On abstract F-systems. A graph-theoretic model for paradoxes involving a falsity predicate and its application to argumentation frameworks
aa r X i v : . [ m a t h . L O ] M a y Noname manuscript No. (will be inserted by the editor)
On abstract F -systems A graph-theoretic model for paradoxes involving a falsitypredicate and its application to argumentation frameworks
Gustavo Bodanza
Received: date / Accepted: date
Abstract F -systems are digraphs that enable to model sentences that pred-icate the falsity of other sentences. Paradoxes like the Liar and Yablo’s canbe analyzed with that tool to find graph-theoretic patterns. In this paperwe present the F -systems model abstracting from all the features of the lan-guage in which the represented sentences are expressed. All that is assumed isthe existence of sentences and the binary relation ‘. . . affirms the falsity of. . . ’among them. The possible existence of non-referential sentences is also consid-ered. To model the sets of all the sentences that can jointly be valued as truewe introduce the notion of conglomerate , the existence of which guaranteesthe absence of paradox. Conglomerates also enable to characterize referentialcontradictions , i.e. sentences that can only be false under a classical valuationdue to the interactions with other sentences in the model. A Kripke’s stylefixed point characterization of groundedness is offered and fixed points whichare complete (meaning that every sentence is deemed either true or false) andconsistent (meaning that no sentence is deemed true and false) are put incorrespondence with conglomerates. Furthermore, argumentation frameworksare special cases of F -systems. We show the relation between local conglom-erates and admissible sets of arguments and argue about the usefulness of theconcept for argumentation theory. Keywords
Liar paradox · Yablo’s paradox · F -system · Conglomerate · Groundedness · Argumentation Framework
This work was partially supported by the National Agency for Scientific and TechnologicalPromotions (ANPCYT) (grant PICT 2017-1702), and Universidad Nacional del Sur (grantPGI 24/I265), Argentina.Departamento de Humanidades, Universidad Nacional del Sur, and Instituto de Investiga-ciones Econ´omicas y Sociales del Sur (IIESS), CONICETBah´ıa Blanca, ArgentinaTel.: +54-291-4595138E-mail: [email protected] Gustavo Bodanza
Semantic paradoxes like the Liar and Yablo’s involve sentences that predicatethe falsity of other sentences. Cook [4] introduced the novelty of using graph-theoretic tools for dealing with semantic paradoxes involving a falsity predi-cate. Rabern, Rabern and Macauley [9] coined the term ‘ F -systems’ to refer“sentence systems which are restricted in such a way that all the sentences canonly say that other sentences in the system are false”. The works by Cook andRabern et al. concentrate on systems where every sentence affirms the falsityof some other sentence(s). That is represented by serial or sink-free digraphs(roughly, every node “shoots” at least one arrow). On the other hand, Beringerand Schindler [2] also consider sentences that do not refer to other sentences.We will follow this last approach since it is more general and, particularly,enables to take into account the interaction among object language sentences(like ‘Snow is white’) and metalanguage sentences (like ‘The sentence ‘Snow iswhite’ is false’). However, we bring the analysis to a more abstract level wherethe specificities of the underlying languages are (or the hierarchy of languagesis) irrelevant to find the graph-theoretic patterns that characterize paradox.The aim is to get the simplest model that enables that. All we need is a set S of nodes, which represent primitive entities we call sentences (indeed, theycan be understood as names of sentences of a given language) and a binaryrelation F ⊆ S × S , i.e. a set of directed edges or arrows such that for any pairof sentences x and y of S , ( x, y ) ∈ F is understood as ‘ x says that y is false’.In this way, for example, we can model the relationship between the (English)sentences ‘Snow is white’ and ‘The sentence ‘Snow is white’ is false’, throughan F -system F = h S, F i , where S = { a, b } and F = { ( b, a ) } , and where a and b represent ‘Snow is white’ and ‘The sentence ‘Snow is white’ is false’,respectively.Semantic paradoxes are sets of sentences to which it is not possible to assigna classical truth-value (true/false) to all of them at the same time. We willrepresent the assignment of truth-values through labellings on the nodes ofthe F -systems, in such a way that every node can be labeled with T (for T rue), F (for F alse) or U (for U ndetermined). Paradoxical F -systems will be such thatevery labelling can only put the label U on some nodes. “Classical” labellings,i.e. those that can put T or F on every sentence, will be put in correspondencewith graph-theoretic patterns that we will call conglomerates . The notion ofconglomerate extends that of kernel used by Cook, which represent a subset ofsentences that can be true together. Kernels are suitable for capturing classicassignments of truth values in systems where each sentence refers to othersentences. But if we use this notion in systems that include sinks, kernels willonly allow them to be represented as true. Conglomerates, although they willlead