Introduction to Structured Argumentation

Transcription

1 Introduction to Structured Argumentation Anthony Hunter Department of Computer Science, University College London, UK September 6, / 134

2 Computational models of argument 1 Introduction 2 Framework for deductive argumentation Definitions with examples of instances General properties 3 Defeasible reasoning 4 Meta-level argumentation Representing argument schema 5 From natural language arguments to formalized arguments Understanding arguments in free text Handling enthymemes 6 Conclusions and further directions 2 / 134

3 Computational models of argument Abstract argumentation Structured Argumentation Decision making Sense making Dialogical Argumentation Persuasion Negotiation Distributed decision making [Besnard + Hunter 2008] 3 / 134

4 Abstract argumentation: Graphical representation Graphical representations of argumentation have a long history (see for example Wigmore, Toulmin, etc. ) A 1 = Patient has hypertension so prescribe diuretics A 2 = Patient has hypertension so prescribe betablockers A 3 = Patient has emphysema which is a contraindication for betablockers [See Dung (AIJ 1995)] 4 / 134

5 Abstract argumentation: Pros and cons Pros An argument graph provides a simple and intuitive representation of a controversial topic. Various dialectical semantics by Dung, and others, provide valuable insights into the nature of argumentation. Various tools have been developed for analysing arguments in terms of abstract argumentation. Cons Arguments in abstract argumentation are atomic. They cannot be broken down. They do not have an internal structure. They cannot be combined. There is no formal definition for arguments or attacks. 5 / 134

6 Approaches to structured argumentation Origins of research in structured argumentation John Pollock pioneered the formalization of arguments and counterarguments using logic (See Pollock 1987). Some frameworks for structured argumentation Deductive argumentation (Hunter, Besnard, Cayrol, Amgoud, et al) Defeasible logic programming (Simari, et al) Assumption-based argumentation (Toni, et al) ASPIC+ (Prakken, et al) See the special issue in Argument & Computation, volume 5 (1), 2014, for tutorials on each of these frameworks. 6 / 134

7 Overview of deductive argumentation Descriptive graphs Generative graphs Counterarguments Arguments Base logic [Besnard + Hunter 2014] 7 / 134

8 Base logic Choice of base logic Here we focus on simple logic and classical logic, but other options include non-monotonic logics, conditional logics, temporal logics, description logics, and paraconsistent logics. A few definitions for base logic Let L be a language for a logic, and let i be the consequence relation for that logic. If α is an atom in L, then α is a positive literal in L and α is a negative literal in L. For a literal β, the complement of β is defined as follows: If β is a positive literal, i.e. it is of the form α, then the complement of β is the negative literal α, if β is a negative literal, i.e. it is of the form α, then the complement of β is the positive literal α. 8 / 134

9 Arguments Definition for deductive argument Given a base logic i, a deductive argument is an ordered pair Φ, α where Φ i α. Φ is the support, or premises, or assumptions of the argument, and α is the claim, or conclusion, of the argument. For an argument A = Φ, α, the function Support(A) returns Φ and the function Claim(A) returns α. Example {report(rain), report(rain) carry(umbrella)}, carry(umbrella) {study(sid, logic), study(sid, logic)}, KingOfFrance(Sid) 9 / 134

10 Arguments The consistency constraint An argument Φ, α satisfies the consistency constraint when Φ is consistent. Example If we assume the consistency constraint, then the following are not arguments. {study(sid, logic), study(sid, logic)}, study(sid, logic) study(sid, logic) {study(sid, logic), study(sid, logic)}, KingOfFrance(Sid) Consistency constraint is not essential If we assume the base logic is a paraconsistent logic (such as Belnap s four valued logic), and we do not impose the consistent constraint, then the following are arguments. {study(sid, logic) study(sid, logic)}, study(sid, logic) {study(sid, logic) study(sid, logic)}, study(sid, logic) 10 / 134

11 Arguments The minimality constraint An argument Φ, α satisfies the minimality constraint when there is no Ψ Φ such that Ψ i α. Example If we assume the minimality constraint, then the following is not an argument. {report(rain), report(rain) carry(umbrella), happy(sid)}, carry(umbrella) 11 / 134

