋀zRx⋁yR1(z,x)(Nz∨y)∧¬⋁zRxz∧⋀yR1(z,x)Ny)→¬x∧¬Nx

Again we use , and to define , and let the semantics be all CN models of . We need to prove a correspondence theorem similar to Theorem 3.2.

CC4. A word of caution

Although we are showing how other types of networks can be translated into CN, we are not just saying “look, our paper is introducing you to another master generalisation of argumentation”. We reserve judgement about until the end of this paper and we might say at the end to the reader “in view of our paper, do not think any more in the old conceptual framework of argumentation of but think in terms of strong negation of just being attacked, and do your argumentation from now on in classical logic with ”. The reason we reserve judgement is because we want to figure out first how the use of affects related systems such as ABA (Assumption Based Argumentation, see [20]) and ASPIC (see [21]). In ASPIC and ABA, the game is to start with a logic theory , define proofs from as the argument set , define a respective suitable attack between proofs, then define as an instantiated network, take extensions and then make sure that is consistent in the logic of . This we perceive as a possibly unnecessary external detour. Our method might say to ABA and ASPIC, “why do you need all this roundabout way involving a multitude of problems? Why not add strong negation directly to the logic of and model your argumentation there and you are done? If CN can swing this and we succeed in working out the details, then ABA and ASPIC would immensely benefit from our conceptual view. See CC7 below.

CC5. Comparing with abstract dialectical framework (ADF)

ADF were introduced in [11] by Professor G. Brewka as a generalisation of argumentation frameworks and immediately caused both excitement and criticism. In this paper we shall use an example from Brewka’s slides [10] to do our comparison.

An ADF has the form where is a set of arguments and is the link relation (Brewka calls them “links” because he does not view them as attacks). is a family of acceptance relations. For each , there is a formula of classical propositional logic , specifying the acceptance (“in”) condition for , based on the acceptance values of .

In our terms, as well as in classical logic terms, what ADF is saying is condition :

 x↔φx, for all x∈S

If we regard as a condition in our logic CN, then we can define all complete extensions in our sense as all extensions obtained from models m of , where the values {in, out, und} are as in the proof of part 1 of Theorem 3.2, namely:

• in if

• out if

• und if .

Brewka, however works only in classical proposition logic with the three valued semantics according to Kleene. So his models give 3 values with

 1∧1=10∧0=01∧u=0∧u=u∧u=u

Brewka derives his extensions for the 3 valued models using some process. See [10, slides 13–18]. We now reproduce in Figure 5 the original slide 18 of Brewka [10] in order to compare ADF with our CN approach.

The Brewka extensions are given in the figure.

Figure 6 lists the models in CN obtained for the ADF theory.

We note that our model is he same as Brewka’s and our model is the same as Brewka’s . However, model gives in, in, in and und, and model gives und and in. We do not get the Brewka’s grounded model .

This model is not one of the model of the theory

 ΔC={x↔φx|x a node in Figure ???}∪Θn.

So how does Brewka get his grounded extension? Look at and , gets 1 in both, but and get value 1 in one of them and 0 in the other. So if we look at with the undecided value given to and for each of these arguments there are extensions which make its value 1 and there are extensions which make its value 0. Therefore their value, according to ADF is undecided.

From our point of view, this way of looking at undecided is just external combinatorics, devoid of conceptual content. According to our view only makes undecided, namely is false but not strongly false.

is not an extension. It is not a model of because of the axiom .

Obviously we could try and add restrictions on the models to get the same results as Brewka (i.e. implement ADF in CN) but why should we do that at all? Our methodology is sound and stronger.

Given that we had in the object language, we can do more than ADF. ADF writes the acceptance conditions in 2-valued classical logic and brings in the undecided value only through the semantic interpretation. We have the undecided value in the language itself and therefore we can put the undecided property into the acceptance conditions. Consider the joint attack described in Figure 7.

Suppose all three attackers are undecided. In this case we traditionally say that is also undecided because we do not know, maybe all three in. This could happen. If, however, we adopt the new view that the chance that all three attackers are “in” can be disregarded, then we want to say it in our acceptance conditions.

So we want to say

 x↔(Na∧Nb∧Nc)⋁(¬a∧¬Na∧¬b∧¬Nb∧¬c∧¬Nc)
 a∨b∨c→Nx
 a↔Nbb↔Na∧Ncc↔Nb
 a∨c→Nbb→Na∧Nc

Can ADF express exactly the same extensions as what CN gets for the above, especially the one und, in?