to similar formal results regarding paradoxes, will enable a more intuitiverepresentation since object language sentences will be considered with anytruth-value. We borrow the term ‘labelling’ from [3]. In [2], the term ‘decoration’ is used instead.n abstract F -systems 3 Another aim of this work is to define a Kripke’s style fixed point operator tocharacterize groundedness in F -systems [8]. Grounded sentences are, roughly,those which truth-value can be tracked through the reference path until asentence with a definite truth-value (the “ground”). We will define completeand consistent fixed points (meaning that every sentence is deemed eithertrue or false and no sentence is deemed true and false) and show that theycorrespond exactly to conglomerates.Conglomerates also enable to characterize referential contradictions (tau-tologies) , i.e. sentences that can only be false (true) under a classical valuation,due to the interactions with other sentences in the model. This is an advantagewith respect to kernels, which are unable to do that.Finally, we define local conglomerates , a notion that enables to cover andextend that of admissibility in Dung’s argumentation frameworks. As shownby Dyrkolbotn [6], argumentation frameworks are special cases of F -systemswhere arguments play the role of sentences and F is interpreted as an attackrelation. Admissibility formalizes the idea of sets of arguments that can be de-fended between them. Local conglomerates cover that idea and give it a twist:they deem “admissible” also sets of arguments that can be defended togetheragainst any argument, except those that promote some non-preferred value,which suggests a new semantics for value-based argumentation frameworks [1].The paper is organized as follows. In Section 2 we define abstract F -systems, labellings, and the notion of conglomerate. In Section 3 we give afixed-point characterization of groundedness, and show the relation of con-glomerates with complete and consistent fixed points. In Section 4 we definereferential contradictions and tautologies, and show that the transitivity of F in non-paradoxical systems is a sufficient condition for their existence. More-over, transitivity –as shown by Cook [4]– can also be a source of paradox aswell as odd-length cycles. We comment on those points in Section 5. In Sec-tion 6 we introduce the notion of local conglomerate and apply it to coverand extend that of admissibility in Dung’s argumentation frameworks. Finalconclusions are summarized in Section 7. F -systemsDefinition 1 An ( abstract ) F - system is a pair F = h S, F i where S is a setwhich elements are primitive entities called sentences and F ⊆ S × S is abinary relation among sentences.For every x ∈ S , we define −→ F ( x ) = { y ∈ S : ( x, y ) ∈ F } and ←− F ( x ) = { y ∈ S : ( y, x ) ∈ F } , and for every subset A ⊆ S , −→ F ( A ) = S x ∈ A −→ F ( x ) and ←− F ( A )= S x ∈ A ←− F ( x ). If −→ F ( x ) = ∅ , x is said to be a sink , and we define sinks ( A ) = { x ∈ A : x is a sink } . In order to avoid misrepresentations, we assume thatnon-sink sentences do not assert anything more than what is represented in F (and, naturally, sink sentences do not assert anything about other sentences).To illustrate the kind of issues we want to avoid, consider the following exam-ple. Let F = ( { x, y } , { ( y, x ) } ). Then we want to interpret that x is true if y is Gustavo Bodanza false and x is false if y is true; moreover, we want to interpret that if x has anundetermined truth value then the value of y is undetermined, too. However,if we accept the interpretation: x = ‘Snow is red’ and y = ‘ x is false and thesnow is blue’, then the above considerations about the truth and falsity of x and y would not be valid, since x and y could both be false. Though it is truethat y affirms the falsity of x , the component ‘the snow is blue’ of y which is“hidden” in the representation can yield anomalous interpretations. The ab-stract level of the model does not allow to represent such molecular sentencessince there are no elements to represent logical connectives. Hence, we leavethat kind of interpretations out of the scope of the model. On the other hand,the only molecular sentences that can be represented in the model, preservingthe intuitions about the assignment of truth values, are conjunctions of falsityassertions about other sentences like, for instance, ‘ x says that both y and z are false’, which can be modeled as { ( x, y ) , ( x, z ) } ⊆ F .Since F -systems define digraphs, we can see the assignment of truth valuesto the sentences as labels on the nodes of a digraph. We consider three labels, T , F and U , for true , f alse and undetermined , respectively. The non-classicalvalue undetermined is intended to express either that the actual value isunknown (as in the case of conjectures) or just the impossibility of assigninga classical truth value (as in the case of paradoxes). Definition 2
Given F = h S, F i , a labelling on F is a total function L suchthat:1. L : S → { T , F , U } , and2. for all x ∈ S \ sinks ( S )(a) L ( x ) = F iff L ( z ) = T for some z ∈ −→ F ( x ), and(b) L ( x ) = T iff L ( z ) = F for every z ∈ −→ F ( x ).Note that the assignment of values to sink nodes is unrestricted. Moreover, forevery F -system there always exist a labelling that labels all the nodes with U . Definition 3
