A formal analysis of enthymematic arguments Sjur K. Dyrkolbotn1 , Truls Pedersen2 , 1 Western Norway University of Applied Sciences 2 University of Bergen sdy@hvl.no, truls.pedersen@infomedia.uib.no Abstract to the shop and says “no, it’s not your lunch break yet, so you should not leave your office”, my assessment of this coun- We provide a simple formalisation of en- terargument depends rather crucially on whether or not I am thymematic arguments, based on formal ar- at home with my kids or at work with my colleagues. Still, gumentation theory. We start from a simple if I have not informed my adviser either way, I can hardly representation of arguments as sequences of blame her for assuming the worst and warning me accord- formulas and rules. Regular arguments are those ingly. Indeed, in a case like this, it seems clear that if I am that explicitly lists the conclusions of all rules misinterpreted, the burden to defend my conclusion falls on applied, while also establishing every premise of me. The AI adviser makes a case against leaving the office. every rule used. An enthymematic argument is What it said will attack any argument for going to the shop then defined as a sequence that does not satisfy that depends on the fact that I may leave my office. this property, but which can be extended to such In other words, the AI adviser attacks some interpretations a sequence in one or more ways. Borrowing of my argument. Since these are interpretations I did not rule terminology from the informal logic literature out, the attack succeeds also as an attack on my underspec- on enthymematic arguments, we then define the ified argument. This, in essence, is how we conceive of en- “crater” of an enthymematic argument as a set of thymematic arguments in this article, as partially explicated arguments, namely those that minimally extend arguments A, B such that A attacks B if some interpretation the enthymematic argument in the appropriate of A attacks some interpretation of B. We formalise this in way. We go on to propose a notion of attack the following, culminating in a Dung-style argumentation se- between enthymematic arguments, allowing us to mantics for enthymematic arguments that also leads to a sim- represent them as nodes in a Dung-style attack ple and natural characterisation of which such arguments we graph. We also prove a characterisation result, can accept. providing necessary and sufficient conditions for The structure of the paper is as follows. In Section 2 we the acceptance of an enthymematic argument under describe some of the philosophical history of the notion of Dung-style semantics based on admissible sets. enthymemes and the importance of them in relation to arti- ficial intelligence. In Section 3 we consider the question of 1 Introduction how enthymematic arguments should be defined, proposing a If I argue that I am hungry, so I should go to the store, you definition based on the theory of structured argumentation. In are unlikely to write me off as irrational. What I said makes Section 4 we define and discuss semantics for enthymematic sense, on the assumption that I am at home without food. It arguments in terms of abstract argumentation frameworks. In also makes sense if I am at home and there is some food, as Section 5 we present the main result of the paper, provid- long as the food I have is not food I want to eat. Similarly, ing necessary and sufficient conditions for the acceptance of what I said makes sense if I am not at home, but at work, if I enthymematic arguments. In Section 6 we offer a short con- forgot my lunch. There are countless variations on this theme, clusion and directions for future work. of course, providing a reasonable interpretation of what I said. I did not present a complete argument for the conclusion that I 2 Enthymemes should go to the store, but any charitable listener will be able The concept of an enthymeme was first discussed in Aristo- to fill in the gaps and form a meaningful hypothesis about tle’s studies of logic and rhetoric. It is meant to capture what my meaning. This is the typical situation in natural language, is “left in the mind” after