CC6. Using Cn for the probabilistic approach

There are many papers on probabilistic argumentation. Our paper [18] offers a comprehensive approach based on probability models of classical logic. We shall therefore compare the probabilistic use of CN with the approach in [18]. We shall use the network of Figure 8 as an example.

The approach of [18] regards as atoms of classical propositional logic. The logic with these atoms has the following models

 m2:a∧b;xm2:a∧¬b;ym3:¬a∧b;zm4:¬a∧¬b;1−x−y−z

We can give a probability distribution on the models

 P(m1)=xP(m2)=yP(m3)=zP(m4)=1−x−y−z.

The fact that these nodes are part of the network of Figure 8 is reflected in the probability having to satisfy the equational approach equation called E3 in [18] (see [39] for the Equational Approach to argumentation).

 P(x)=P(⋀yRx¬y).

In our case this means

where

Therefore the equations we get are

These two are the same equation, yielding

 1−x−y−z=P(m4)=x.

Therefore any probability distribution of the form

 P(a∧b)=xP(a∧¬b)=yP(¬a∧b)=zP(¬a∧¬b)=x

with is a good one, respecting the attack relation.

We have .

Let us now check how the models of CN relate to probability. For the language with atoms satisfying the CN axiom

 Nx→¬x

we can have the following models where

 α1=a∧¬Naα2=¬a∧¬Naα3=¬a∧Naβ1=b∧¬Nbβ2=¬b∧¬Nbβ3=¬b∧Nb.

Here we have 9 models as opposed to 6 models of the previous approach. Let the probabilities on these models be

 π(αi∧βj)=Pi,j

with

 ∑i,jPi,j=1.

The models must satisfy the theory for the network of Figure 8. These are

 Na→¬aNb→¬ba↔Nbb↔Na¬a∧¬Na→¬b∧¬Nb¬b∧¬Nb→¬a∧¬Na.

So only 3 models are models of . These are

 h1=a∧¬Na∧Nb∧¬bh2=¬a∧Na∧b∧¬Nbh3=¬a∧¬Na∧¬b∧¬Nb.

Let the relative probabilities be respectively with .

Note that is the extension in, out is the extension in, out and is the extension und.

The first difference between the approaches is that while the approach in [18] gives probability to arguments (called in [18] “the internal approach”), the CN approach ends up giving probabilities to extensions (called in [18] “the external approach”).

Giving probability to extensions is not new. It is supported by many authors (see references/discussion in [18]). Let us calculate the probabilities and . We get

 π(a)=p1,π(b)=p2,π(¬a∧¬b)=p3.

Comparing the two approaches, it makes sense to equate

 p3=x,p2=x+z,p1=x+y.

The main difference is that we are giving probabilities to 2-valued models in [18] and using CN we are using 3-valued models.

In both cases we give probabilities to models. However, in case of CN, the models are extensions and so we are giving probabilities to extensions also.

Cc7. Cn and bipolar networks

Note that in CN we get support and contrary arguments for free. Since we have implication “” in the logic, we can write “” to mean “ supports ” and since we have negation “” in the logic, we can view as the contrary of . We need not introduce additional atoms into the argumentation network for contraries, nor do we have to introduce an additional arrow for support into the network. Furthermore, since we have negation, we have the additional option to represent “ supports ” as “” namely as .

Let us do this in a systematic manner.

Definition 4.1
1. {540-DB1}

2. A bipolar network has the form , where is the attack relation (also denoted by ) and is the support relation (also denoted by ).

3. Given a bipolar network , we offer two possible translations into CN.

4. Note that the complete extensions of the bipolar networks will be obtained from all the models of CN.

Remark 4.2
1. {540-RB2}

2. We note that implies . This holds because, as we shall see in Section 5, and because both hold. Thus is stronger than .

3. When we have a translation from one system (e.g. ) into another (e.g. CN), we need to examine the properties of soundness and completeness.

Soundness in our case means that whatever we consider as a bipolar complete extension for will turn out to be a complete extension according to CN. Completeness means that CN does not give any additional complete extensions.

The problem in this case is that there is disagreement in the community about how to define the complete extensions for . The main papers of C. Cayrol and M. C. Lagasquie-Schiex are [28, 29]. These have been criticised by G. Brewka and S. Woltran [11] and a solution was proposed in our paper [26]. Thus a detailed analysis of soundness and completeness for our translations must be postponed to a continuation paper [24]. However, we can point out in this paper some key properties involved in [26].