A labelling L on F is classical iff for every x ∈ S , L ( x ) = U .Paradoxes in F -systems can be characterized as follows: Definition 4 An F -system is paradoxical iff it has no classical labellings.Moreover, a sentence x is paradoxical iff L ( x ) = U for every labelling L . Example 1
Let F = h{ a k } k ∈ N , { ( a k , a m ) } k Given F = h S, F i , a conglomerate is a subset A ⊆ S that sat-isfies:1. ←− F ( A ) ⊆ S \ A , and2. ( S \ A ) \ sinks ( S ) ⊆ ←− F ( A )The idea is that a conglomerate coalesce all the sentences that can sharethe true value, leaving outside all and only the sentences that can share thefalse value. So, conglomerates can only exist in systems which sentences canbe “polarized” into true and false. A conglomerate can also be understoodin a Kripkean way as defining the extension of the truth predicate of theunderlying language, being its complement in the system the anti-extension(we will return to this point in Section 3). If a conglomerate exists, then wecan say that the truth predicate is completely defined in the system. Since aconglomerate A is supposed to comprise all the true sentences, the condition 1says that it cannot contain two sentences such that one affirms the falsity of theother (i.e. A is independent ). And S \ A is supposed to comprise all the falsesentences, so the condition 2 says that every non-sink sentence must assertthe falsity of at least one true sentence (i.e. A absorbs every external non-sink node). This is different from kernels, which absorb every outer node. Thisimplies that kernels comprise all the sinks, so these can only be interpretedas true sentences in such a model. In this sense, conglomerates seem to bemore suitable than kernels to represent the Kripkean view: sinks may or maynot belong to the conglomerates, representing object language sentences thatmay or may not be true. Furthermore, the notion of conglomerate clearly alsoencompasses that of kernel. Example 2 Let F = h{ a, b } , { ( b, a ) }i . Assume that F represents the relationbetween a : ‘I am wearing a hat’ and b : ‘The sentence ‘I am wearing a hat’ isfalse’. F has only one kernel, { a } , but it has two conglomerates, { a } , deeming‘I am wearing a hat’ as true and ‘The sentence ‘I am wearing a hat’ is false’ asfalse, and { b } , deeming ‘I am wearing a hat’ as false and ‘The sentence ‘I amwearing a hat’ is false’ as true. Every kernel is a conglomerate, but not viceversa.The notion of conglomerate is not well-defined, in the sense that some F -systems have no conglomerates. As expected, those systems are the paradoxicalones. Example 3 Let F = h{ a } , { ( a, a ) }i (the Liar paradox). Then F does not haveany conglomerate.The correspondence between conglomerates and classical labellings is easy toprove: Theorem 1 L is a classical labelling iff A = { x : L ( x ) = T } is a conglomerate.Proof Let F = h S, F i .(If) Let A be a conglomerate of F . Let L be such that ∀ x ( x ∈ A → L ( x ) = T ) Gustavo Bodanza and ∀ x ( x ∈ S \ A → L ( x ) = F ). Then L trivially satisfies the conditions of aclassical labelling.(Only if) Let L be a classical labelling of F and let A = { x : L ( x ) = T } and B = { x : L ( x ) = F } . (i) By definition, if L ( x ) = T then for all z such that z ∈ −→ F ( x ), L ( z ) = F . Hence, by construction, x ∈ A and z ∈ B . Then, forall x, z ∈ A , z / ∈ −→ F ( x ). (ii) By definition, for all x ∈ S , if L ( x ) = F and x isnot a sink, then there exists some z ∈ −→ F ( x ) such that L ( z ) = T . Hence, byhypothesis, x ∈ B and z ∈ A . Therefore, given (i) and (ii) we have that A isa conglomerate. Corollary 1 F is paradoxical iff it does not have any classical labelling. The truth value of sentences asserting the falsity of other sentences depends onthe truth value of the referred sentences. If the truth value of a sentence doesnot depend on that of other sentences, “so that the truth value of the originalstatement can be ascertained, we call the original sentence grounded , otherwise ungrounded ” (Kripke, 1975: 694). In our framework, sentences at sink nodes(for instance, object language sentences) do not depend on other sentences inthat sense, so their truth value depend on material (contingencies) or formal(tautologies or contradictions) facts that are exogenous to the model. Takingas grounded all the sinks that are either true or false, the groundedness ofall the remaining sentences of an F -system will be determined in an iteratedprocess very similar to Kripke’s. In the base case, all the sinks determinedas true belong to a set S +0 and all those determined as false belong to a set S − . That is, the partial set ( S +0 , S +0 ) models the interpretation of the sinksentences. The systems considered by Cook and Rabern et al. are sink-free,hence no sentence is grounded in the above sense in those systems. Beringerand Schindler, on the other hand, consider the existence of sinks (representingtrue arithmetical sentences) and so their system constitute a special instanceof an abstract F -system in which groundedness can be tracked by dependenceon the sinks. Definition 6 Given F = h S, F i , a pair ( S +0 , S − ) is a ground base iff S +0 ∪ S − = sinks ( S ) and S +0 ∩ S − = ∅ .Then, we can find the other grounded sentences by iterated applications ofthe following operator: Definition 7 Given two subsets S + , S − ⊆ S , we define φ (( S + , S − )) = ( S ′ + , S ′− ),where S ′ + = sinks ( S + ) ∪ { x : ∅ 6 = −→ F ( x ) ⊆ S − } , and S ′− = sinks ( S − ) ∪ { x : ∅ 6 = −→ F ( x ) ∩ S + } . n abstract F -systems 7 That is, S ′ + includes the sinks that are already known as true plus all thesentences that only affirm the falsity of sentences already known as false; and S ′− includes the sinks that are already known as false plus all the sentencesthat affirm the falsity of some sentence already known as true. So, starting fromany ground base ( S +0 , S − ), iterated applications of φ will lead to a fixed point.A fixed point is any pair ( S + , S − ) = φ (( S + , S − )). The fixed point ( S + , S − )= φ ∞ (( S +0 , S − )) reached by the above mentioned iteration procedure is theleast one relative to the ground base ( S +0 , S − ), in the sense that any otherfixed point ( S ′ + , S ′− ) where S +0 ⊆ S ′ + and S − ⊆ S ′− is such that S + ⊆ S ′ + and S − ⊆ S ′− . The operator φ is monotone. Let ( S + , S − ) ≤ ( S ′ + , S ′− ) iff S + ⊆ S ′ + and S − ⊆ S ′− . Then: Remark 1 (Monotony) If ( S + , S − ) ≤ ( S ′ + , S ′− ) then φ (( S + , S − )) ≤ φ (( S ′ + ,S ′− )). Proof Assume ( S + , S − ) ≤ ( S ′ + , S ′− ). Let φ (( S + , S − )) = ( T + , T − ) and φ (( S ′ + , S ′− )) = ( T ′ + , T ′− ). Assume now w ∈ T + and z ∈ T − . We have toprove that 1) w ∈ T ′ + and 2) z ∈ T ′− .1) If w ∈ S + the result is obvious. Assume now w S + . Then w ∈ { x : x S and ∀ y ( y ∈ −→ F ( x ) → y ∈ S − ) } . Then, w S and ∀ y ( y ∈ −→ F ( w ) → y ∈ T − ).Therefore, w ∈ T ′ + .2) If z ∈ S − the result is obvious. Assume now z S − . Then z ∈ { x : x S and ∃ y ( y ∈ −→ F ( x ) ∧ y ∈ S + ) } . Then, z S and ∃ y ( y ∈ −→ F ( z ) ∧ y ∈ T + ).Therefore, z ∈ T ′− .The existence of the least fixed point is guaranteed by the monotony of φ andthe fact that all the fixed points (relative to the same ground base) form acomplete lattice (by Tarski’s fixed points theorem).Now, some ground bases can deem all the sentences grounded and othersnot. Consider the following example: Example 4 Let F = h{ a, b, c } , { ( b, a ) , ( b, c ) , ( c, b ) }i . The only sink a determinestwo possible ground bases: (1) ( { a } , ∅ ), and (2) ( ∅ , { a } ). For (1), all a , b , and c are grounded (true, false, and true, respectively), and for (2), only a isgrounded (false). By way of illustration, the least fixed point in each case isreached as follows:(1) φ (( { a } , ∅ ))=( { a } , { b } ) φ (( { a } , ∅ ))=( { a, c } , { b } )... φ ∞ (( { a } , ∅ ))=( { a, c } , { b } )(2) φ (( ∅ , { a } ))=( ∅ , { a } )... φ ∞ (( ∅ , { a } ))=( ∅ , { a } ) Gustavo Bodanza Since an F -system can have different ground bases, some of them can deem allthe sentences grounded but others not. The following notion of groundednesstakes into account those possibilities: Definition 8 F = h S, F i is ( relatively ) grounded iff for every x ∈ S and for(some) every ground base ( S +0 , S − ), x ∈ S + ∪ S − , where ( S + , S − ) = φ ∞ (( S +0 , S − )).The system in Example 4 is relatively grounded but not grounded. Onthe other hand, F -systems representing paradoxes as the Liar or Yablo’s areneither grounded nor relatively grounded, as expected. Other systems are notgrounded nor relatively grounded either, though they are not paradoxical.For instance, F = h{ a, b } , { ( a, b ) , ( b, a ) }i . Since there are no sinks, the onlyground base is ( ∅ , ∅ ), and φ ∞ (( ∅ , ∅ )) = ( ∅ , ∅ ) is the least fixed point. However,other fixed points like ( { a } , { b } ) and ( { b } , { a } ) imply that a and b can beconsistently assigned a classical truth value. This makes the difference withparadoxical systems, where no fixed point represents a consistent assignmentof truth values to all the sentences.The conglomerates of an F -system can be characterized by means of thefixed points of φ as follows: Definition 9 We say that ( S + , S − ) is complete iff for every x ∈ S , x ∈ S + ∪ S − , and consistent iff S + ∩ S − = ∅ . Theorem 2 A is a conglomerate iff ( A, S \ A ) is a complete and consistentfixed point of φ .Proof (Only if) Let A be a conglomerate of F = h S, F i . Let ( S + , S − ) = φ (( A, S \ A )). Then we have:(i) S − = sinks ( S \ A ) ∪ { x : ∃ y ( y ∈ −→ F ( x ) ∧ y ∈ A ) } , and(ii) S + = sinks ( A ) ∪ { x : x sinks ( S ) and ∀ y ( y ∈ −→ F ( x ) → y ∈ S \ A ) } .To see that ( A, S \ A ) is a fixed point of φ we have to prove that S + = A and S − = S \ A . From (i), it follows that S − = S \ A , since A absorbs everynon-sink node and, obviously, every sink of S \ A belongs to S \ A . Let usprove now that S + = A . (1) S + ⊆ A : By reductio , assume that there exists x A such that x is not a sink and ∀ y ( y ∈ −→ F ( x ) → y ∈ S \ A ). Butthen x is not absorbed by A , which contradicts that A is a conglomerate. (2) A ⊆ S + : Of course, sinks ( A ) ⊆ S + . Let now x ∈ A be a non-sink node.From the definition of conglomerate, ←− F ( A ) ⊆ S \ A . Therefore, it is clear that ∀ y ( y ∈ −→ F ( x ) → y ∈ S \ A ). Finally, the fact that ( A, S \ A ) is a complete andconsistent fixed point is obvious.