an argument has been put forward when people argue and give reasons for actions. Complete (from Greek: en- “in” and thymos “mind”). Today, the term specifications are not feasible, but incomplete approximations is often associated with rhetoric, but it has also been stud- are commonplace and easily understood in most cases. ied in argumentation theory (see e.g., [Walton, 2008]). En- However, trouble can arise in case of misaligned interpreta- thymemes are informally described in various ways, such as tions. If my Google AI adviser hears my argument for going [Gilbert, 1991]: “Everyone agrees that an enthymeme is an argu- sions ought or must be left underspecified, these cases are not ment. Most writers also agree that enthymemes, the subject of our investigation. We believe they have more to even though they are formally invalid, are not do with ambiguity than with incompleteness, and we prefer to bad arguments simply as a result of being en- keep these distinct sources of uncertainty apart in the formal thymematic, but rather lack something that non- analysis. enthymemes do not.” What enthymemes lack that non-enthymemes do not vary 3 Defining enthymematic arguments according to different authors. Premises are particularly of- We assume that we are working with a propositional language ten said to be lacking, either because they are “unexpressed”, L , reasoning about formulas from this language using a set “suppressed”, “implicit”, “hidden”, or “unstated”. These nu- of strict rules S and a set of defeasible rules D. We assume ances aside, an enthymeme may also lack other elements than that every rule ri ∈ S ∪ D has the form ri = (Pi , ci ) where premises, such as the conclusion or even the rules applied. Pi ⊆ L is the set of premises of ri and ci ∈ L is its conclu- Such a lack may be similar to lacking part of what would con- sion. We introduce some simple projections for the premises stitute “warrant” in Toulmin’s framework [Toulmin, 2003]. and conclusions of rules, and permit them also to apply to Not everybody agrees that enthymemes are arguments. In formulas, mapping D ∪ S ∪ L → 2L and D ∪ S ∪ L → L , [Goddu, 2016] it is argued that lacking something essential respectively: is essential to enthymemes, making them incompatible with   what it means to be an argument. P if x = (Px , cx ) c if x = (Px , cx ) Metaphysical questions aside, enthymemic arguments are P(x) = x c(x) = x {x} otherwise x otherwise interesting for many reasons. To us, their potential applica- tion in artificial intelligence is of particular interest. Here, formalising enthymemes can provide a natural approach to Example 1. Consider the example from the intro- reasoning about agents who are unable or unwilling to pro- duction. Here are some propositions denoting the vide complete arguments and explanations for why they be- claims involved, as well as some additional claims lieve certain things or act in certain ways. A general inabil- that we will use to fill in the “gaps” of our argument: ity to completely specify one’s point of view seems endemic p1 : “I should go to the store” p5 : “I am hungry” to complex reasoners, so an ability to deal with incomplete p2 : “Fresh food in the fridge” p6 : “I am at home” specifications seems like a crucial feature for complex social p3 : “I have food I want to eat” p7 : “I am at work” agents. Furthermore, efficiency gains can be made by taking p4 : “I forgot my lunch” p8 : “Lunch break” an economic approach to reasoning and explanation, relying The rules we rely on are the following: on our ability to process enthymematic arguments rather than p3 , ¬p5 p7 , ¬p8 p3 , p5 asking always for the most “complete” picture possible. ∼∼∼∼∼∼ r1 , ∼∼∼∼∼∼ r2 , ∼∼∼∼∼ r3 p1 ¬n(r5 ) ¬p1 We should clarify at the outset that there is some disagree- p2 , ¬p6 p4 , p7 ment about whether an enthymeme is an argument that is ∼∼∼∼∼∼ r4 , ∼∼∼∼∼ r5 ¬p3 ¬p3 (i) implicitly describing premises, (ii) missing premises (iii) missing premises or a conclusion, or (iv) missing something The enthymeme the reader hypothetically accepted in the less specific (e.g., lacking in clarity or precision). We take introduction