The possible properties are as follows:

4. We need to check these properties for both

 τ(x↠y)=x→y

and

 τ(x⇒y)=x→N¬y.

What we need is consistency. Can we add the translation of these rules to CN and remain consistent?

The answer is positive. So we can hope to have something like the following theorem:

Theorem 4.3

Let be a set of properties for a bipolar network (e.g. (P1)–(P3) of Remark 4.2). Let be a translation of into CN, and let be the translation of into CN. Then the translation of into is sound and complete.

As we said, we shall address this theorem in Part 2 of this paper.

CC8. Limitations of the Cn approach

The CN approach transforms the geometrical representation of an argumentation network into a theory of the logic CN. Theorem 3.2 ensures that the correspondence between complete extensions of and models of is sound and complete. We are trading off here, however, geometry for model theory.

The previous CC1–CC4 discussed the advantages of the CN approach. The limitations come from the fact that in the CN approach we obscure/lose the geometrical aspects of . Therefore any moves in argumentation theory which use the geometry (e.g. the strongly connected components, SCC of Baroni et al. [14]) will become less transparent. We can mathematically do them in CN, but we would have to extract the geometry of back out of !

The following Figure 9 can be used to illustrate our point using the CF2 semantics [14, 15].

The CN semantics gives the complete extension of all undecided to the network of Figure 9 in agreement, of course, with the traditional Dung approach. The CF2 semantics takes maximal conflict free subsets of the top SCC and therefore yields the extensions . CF2 relies on identifying the SCCs. It relies on the geometry of .

The CN theory for this network is

 ΔA=Θn∪{a→Nb,b→Nc,c→Na,c→Nd,d→Ne,e↔Nd,d↔Nc,c↔Nb,b↔Na,a↔Nc}

Looking at , we have to define/extract the cycles in order to define the CF2 semantics for .

Gabbay’s approach using annihilators [14] also requires the use of geometry but to a lesser extent. We can identify syntactically a cycle, say

 x1→Nx2,x2→Nx3,…,xn→Nx1

and break the cycle by applying an annihilator, say to annihilate the point , i.e. add a new point to with , and look instead at

 Δz(xi)A=ΔA∪{z(xi),z(xi)→Nxi}.

We still need some geometric intuition in doing this.

5 Introducing the logic Cnn

5.1 A meta-level object level short discussion

The perceptive reader may be aware of Section 5 of my 2009 paper with Caminada [1]. In this paper we describe several options of looking at an abstract argumentation network . Since 2009, there were many other papers, which put forward different representations of Abstract Argumentation Networks in terms of well known logics, see for example [32][37]. We shall compare and discuss these papers in our Comparison with the Literature Section 9. Meanwhile in this section, we want to make a critical point about the difference between Object Level Vs Meta-Level representation of Abstract Argumentation Networks, and so we consider now one of the possible options of Section 5 of [1]. This option is to describe completely in classical predicate logic. We consider as the domain of the logic and we consider the attack relation as a binary relation on . In addition to that, we need 3 unary predicates, an , representing the 3 Caminada labels for the elements of , namely is “in”, is “out” and is “undecided”, respectively.

We write axioms in predicate logic basically expressing the properties of the labelling relative to making it a legitimate labelling. These are the following:

Consider the following classical theory .

Any model of with domain defines an argumentation framework with the set of argument , and the attack relation is and the labelling is what we obtain from the elements satisfying the respective predicates and . Notice that we are not using “”.

If we want to characterise any specific argumentation framework we need equality and we need constant names for every element of . We write the following additional axioms

The use of predicate logic to talk about is meta-level. The predicates and are not logical connectives. We cannot write, for example, the expression . In contrast, the logic CN is object level. It can express the predicates and in the object level, as well as the relation , as follows (see, however, Remark 3.6 of Section 3):

 Q1(x)=def xQ0(x)=def NxQ?(x)=def ¬x∧¬NxxRy=def x→Ny.

Note the difference between object level and meta-level. Suppose we want to instantiate with arguments which are formulas of predicate logic, say we have and we instantiate and . In the meta-level language we cannot write . Even if we allow for the use of names “” and “” and add appropriate axioms for the correct handling of names, we still do not know what “” means in terms of semantics. The meta-level translation does not give any meaning to it is just a translation. On the other hand , we can write in our object level system the formula , and if we use predicate logic with , we can let the models of this predicate logic with provide a proposed semantics for the instantiation by using Theorem 3.2 as a definition. Furthermore we can write additional axioms for . For example we can also write and add an axiom (which is valid)

 x↔NNx