(If) Let ( S + , S − ) be a complete and consistent fixed point of F = h S, F i .By way of the absurd, assume z ∈ −→ F ( x ) and x, z ∈ S + . Then φ (( S + , S − ))= ( S ′ + , S ′− ) is such that x ∈ S ′ + and x ∈ S ′− . But this contradicts theconsistency property. Therefore, (i) for all x, z ∈ S + , z / ∈ −→ F ( x ). Now, since( S + , S − ) is complete and consistent it follows that S − = S \ S + . By this and n abstract F -systems 9 because ( S + , S − ) is a fixed point of φ , we have that S \ S + = sinks ( S − ) ∪{ x : ∃ y ( y ∈ −→ F ( x ) ∧ y ∈ S + ) } , which in turn implies that (ii) ( S \ S + ) \ sinks ( S )= { x : ∃ y ( y ∈ −→ F ( x ) ∧ y ∈ S + ) } ⊆ ←− F ( A ). Therefore, from (i) and (ii) and bydefinition, S + is a conglomerate. Example 5 (Continuation of Example 4) There are three conglomerates, { a, c } , { b } , and { c } , that can be put in correspondence with the fixed points ( { a, c } , { b } ), ( { b } , { a, c } ), and ( { c } , { a, b } ), respectively. We have seen that some F -systems, by their own structural nature, have sen-tences that can only be labeled as undetermined ( U ): those just characterizedas paradoxical. In addition, some systems have sentences that can only be la-beled as true ( T ) and others as false ( F ) by every classical labelling. We will saythat those sentences are referential tautologies / contradictions , meaning thattheir truth/falsity is due to structural conditions of the system. Definition 10 Given F = h S, F i , x ∈ S is a referential contradiction ( tautol-ogy ) iff L ( x ) = F ( L ( x ) = T ), for every classical labelling L .As a consequence, referential contradictions and tautologies are related toconglomerates in the following way: Proposition 1 Let F = h S, F i be non-paradoxical. Then x ∈ S is a referentialcontradiction (tautology) iff for every conglomerate A , x / ∈ A ( x ∈ A ) .Proof It follows immediately from Theorem 1.Referential tautologies are just sentences that assert the falsity of a contradic-tion. However, referential contradictions can exist independently of referentialtautologies. Proposition 2 Given F = h S, F i , if x ∈ S is a referential tautology, thenthere exists some z ∈ S such that z is a referential contradiction (and z ∈−→ F ( x ) ).Proof By definition, if x is a referential tautology then L ( x ) = T for everyclassical labelling L . By definition of labelling, x cannot be a sink (otherwiseit could be labeled with T by some classical labelling). Hence, there exists z ∈ −→ F ( x ) and, obviously, L ( z ) = F in every classical labelling L .Note that referential contradictions and tautologies are not related to ker-nels in the same way as to conglomerates. Example 6 Let F = h{ a, b, c } , { ( a, b ) , ( b, c ) , ( a, c ) }i . There exist two conglom-erates, { c } and { b } , deeming a , which is not paradoxical, as a referential con-tradiction. The only kernel { c } is useless to capture that fact. The previous example also shows that transitivity is a source of referentialcontradictions when the antecedent conditions of the transitive property aremet in non-paradoxical systems. Proposition 3 Let F = h S, F i be non-paradoxical and F be transitive. If x, y, z ∈ S are such that ( x, y ) , ( y, z ) ∈ F , then x is a referential contradiction.Proof Assume the antecedent of the claim. By the transitivity of F , ( x, z ) ∈ F .Let L be a classical labelling of F (which exists due to the non-paradoxicalityof F ). Then either (i) L ( z ) = T or (ii) L ( z ) = F . If (i) is the case then L ( x ) = F .If (ii) is the case then either (a) L ( y ) = T or (b) L ( y ) = F . If (a) is the casethen L ( x ) = F , and if (b) is the case then there exists w ∈ S such that( y, w ) ∈ F and L ( w ) = T . But then, by transitivity, ( x, w ) ∈ F . That impliesthat L ( x ) = F . Hence, in any case L ( x ) = F . Therefore, x is a referentialcontradiction.In addition, if transitivity is satisfied by systems where every sentence refersto other sentences we get paradox, as we will see in the next section. Rabern et al. ([9]) identify structural properties of the digraphs as necessaryconditions for paradox. The conditions are, basically, the existence of directedcycles (as in the case of the Liar) or double paths (as in the case of the Yablo’sparadox). However, those conditions are not sufficient. In this section we showsome sufficient conditions present in the literature and establish another onerelated to odd-length cycles.5.1 Transitivity F is transitive iff for all x, y, z ∈ S , if y ∈ −→ F ( x ) and z ∈ −→ F ( y ) then z ∈ −→ F ( x ).For example, F is transitive in Example 6. In that system, note that a cannotbe labelled with T , but it can be labeled with F whenever c is labeled with T or F (i.e. a is a referential contradiction). Moreover, if every sentence refers tothe falsity of other sentence, then transitivity will also prevent the assignmentof the F label. This result was showed by Cook [4]. Definition 11 F = h S, F i is unlimited transitive iff (i) F is transitive and(ii) F is a serial digraph (i.e. no node is a sink). Proposition 4 (Cook) If F is unlimited transitive then it is paradoxical. A double path is a graph consisting of two non-trivial paths, both with common originand end.n abstract F -systems 11 Proof Assume F = h S, F i is not paradoxical. Then it has a classical labelling L . Then, for every x ∈ S , either L ( x ) = T or L ( x ) = F . Assume L ( x ) = T .Then, for all y ∈ −→ F ( x ), L ( y ) = F . Let now y ∈ −→ F ( x ). Then, for some z ∈−→ F ( y ), L ( z ) = T . But, by transitivity, z ∈ −→ F ( x ), which implies that L ( z ) = F .Contradiction. Assume now L ( x ) = F . Then, for some y ∈ −→ F ( x ), L ( y ) = T .Then we can apply on y the same argument as before to get a contradiction.Therefore, F is paradoxical.Both the Liar and Yablo’s paradoxes can be modeled as unlimited transitive F -systems.5.2 Odd-length cyclesUnlike the Yablo’s paradox, the Liar paradox contains a referential cycle. Butnot every referential cycle leads to paradox. Next we define some conditionsinvolving cycles that suffice to yield paradox. Definition 12 Given F = h S, F i , the subset O ⊆ S is an odd core of F iff for some n ≥ x , . . . , x n +1 ∈ S such that O = { x , . . . , x n +1 } , { ( x i , x i +1 ) : 1 ≤ i ≤ n } ∪ { ( x n +1 , x ) } ⊆ F , and for all x ∈ O , | F ( x ) | = 1. Moreover, F is odd iff it has an odd core.Informally, the definition says that an odd F -system is such that there existsan odd-length cycle in F , the nodes of which can shoot exactly one arrow each(no matter how many arrows point to them). Proposition 5 If F is odd then it is paradoxical.Proof Let F = h S, F i be odd and let O = { x , . . . , x n +1 } be an odd coreof F . By way of contradiction, let us assume that A is a conglomerate of F .Assume, without lost of generality, that x ∈ A . Then, since for all x ∈ O , | F ( x ) | = 1, by the absorption condition we have { x , x , . . . , x n +1 } ⊆ A .But, since ( x n +1 , x ) ∈ F , that contradicts the independence property of A .Hence, we should have x ∈ S \ A . This implies that { x , x , . . . , x n , x } ⊆ A .But, since ( x , x ) ∈ F , we get again a contradiction with the independenceproperty. Therefore, F cannot have any conglomerate, which means that it isparadoxical.As a consequence, we can see that the source of paradox in the Liar paradoxis twofold: the F -system is both unlimited transitive and odd. Yablo’s paradox,in turn, only suffers from unlimited transitivity.Finally, it is worth mentioning that Dyrkolbotn [6] resumes some knownsufficient conditions to avoid paradox in presence of odd-length cycles. In ourframework, that result can be expressed as follows: Proposition 6 (Dyrkolbotn) Any F -system has a conglomerate if every odd-length cycle has one of the following 1. at least two symmetric nodes2. at least two crossing consecutive chords (a chord is an arrow on a cycleconnecting two non-consecutive nodes)3. at least two chords with consecutive targets. Conglomerates capture all the sentences that can be true together in systemsthat are free of paradoxes. In terms of Dyrkolbotn [6], the absorption conditionon conglomerates is global , and that inhibit some systems of having conglom-erates. But we can still want to know what sentences can be true together insystems containing paradoxes, even if that class is empty. That can be doneby defining a local version of the absorption condition. To that aim, in thissection we define the notion of local conglomerate .P.M. Dung [5] has defined argumentation frameworks , which are just spe-cial cases of F -systems where S is interpreted as a set of arguments and F asan attack relation. The notion of admissibility captures the idea of sets of ar-guments that can be defended jointly: A ⊆ S is admissible iff 1) −→ F ( A ) ⊆ S \ A (i.e. the arguments that A attacks are outside A ), and 2) ←− F ( A ) ⊆ −→ F ( A ) (i.e. A attacks all its attackers). In graph theory, and replacing F with F − , thatnotion is known with the name of semikernel [7]. Following the same motiva-tion that led us to replace the notion of kernel with that of conglomerate, weare going to replace the notion of semikernel with the following: Definition 13 Given F = h S, F i , a local conglomerate is a subset A ⊆ S thatsatisfies:1. ←− F ( A ) ⊆ S \ A , and2. −→ F ( A ) \ sinks ( S ) ⊆ ←− F ( A ).It is easy to see that conglomerates are also local conglomerates, whichenables to establish a hierarchy among the notions, including kernels. Localconglomerates only absorb those non-sink nodes to which they point at. Asa consequence, the empty set of nodes is always a local conglomerate, whichimplies that the notion is well-defined: every F -system has at least one localconglomerate. Moreover, due to the fact that local conglomerates are local“absorbers”, they capture all the sentences that can have a classical truthvalue in contexts where paradoxes can be isolated. Example 7 Let F = h S, F i = h{ a, b, c, d } , { ( a, b ) , ( b, c ) , ( a, c ) , ( d, d ) }i . Then F has no conglomerates but has two non-empty local conglomerates, { b } and { c } . They can be matched with the fixed points ( { b } , { a, c } ) and ( { c } , { a, b } ),respectively. Clearly, the paradoxical sentence d , which is not connected to theother sentences, is not present in those fixed points. n abstract F -systems 13 In Dung’s argumentation frameworks, maximally (w.r.t. ⊆ ) admissible setsof arguments are called preferred extensions . Analogously, we can think of themaximal (w.r.t. ⊆ ) local conglomerates of an F -system, i.e. subsets of sen-tences that can all be true together at the same time and, if so, the remainingsentences of the system must necessarily be untrue. These can be put in cor-respondence with consistent fixed points that are maximal w.r.t. ≤ . Theorem 3 A is a maximal (w.r.t. ⊆ ) local conglomerate iff ( A, −→ F ( A ) ∪←− F ( A )) is a consistent maximal (w.r.t. ≤ ) fixed point of φ .Proof (If) Let ( A, B ) be a consistent maximal (w.r.t. ≤ ) fixed point of φ . Then(i) ←− F ( A ) ⊆ S \ A . To prove this, assume x ∈ ←− F ( A ). By way of contradiction,assume now that x ∈ A and let z ∈ A be such that z ∈ −→ F ( x ). Since ( A, B ) isa fixed point, we have that B = sinks ( B ) ∪ { x : ∅ 6 = −→ F ( x ) ∩ A } . Then, z ∈ B .So, z ∈ A ∩ B , which contradicts the consistency of ( A, B ). Therefore, x A ,which implies that x ∈ S \ A .