is E ? = (p5 , p1 ): “I am hungry, so I (should) go the position here that enthymemes may be missing premises to the store”. We are not mentioning the rule(s) we are ap- or explicit references to inference rules. However, we insist plying. Indeed, from the little information we have expressed that the conclusion is at least implicitly provided. There are we are not even able to apply any rules. These utterances technical and philosophical reasons for this assumption. The are sufficient for the recipient to “fill in the gaps”. Suppose technical reasons will appear when we model enthymemes our theory includes the observations that ¬p2 : “(There is no) in ways compatible with ASPIC+ and related argumentation fresh food in the fridge”, p4 : “I forgot my lunch”, p5 : “I frameworks [Modgil and Prakken, 2014]. In these frame- am hungry”, and p7 : “I am at work”. Then the theory sup- works, explicitly refering to the inference rules which the ar- ports the application of r5 yielding ¬p3 . Together with p5 guer applies in her argument exposes the argument for under- from the theory, we may now apply r1 and obtain p1 . The cutting arguments. enthymeme we stated can be expanded by making these ob- One philosophical reason is that for the kind of arguments servations explicit to form, for example, the explicit argument we are interested in trying to capture, it is a reasonable ex- E1 = (p5 , p4 , p7 , r5 , ¬p3 , p5 , r1 , p1 ). pectation that the arguer makes the conclusion known to his As is usual in the theory of structured argumentation, we audience. We believe the arguer has an obligation to make the rely on the definition of contrary formulas, : L → 2L , de- conclusion known in order to make the proposed argument noting the set of formulas that is contrary to a given formula. honestly open to refutation. If we permit missing conclusions Since the order of the reasoning rules we invoke can be it may become unreasonably difficult to attack a proposal, significant when evaluating enthymematic arguments, we will since a chain of “that is not what I meant”-defences can then represent arguments as sequences of rules rather than as proof be extended indefinitely. We believe these considerations are trees. Furthermore, we will also be interested in arguments important particularly in advanced AI-agents [Rahwan and that include redundant premises and rules, so we will not stip- Simari, 2009], as we will allude to in the running example. ulate that the rules occurring in an argument are all needed to While we do not deny that there are cases in which conclu- establish premises required by subsequent rules. In order to accomodate undercutting attacks, we name all affirms the conclusion of it. If it is a formula, then it affirms defeasible rules by a naming function similar to how ASPIC+ itself. This forces the conclusions of the applied rules to models this. For uniformity, we define the function n : D ∪ be explicitly listed in the argument after they are derived. S ∪ L → L , where we require that Finally, all of P(r5 ) and P(r1 ) occur in the sequence before • n(x) ∈ L if x ∈ D, the respective rules occur. • n(x) = > if x ∈ S, and Suppose we added p2 to E1 in Example 4. We get a new argument E10 = (p2 , p4 , p7 , r5 , ¬p3 , p5 , r1 , p1 ) which is identi- • n(x) = x if x ∈ L . cal to E1 but with the proposition p2 appended to it. Since p2 This means that every defeasibly rule is named, strict rules is in the theory, E10 is still complete. However, p2 does not oc- have a vacuous name and every formula names itself. This cur as a premise of any of the applied rules, nor is if otherwise notion is altered slightly from ASPIC+ ’s terminology, but not connected with the conclusion. Furthermore, E10 has E1 as a substantially. strict complete subsequence. In what follows, we will gener- Formally speaking, any argument A in our formalism will alise this observation to formalise the notion of a minimally be instantiated by a sequence A = (x1 , x2 , . . . , xn ) of rules and complete argument. This, in turn, will serve as a basis for formulas, satisfying the constraints in Definition 1. our formal definition of what it means to be an enthymematic Definition 1. Given a theory T argument. For any positive integer n, we let [n] denote the set of natu- • An argument based on T (using D and S) is a sequence ral numbers between 1 and n. A sequence A = (x1 , x2 , . . . , xn ) A = (x1 , x2 , . . . , xn ) such that ∀1 ≤ i ≤ n : xi ∈ D ∪ S ∪ L . can then be conventionally written as (xi )i∈[n] . We say that • An argument A = (x1 , x2 , . . . , xn ) is said to be complete a function f : [n] → [m] from sets of positive integers to if sets of positive integers is strictly increasing if f (x) < f (y)  x ∈ T, or whenever x < y, for x, y ∈ [n]. Then the notion of a sub- – ∀1 ≤ i ≤ n : i sequence is formally defined by the condition that A  B ∀φ ∈ P(xi ) : φ ∈ {c(x j ) | 1 ≤ j < i} and for arguments A = (x1 , x2 , . . . , xn ) and B = (y1 , y2 , . . . , ym ) iff there is a strictly increasing function f : [n] → [m] such that – ∀1 ≤ i ≤ n : ∃i ≤ j ≤ n : c(xi ) = x j . ∀i ∈ [n] : xi = y f (i) . That is, A  B if A can be obtained from The conditions on complete argument can be intuitively jus- B by deleting rules or formulas. Equivalently, B is obtained tified as follows. The first condition requires that an element from A by filling in the gaps with rules or formulas. is either the explication of a formula in the theory, or that This points to a formalisation of the intuition we had about all formulas/premises this element relies on has already been the meaning of enthymematic arguments. Specifically, we ar- established earlier in the sequence. The second requirement rive at the following general definition of the crater of an en- states that the conclusion of every rule must be explicated at thymematic argument (see for example [Paglieri and Woods, some point after the rule has been applied. 2011]). We define the conclusion of A = (x1 , x2 , . . . , xn ) as c(A) = c(xn ). That is, if A ends with a rule then the conclusion of Definition 2. For all arguments A ∈ A, we say that A is min- A is the conclusion of the final rule applied in A. If, on the imally complete if there is no complete argument B ≺ A such other hand, A ends with a formula, the conclusion of A is this that c(A) = c(B). formula. The set of all arguments based on T is denoted by Definition 3. Given an argument A = (x1 , x2 , . . . , xn ) based on A, while the set of complete arguments is denoted by A. T, the crater of A, denoted I(A) contains all minimally com- When we do not need to reference individual rules, we gen- plete arguments B = (y1 , y2 , . . . , ym ) such that c(A) = c(B) and erally use upper-case letters like A, B,C etc. to denote argu- either A  B or B  A. ments. However, any such abstract argument corresponds to Definition 4. For any argument A, we say that A is an actual sequence of rules meeting the requirements from Definition 1. • incoherent if I(A) = 0. / Example 2 (Example 1 continued). Continuing from the pre- • enthymematic if ∃B ∈ I(A) : A ≺ B, vious example, it is easy to verify that the uttered E ? complies • regular if I(A) = {A} with the weakly constrained definition of an argument: every element in the sequence is either a rule or a formula. It is not • superfluous if ∃B ∈ I(A) : B ≺ A. a complete argument, however. We have P(p5 ) = {p5 } ⊆ T, That is, A is an enthymematic argument if its crater con- but P(p1 ) = {p1 }, and p1 is neither in the theory nor is it (the tains a minimally complete argument that extends A. This conclusion of) an earlier element. definition rules out other forms of incompleteness in argu- After we filled in the gaps to obtain E1 = mentation, e.g., cases where the expression used to express (p5 , p4 , p7 , r5 , ¬p3 , r1 , p1 ) we obtained an argument sat- a rule is ambiguous so that two or more interpretations are isfying every condition used to characterise a complete possible. At the same time, the definition rules out arguments argument. Every element in the sequence is such that, if it is that cannot be completed, as well as complete arguments that a formula, then it is the conclusion of a previously applied contain redundant rules or formulas (i.e., arguments that are rule or already a part of the