(ii) −→ F ( A ) \ sinks ( S ) ⊆ ←− F ( A ). To prove this, let u ∈ A and w ∈ −→ F ( u ) \ sinks ( S )(i.e. w ∈ −→ F ( A ) \ sinks ( S )). Since ( A, B ) is a fixed point, the hypothesis w F ( A ) implies that w B , since B = sinks ( B ) ∪ { x : ∅ 6 = −→ F ( x ) ∩ A } . But then u A , since A = sinks ( A ) ∪ { x : ∅ 6 = −→ F ( x ) ⊆ B } . Contradiction. Therefore, w ∈ ←− F ( A ).From (i) and (ii) it follows that A is a local conglomerate. Assume now that A is not maximal, i.e. there exists some local conglomerate A ′ , such that A ⊂ A ′ .Let x ∈ A ′ \ A . If x is a sink then x ∈ sinks ( A ′ ) ∪ { x : ∅ 6 = −→ F ( x ) ⊆ B } = A ′ . But this contradicts that ( A, B ) is a maximal fixed point. Hence, x is nota sink. Let z ∈ −→ F ( x ). Then z ∈ B ′ , for some B ′ such that B ⊆ B ′ . But then z ∈ ←− F ( A ′ ). Then ( A, B ) ≤ φ (( A ′ , B ′ )) but not φ (( A ′ , B ′ )) ≤ ( A, B ). This alsocontradicts that ( A, B ) is a maximal fixed point.(Only if) Let A be a maximal (w.r.t. ⊆ ) local conglomerate. We have to provethat φ (( A, −→ F ( A ) ∪ ←− F ( A ))) = ( A, −→ F ( A ) ∪ ←− F ( A )). This follows from: (i) A = { sinks ( A ) ∪ { x : ∅ 6 = −→ F ( x ) ⊆ −→ F ( A ) ∪ ←− F ( A ) } . For this, just observe thatsince A is a maximal local conglomerate, if x is not a sink then x is such that ∅ 6 = −→ F ( x ) ⊆ −→ F ( A ) ∪←− F ( A ) iff x ∈ A . (ii) −→ F ( A ) ∪←− F ( A ) = sinks ( −→ F ( A ) ∪←− F ( A )) ∪{ x : ∅ 6 = −→ F ( x ) ∩ A } . To prove this, note that since { x : ∅ 6 = −→ F ( x ) ∩ A } = ←− F ( A ),it is obvious that sinks ( −→ F ( A ) ∪ ←− F ( A )) ∪ { x : ∅ 6 = −→ F ( x ) ∩ A } ⊆ −→ F ( A ) ∪ ←− F ( A ).Now, since A is a local conglomerate, −→ F ( A ) \ sinks ( −→ F ( A )) ⊆ ←− F ( A ), hence wehave that −→ F ( A ) ∪ ←− F ( A ) ⊆ sinks ( −→ F ( A ) ∪ ←− F ( A )) ∪ { x : ∅ 6 = −→ F ( x ) ∩ A } .The class of all the admissible sets of an argumentation framework form acomplete partial order with respect to set inclusion (i.e. reflexive, transitive,antisymmetric, and every increasing sequence has a least upper bound). The“fundamental lemma” from which the result follows immediately in [5] can beparaphrased in our framework with the help of the operator φ . Lemma 1 Let φ (( A, −→ F ( A ) ∪ ←− F ( A ))) = (( S + , S − )) and A be a local conglom-erate. Then for every x, z ∈ S + , the set A ′ = A ∪ { x } is such that A ′ is a local conglomerate.2. Let φ (( A ′ , −→ F ( A ′ ) ∪ ←− F ( A ′ ))) = ( S ′ + , S ′− ) . Then z ∈ S ′ + .Proof 1. We only have to prove that ←− F ( A ′ ) ⊆ S \ A ′ . Assume the contrary.Since ←− F ( A ) ⊆ S \ A , there exists some w ∈ A such that either x ∈ −→ F ( w )or w ∈ −→ F ( x ). Assume x ∈ −→ F ( w ). Since x ∈ S + , by definition of φ , ∅ 6 = −→ F ( x ) ⊆ −→ F ( A ) ∪ ←− F ( A ). Hence −→ F ( x ) ∩ A = ∅ , which implies that A doesnot absorb w . This contradicts that A is a local conglomerate. Assume nowthat w ∈ −→ F ( x ). Then x S + . Contradiction.2. Since z ∈ S + , we have that −→ F ( z ) ⊆ −→ F ( A ) ∪ ←− F ( A ). Then, given that A ⊆ A ′ , −→ F ( z ) ⊆ −→ F ( A ′ ) ∪ ←− F ( A ′ ). Therefore, z ∈ S ′ + .Caminada [3] showed that preferred extensions correspond to labellingsthat maximize T (or, equivalently, F ). We can consequently put maximal localconglomerates also in correspondence with such labellings in our framework. Theorem 4 L is a labelling that maximizes T iff A = { x : L ( x ) = T } is amaximal local conglomerate.Proof (If) Let A = { x : L ( x ) = T } be a maximal local conglomerate. Let usassume, by way of the absurd, that there exists z A such that L ( z ) = T .Then for all w ∈ −→ F ( z ), L ( w ) = F . Then, since −→ F ( A ) \ sinks ( S ) ⊆ ←− F ( A )and L ( x ) = T for every x ∈ A , it follows that z 6∈ −→ F ( A ). Moreover, z 6∈ ←− F ( A )either, otherwise we would have L ( z ) = F . Then, A ∪{ z } is a local conglomerategreater than A , contradicting the hypothesis. Therefore, for every z A , L ( z ) = T , which implies that L is a labelling that maximizes T .(Only if) Let L be a labelling that maximizes T . First, it is easy to see that A = { x : L ( x ) = T } is a local conglomerate. Now assume, by way of theabsurd, that A is not maximal. Then there exists some local conglomerate A ′ such that A ⊂ A ′ . Let x ∈ A ′ \ A . Since L maximizes T , then L ( x ) = T . If L ( x ) = F then there exists some z ∈ A such that z ∈ −→ F ( x ). This contradictsthat A ′ is a local conglomerate. And if L ( x ) = U then x is paradoxical, whichalso contradicts that A ′ is a local conglomerate. Therefore, A is a maximallocal conglomerate.So, from the two previous theorems we get the following: Corollary 2 For every set of sentences A ⊆ S , the following three statementsare equivalent:1. A is a maximal local conglomerate.2. There exists a labelling L that maximizes T and A = { x : L ( x ) = T } .3. ( A, −→ F ( A ) ∪ ←− F ( A )) is a consistent maximal (w.r.t. ≤ ) fixed point of φ . Dung