theory. Also, every element superfluous). Such arguments A have craters that are empty xi satisfies the condition that it or some later element x j or contain subsequences of A. As a first step towards unpacking the definition further, we AI perspective on E ? . Suppose the theory of the AI con- record the following simple claim. tains ¬p8 (no lunch break), because the AI knows that it Proposition 1. An argument is regular if, and only if, it is is not lunch time. Then the AI can form the argument minimally complete. G = (¬p8 , r7 , p7 , r2 , ¬n(r5 )). This argument involves the rule r2 , which can be used to undercut r5 (intuitively, the argument Proof. By Definition 3, I(A) contains all minimally complete tells me not to think about food at all when it is not my lunch B such that A  B or B  A. By Definition 2, an argument break). It is easy to verify that G is regular, i.e., its crater A is minimally complete if, and only, if there is no complete consists of G itself. In the next section, we define a seman- B ≺ A such that c(A) = c(B). Hence, if A is minimally com- tics according to which G also attacks E ? in this case, since it plete, there is no minimally complete B ≺ A. Furthermore, attacks F1 . there is no minimally complete B such that A ≺ B, since A being minimally complete contradicts any such B being so. 4 Semantics Since A  A, it follows that I(A) = {A} if, and only if, A is minimally complete. We let A denote the set of all regular arguments, namely all A such that I(A) = {A}. The set A, meanwhile, denotes all It is also easy to show that any argument belongs to exactly arguments (so that A ⊆ A). one of the categories listed in Definition 4. Specifically, this follows as a corollary of the following simple observation. Definition 5. We define two relations of attack as follows: Proposition 2. For all A ∈ A, we have: • For all A, B ∈ A we define R such that (A, B) ∈ R if, and only if, • a) ∃B ∈ I(A) : B ≺ A ⇒ ∀B ∈ I(A) : B ≺ A and ∃x ∈ B : c(A) ∈ c(x) ∪ n(x) • b) ∃B ∈ I(A) : A ≺ B ⇒ ∀B ∈ I(A) : A ≺ B Proof. For a), assume B ∈ I(A) with B ≺ A. Assume towards • For all A, B ∈ A we define R such that (A, B) ∈ R if, and contradiction that there is some C ∈ I(A) with A  C. By Def- only if, inition 3, this means that C is minimally complete. Since ≺ is – I(B) = 0/ or transitive, we get B ≺ C. But by Definition 2, this contradicts – ∃A0 ∈ I(A) : ∃B0 ∈ I(B) : (A0 , B0 ) ∈ R the fact that C is minimally complete. The argument for b) is similar. It is easy to see that R and R agree on the notion of attack for regular arguments. Corollary 1. Any argument A is exactly one of the following: incoherent, enthymematic, regular, or superfluous. Proposition 3. For all A, B ∈ A, we have (A, B) ∈ R if, and only if (A, B) ∈ R. Proof. Consider arbitrary A ∈ A. Obviously, A belongs to at least one of the categories defined in Definition 4. Fur- Proof. Since A, B ∈ A, we have I(A) = {A} and I(B) = {B}. thermore, the claim that A belongs to only one of these cat- Hence, the claim follows by Definition 5. egories is obviously true if A is incoherent or regular. If A is enthymematic, then it is clearly not incoherent or regular. In other words, R ⊆ R, so that R extends the attack rela- Moreover, it follows by Proposition 2 a) that A is not super- tion to enthymematic arguments. Viewing the set (A, R) as an fluous either. Similarly, if A is superfluous, it is obivously not abstract argumentation framework, this means that we obtain incoherent or regular. Furthermore, it follows by Proposition semantics also for enthymematic arguments. 2 b) that it is not enthymematic either. Definition 6. Assume given a pair (X, R) where R ⊆ X × X. Example 3 (Example 1 continued). Continuing from the pre- Then we have the following argumentation semantics for vious example, it is easy to verify that E ? is indeed en- (X, R): thymematic according to Definition 4. First, notice that its Admissible Adm(X, R) = {S | ∀A, B ∈ S : (A, B) 6∈ R & ∀A ∈ crater is I(E ? ) = E1 where E1 is a set of minimally com- S : ∀B ∈ R− (A) : ∃C ∈ S : (C, B) ∈ R}. plete arguments that contain all permutations of the inter- nal elements of E1 that still result in an acceptable elab- Complete Com(X, R) = {S ∈ Adm(X, R) | S = S} where for oration of E ? (the order of the “missing” elements does all S ⊆ X, not matter, as long as we get a minimally complete ar- gument that extends E ? ). To illustrate when we can en- S = S ∪ {A ∈ X | ∀B ∈ R− (A) : ∃C ∈ S : (C, B) ∈ R}. counter craters with semantically distinct objects, assume that we replace p6 , p7 by the default rules r6 : (>, p6 ), r7 : Preferred Pref(X, R) = {S ∈ Adm(X, R) | ∀S0 ∈ Adm(X, R) : (>, p7 ). This is a possible encoding of the state of an AI S 6⊂ S0 }. adviser who has defeasible reasons to think both that I am Grounded Ground(X, R) = {S ∈ Com(X, R) | ∀S0 ∈ at home and that I am at work (this encodes uncertainty, Com(X, R) : S0 6⊂ S}. in argumentative terms). In this case, the crater of E ? in- cludes variants of both F1 = (p5 , p4 , r7 , p7 , r5 , ¬p4 , r1 , p1 ) Proposition 4. Let A, B ∈ A and assume there is some A0 ∈ and F2 = (p5 , r6 , p6 , ¬p2 , r4 , ¬p3 , r1 , p1 ). This encodes the I(A) that attacks B. Then every A0 ∈ I(A) attacks B. Proof. First notice that we have c(A1 ) = c(A2 ) for all A1 , A2 ∈ A0 , B0 . Since S is admissible, it follows that there is some I(A). That is, all A0 ∈ I(A) have the same conclusion. Notice, C ∈ S such that (C, A0 ) ∈ R. However, since A ∈ Ext(S) it moreover, that for all A1 , A2 , B ∈ A, if c(A1 ) = c(A2 ), then follows by Definition 7 that I(A) ⊆ S. Hence, A0 ,C ∈ S, con- (A1 , B) ∈ R ⇔ (A2 , B) ∈ R. This is because the conclusion of tradicting independence of S. A uniquely determines which arguments A attack. From this, Self-defence: Let (A, B) ∈ R for some arbitrary B ∈ Ext(S). the claim follows: if there is some argument in the crater of A We have to show that Ext(S) attacks A. By Definition 5 and that attacks B, then every argument in the crater of A attacks the fact that A attacks B, we know there is some A0 ∈ I(A) B. that attacks some B0 ∈ I(B). By Definition 7, we know that I(B) ⊆ S. Since S is admissible it then follows that there Example 4 (Example 1 continued). Once again, consider E ? , is C ∈ S such that (C, A0 ) ∈ R. By Definition 7, we have from the perspective of the AI (such that default rules r6 , r7 C ∈ Ext(S). Moreover, by Definition 5, we get (C, A) ∈ R. can be used to argue for p6 , p7 respectively). The crater con- Hence, Ext(S) attacks A as desired. sists of variants of F1 and F2 . Since F1 involves the rule r5 , 2) Assume that S is an admissible set in (A, R). We have to G = (¬p8 , r7 , p7 , r2 , ¬n(r5 )) attacks F1 , so by Definition 5 it show that Res(S) = S ∩ A is independent and defends itself in also attack E ? . Notice how adding ¬p7 to the knowledge (A, R). base of the AI would prevent this attack on E ? . After the ad- Independence: Assume towards contradiction that there is ditional knowledge is added, G is no longer a minimally com- A, B ∈ Res(S) such that (A, B) ∈ R. By Definition 7 we have plete argument. However, if I extend my enthymematic argu- A0 , B0 ∈ S such that A ∈ I(A0 ), B ∈ I(B0 ). It follows by Defini- ment E ? to another enthymematic argument F ? = (p5 , p6 , p1 ) tion 5 that (A0 , B0 ) ∈ R, contradicting independence of S. I achieve the same effect, since now only variants of F1 is in Self-defence: Consider arbitrary (A, B) ∈ R such that B ∈ the crater. This shows how a formal model of enthymematic Res(S). By Definition 7 there is some B0 ∈ S such that argumentation will allow us to deal with more economically B ∈ I(B0 ). Furthermore, by Definition 5 we have (A, B0 ) ∈ R. expressed arguments in a systematic way, accounting for the semantic effects of leaving arguments underspecified. Since S defends itself, there is some C ∈ A such that (C, A) ∈ R. By Definition 5 this means that there is some C0 ∈ I(C) and some A0 ∈ I(A) such that (C0 , A0 ) ∈ R. Since A ∈ A, we have 5 A characterisation result I(A) = {A}, which implies A = A0 . Furthermore, by Defini- Definition 7. Given a set S ⊆ A, we define the following sets tion 7 we have C0 ∈ Res(S). Hence, Res(S) defends B against of corresponding arguments. A. Since (A, B) as arbitrarily chosen, the claim follows. 