argued that argumentation frameworks with no stable semantics arenot necessarily “wrong”. Analogously, there is nothing necessarily “wrong” in F -systems having no conglomerates. They may have many meaningful, nonparadoxical parts, and those parts can be captured via local conglomerates. n abstract F -systems 15 Finally, the notion of local conglomerate clearly extends that of admissi-ble set (provided that F is changed to F − appropriately). Let us argue whylocal conglomerates not corresponding to admissible sets can make sense inargumentation frameworks. Consider the system h{ a, b } , { ( a, b ) }i . Interpretedas an argumentation framework, we have that argument a attacks argument b ,and the only maximally admissible set is { a } (by the way, ∅ is also admissible,but not maximally). On the other hand, there are two maximal local conglom-erates: { a } and { b } . In what sense can { b } be “admissible”? The notion ofadmissibility was introduced by Dung as a model of the “principle” “The onewho has the last word laughs best” . As such, it is a good model. But we canalso think of situations in which the last word is wrong. For example, assume b is an argument without objections except for a , which is an argument thateverybody in the audience would reject. Then that audience would possiblydeem the argument b in inasmuch a is out , and even if b is the last word. Thelocal conglomerate { b } enables to capture that possibility. A similar intuitionis also modeled by value-based argumentation frameworks (VAF’s) [1]: if a attacks b but b promotes a value that is preferred to the value promoted by a ,then a does not defeat b . For example, consider, on the one hand, an argumentthat promotes the legalization of abortion as a public health necessity in viewof a significant magnitude of women’s deaths as a result of clandestine abortionpractices and, on the other hand, an argument that promotes the prohibitionof abortion as a need for protection of the fundamental human right to thelife of an embryo. Suppose the second argument is advanced as an attack onthe first. However, if the audience values public health over the right to lifeof a human embryo, it will result that the expected defeat will have no effect(the same applies in the opposite case). In this setting, we can capture theopposing opinions with two different (local) conglomerates. The notion of conglomerate enabled us to essentially capture the same re-sults as that of kernel regarding semantic paradoxes. But while kernels canonly represent situations in which all the sink (object language) sentences inthe system are true, conglomerates also enable to represent the false cases.Another important result is that conglomerates enable to characterize refer-ential contradictions and tautologies, i.e. sentences that can be assigned onlythe false value or only the true value, respectively. Referential contradictionscannot belong to any conglomerate. And more generally, as a consequence ofTheorem 3, it is easy to see that referential contradictions are excluded fromevery maximal local conglomerate.Inspiration for an abstract treatment of F -systems came from Dung’sabstract argumentation frameworks [5]. Caminada’s labelling semantics forDung’s systems [3] were adapted for our purpose here. Dyrkolbotn [6] has pre-viously treated paradox, kernel theory and argumentation frameworks on acommon ground. He showed the connection between local kernels and admis- sible sets. We extended here that connection with the notion of local conglom-erate, which in turn gives a twist to the notion of admissibility. In that respect,we plan to explore in the future the use of local conglomerates as a semanticsfor value-based argumentation frameworks [1], as suggested in Section 6. Acknowledgements The author appreciates the invitation of the Department of Mathe-matics of the Universidad de Aveiro, Portugal, for a visit in September 2019, during whicha previous version of this paper was presented. References 1. Bench-Capon, T.J.M.: Persuasion in practical argument using value-based argumentationframeworks. Journal of Logic and Computation (3), 429–448 (2003). DOI 10.1093/logcom/13.3.429. URL https://doi.org/10.1093/logcom/13.3.429 2. Beringer, T., Schindler, T.: A graph-theoretic analysis of the semantic paradoxes. TheBulletin of Symbolic Logic (4), 442–492 (2017). DOI 10.1017/bsl.2017.373. Caminada, M.: On the Issue of reinstatement in argumentation. In: Logics in ArtificialIntelligence, 10th European Conference, JELIA 2006, Liverpool, UK, September 13-15,2006, Proceedings, pp. 111–123 (2006). DOI 10.1007/11853886-114. Cook, R.T.: Patterns of paradox. Journal of Symbolic Logic (3), 767–774 (2004).DOI 10.2178/jsl/10969017655. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonicreasoning, logic programming and n-person games. Artificial Intelligence (2), 321–357(1995)6. Dyrkolbotn, S.: Argumentation, paradox and kernels in directed graphs. PhD Thesis,University of Bergen, Norway (2012)7. Galeana-S´anchez, H., Neumann-Lara, V.: On kernels and semikernels of digraphs. Dis-crete Mathematics (1), 67–76 (1984)8. Kripke, S.: Outline of a theory of truth. Journal of Philosophy (19), 690–716 (1975).DOI 10.2307/20246349. Rabern, L., Rabern, B., Macauley, M.: Dangerous reference graphs and semanticparadoxes. Journal of Philosophical Logic42