1. Ext(S) = {A | 0/ ⊂ I(A) ⊆ S}. S Theorem 2. For all theories T and all (A, R) and (A, R) 2. Res(S) = A∈S I(A) based on T, if ε ∈ {Com, Pref, Ground} have the following: Hence, Ext(S) collects all arguments whose craters are subsets of S. In general, we may have S 6⊆ Ext(S), namely 1) S ∈ ε(A, R) ⇒ Ext(S) ∈ ε(A, R) if, and only if, (?) there is some A ∈ S such that I(A) 6⊆ S. 2) S ∈ ε(A, R) ⇒ Res(S) ∈ ε(A, R) However, if S ⊆ A, then S ⊆ Ext(S), since I(A) = {A} for Proof. ε = Com: 1) Assume S ∈ Com(A, R). We have to all A ∈ A. In this case, Ext(S) is an extension of S. Res(S), show Ext(S) ∈ Com(A, R). By Theorem 1 we know that meanwhile, takes S ∈ A and returns the union of all craters Ext(S) is admissible, so we only have to show that it is of elements in S. Hence, Res(S) ⊆ A, providing in all cases complete. Assume towards contradiction that there is some a projection of S onto the set of regular arguments. Notice, A ∈ A \ Ext(S) such that moreover, that we have S ∩ A ⊆ Res(S), since elements of A are their own craters. It should also be noted that (?) is the ∀(B, A) ∈ R : ∃C ∈ Ext(S) : (C, B) ∈ R. case if, and only if, Res(S) 6⊆ S ∩ A (which is equivalent to Res(S) 6= S ∩ A). By A 6∈ Ext(S) and Definition 7 we must have A 6∈ S and We can now prove the following characterisation theorem, some A0 ∈ I(A) such that A0 6∈ S. Consider arbitrary B ∈ A showing us how to get from admissible sets of arguments such that (B, A0 ) ∈ R. Then by Definition 5 we get (B, A) ∈ R. Since Ext(S) defends A there must be some C ∈ Ext(S) such in (A, R) to admissible sets of arguments in (A, R) and vice that (C, B) ∈ R. But then there is also C0 ∈ I(C) such that versa. (C0 , B) ∈ R. From Definition 7 it follows that C0 ∈ S, so that Theorem 1. For all theories T and all (A, R) and (A, R) S defends A0 . Since S is complete and B was arbitrary this based on T, we have the following: implies A0 ∈ S, contrary to assumption. 1) S ∈ Adm(A, R) ⇒ Ext(S) ∈ Adm(A, R) 2) Assume S ∈ Com(A, R). We have to show Res(S) ∈ Com(A, R). By Theorem 1 we know that Res(S) ∈ Adm(A, R) 2) S ∈ Adm(A, R) ⇒ Res(S) ∈ Adm(A, R) so we only have to show completeness. Assume towards con- Proof. 1) Assume that S is an admissible set in (A, R) and tradiction that there is some A ∈ A \ Res(S) such that: consider Ext(S). We have to show that Ext(S) is independent ∀(B, A) ∈ (A, R) : ∃C ∈ Res(S) : (C, B) ∈ (A, R) and defends itself in (A, R). Independence: Let A, B ∈ Ext(S) and assume towards con- By Definition 7 we must have A 6∈ S, since otherwise tradiction that (A, B) ∈ R. By Definition 5 this means that I(A) = {A} forcing A ∈ Res(S). Consider arbitrary B ∈ A some A0 ∈ I(A) attacks some B0 ∈ I(B). We choose some such such that (B, A) ∈ R. By Definition 5 there must be B0 ∈ I(B), A0 ∈ I(A) such that (B0 , A0 ) ∈ R. Furthermore, sequence can be extended to one or more regular arguments, since A ∈ A, we have I(A) = {A} so A0 = A. It follows that we took it to be an enthymematic argument. We showed that (B0 , A) ∈ R. By the assumption that Res(S) defends A we this fits well with the traditional view on such arguments, have C ∈ Res(S) such that (C, B0 ) ∈ R. Since C ∈ Res(S), whereby they are conceived of as arguments that are miss- there must be some C0 ∈ S such that C ∈ I(C0 ). But then by ing certain components (typically premises). In addition, we Definition 5 we have (C0 , B0 ) ∈ R. Since B was arbitrary and showed how this formalisation allows us to provide a seman- C0 ∈ S, S defends A in (A, R). Hence, by completeness of S tics for enthymematic arguments in terms of abstract argu- we get A ∈ S, contrary to assumption. mentation frameworks. Furthermore, we characterised the class of acceptable argument sets under admissible, complete, ε = Pref: 1) Assume S ∈ Pref(A, R). We have to show preferred, and grounded semantics, by relating acceptability that Ext(S) ∈ Pref(A, R). Assume towards contradic- over the regular argument with acceptability over the set of all tion that there is some admissible S0 ⊃ Ext(S). Then by sequences of rules and formulas (including incoherent and su- Definition 7 there is some A ∈ S0 with I(A) 6⊆ S. Since perfluous arguments alongside the regular and enthymematic S ⊆ A it follows from Definition 7 that S ⊆ Ext(S) so that ones). Res(Ext(S)) = S. Hence, Res(S0 ) ⊃ S (the inclusion is strict The basic intuition behind our semantics is that an en- since S 6⊇ I(A) ⊆ Res(S0 )), which by Theorem 1 contradicts thymematic argument corresponds to a set of regular argu- the fact that S ∈ Pref(A, R). ments, namely the set of arguments corresponding to ways in 2) Assume S ∈ Pref(A, R). We have to show that which to “fill the gaps”. This lead to the notion of a crater, Res(S) ∈ Pref(A, R). Assume towards contradiction which is based on a similar notion from the informal logic that there is some admissible S0 ⊃ Res(S). By Definition 7 literature. In our opinion, the idea that enthymematic argu- we have Ext(Res(S)) ⊇ S. Hence, Ext(S0 ) ⊇ S. Furthermore, ments have craters is very natural. Moreover, it leads natu- since S0 ⊃ Res(S) there must be A ∈ S0 \ Res(S). Hence, from rally to a semantics where you attack an enthymematic argu- Definition 7 we get Ext(S0 ) ⊃ S. But by Theorem 1 we have ment if you attack at least one of the regular arguments in its crater. It seems to us that the resulting formalism has sig- that Ext(S0 ) ∈ Adm(A, R), contradicting S ∈ Pref(A, R). nificant potential when it comes to integrating enthymematic arguments into the computational theory of argumentation. ε = Ground: 1) Assume S ∈ Ground(A, R). We have In future work, we plan to explore the definitions we have to show that Ext(S) ∈ Ground(A, R). Assume towards presented here in further depth, including an investigation of contradiction that there is some complete S0 ⊂ Ext(S). what happens when we add preferences to the formalism, in Consider Res(S0 ). We have Res(Ext(S)) = S, so Res(S0 ) ⊆ S. the style of ASPIC+ . Since Res(S0 ) is complete, we must have Res(S0 ) = S. Now, consider A ∈ Ext(S) \ S0 . Consider arbitrary B ∈ A such References that (B, A) ∈ R. By Definition 7 and A ∈ Ext(S), we have I(A) ⊆ S. By Definition 5, we have some A0 ∈ I(A), B0 ∈ I(B) [Gilbert, 1991] Michael A. Gilbert. The enthymeme buster: such that (A0 , B0 ) ∈ R. But since S is admissible and I(A) ⊆ S, A heuristic procedure for position exploration in dialogic this implies that there is C ∈ S such that (C, B0 ) ∈ R. However, dispute. Informal Logic, 13(3), 1991. from I(C) = {C} and Res(S0 ) = S it follows that C ∈ S0 . [Goddu, 2016] G. C. Goddu. On the very concept of an en- Hence, S0 defends A against the attack from B. Since B thymeme. 2016. was arbitrary and S0 is complete, we get A ∈ S0 , contrary to [Modgil and Prakken, 2014] Sanjay Modgil and Henry assumption. Prakken. The ASPIC+ framework for structured argumen- 2) Assume S ∈ Ground(A, R). By Definition 7, we see that tation: a tutorial. Argument & Computation, 5(1):31–62, Ext(Res(S) ⊇ S. Note that Ext(Res(S)) ⊆ S. To see this, 2014. let A ∈ Ext(Res(S) be arbitrary. By Definition 7 we have I(A) ⊆ Res(S). Hence, for every A0 ∈ I(A) there is some [Paglieri and Woods, 2011] Fabio Paglieri and John Woods. A00 ∈ S such that A0 ∈ I(A00 ). Consider arbitrary B ∈ A Enthymemes: From reconstruction to understanding. Ar- such that (B, A) ∈ R. Then by Definition 5 there is some gumentation, 25(2):127–139, May 2011. A0 ∈ I(A), B0 ∈ I(B) such that (B0 , A0 ) ∈ R. But then there is [Rahwan and Simari, 2009] Iyad Rahwan and Guillermo also A00 ∈ S such that (B, A00 ) ∈ R, which in turn means there Simari, editors. Argumentation in artificial intelligence. is C ∈ S such that (C, B) ∈ R. Hence, S defends A, so that Springer, 2009. A ∈ S follows from the fact that S is complete. [Toulmin, 2003] Stephen Toulmin. The Uses of Argument. Cambridge University Press, 2 edition, 2003. First edition 6 Conclusion from 1958. [Walton, 2008] Douglas Walton. The three bases for the en- In this paper, we have proposed a formalisation of en- thymeme: A dialogical theory. J. Applied Logic, 6(3):361– thymematic arguments in a framework of structured argu- 379, 2008. mentation with Dung-style semantics. Representing argu- ments as sequences made up of formulas and rules, we con- sidered sequences that for some reason do not satisfy those properties “regular” arguments are expected to satisfy. If a