12 Arguments based on simple logic Simple logic Simple logic is based on a language of literals and simple rules where each simple rule is of the form α 1... α k β where α 1 to α k and β are literals. The consequence relation is modus ponens (i.e. implication elimination) as defined next. s β iff there is an α 1 α n β and for each α i {α 1,..., α n} either α i or s α i Example Let = {a, b, a b c, c d}. Hence, s c and s d. However, s a and s b. 12 / 134

13 Arguments based on simple logic Simple argument Let be a simple logic knowledgebase. For Φ, and a literal α, Φ, α is a simple argument iff Φ s α and there is no proper subset Φ of Φ such that Φ s α. Example Let p 1, p 2, and p 3 be the following formulae. p 1 = oilcompany(bp) p 2 = goodperformer(bp) p 3 = oilcompany(bp) goodperformer(bp)) goodinvestment(bp) Then {p 1, p 2, p 3}, goodinvestment(bp) is a simple argument. 13 / 134

14 Arguments based on classical logic Classical logic argument A classical logic argument from a set of formulae is a pair Φ, α such that 1 Φ 2 Φ 3 Φ α 4 there is no Φ Φ such that Φ α. Example The following classical argument uses a universally quantified formula in contrapositive reasoning to obtain the following claim about number 77. { X.multipleOfTen(X) even(x), even(77)}, multipleoften(77) 14 / 134

15 Counterarguments based on simple logic Rebut and undercut for simple logic For simple arguments A and B, we consider the following type of simple attack: A is a simple undercut of B if there is a simple rule α 1 α n β in Support(B) and there is an α i {α 1,..., α n} such that Claim(A) is the complement of α i A is a simple rebut of B if Claim(A) is the complement of Claim(B) Example A 1 = {efficientmetro, efficientmetro usemetro}, usemetro A 2 = {strikemetro, strikemetro efficientmetro}, efficientmetro A 3 = {govdeficit, govdeficit cutgovspending}, cutgovspending A 4 = {weakeconomy, weakeconomy cutgovspending}, cutgovspending 15 / 134

16 Counterarguments based on simple logic Example (Defeasible reasoning) The first argument A 1 captures the general rule that if workday holds, then usemetro(sid) holds. A 1 = {workday, normal, workday normal usemetro(sid)}, usemetro(sid) A 2 = {workathome(sid), workathome(sid) normal}, normal Here we use normal as an assumption of normality for using the rule. 16 / 134

17 Counterarguments based on classical logic Counterarguments for classical logic If Φ, α and Ψ, β are arguments, then Φ, α rebuts Ψ, β iff α β Φ, α undercuts Ψ, β iff α Ψ for some Ψ Ψ Direct undercut A direct undercut for an argument Φ, α is an argument of the form Ψ, φ i where φ i Φ. Example (Using classical logic) {β, β α}, α rebuts {γ, γ α}, α {γ, γ β}, (β (β α)) undercuts {β, β α}, α {δ β}, β is a direct undercut for {α, β}, α β 17 / 134

18 Counterarguments based on classical logic A rebut denotes a disagreement with the claim, whereas an undercut denotes a disagreement with the support (i.e. a disagreement of the explanation or justification). Example a = garlic is horrible b = this dish contains garlic c = this dish is horrible {a, b, a b c}, c { c}, c { a}, a { a c}, a 18 / 134

19 Counterarguments based on classical logic Example (Integrity constraint as a counterargument) Essentially, the attack says that the flight cannot be both a low cost flight and a luxury flight. {lowcostfly, luxuryfly, lowcostfly luxuryfly goodfly}, goodfly { lowcostfly luxuryfly}, lowcostfly luxuryfly 19 / 134

20 Counterarguments based on classical logic Example (Using first-order predicate formulae) Because Tweety is a bird, and birds fly, there is a bird that flies. But, there is a bird that doesn t fly, and so it is not the case that all birds fly. {bird(tweety), X.bird(X) fly(x)}, X.bird(X ) fly(x) { X.bird(X ) fly(x )}, X.bird(X) fly(x) Example (Also using first-order predicate formulae) Some students know nothing. But, they all know their own name { X. Y. knows(x, Y)}, X. Y. knows(x, Y) { X.knows(X, name(x))}, X, Y.knows(X, Y) 20 / 134

21 Counterarguments based on classical logic Some kinds of classical attack Let A and B be two classical arguments. A is a classical defeater of B if Claim(A) Support(B). A is a classical direct defeater of B if φ Support(B) s.t. Claim(A) φ A is a classical undercut of B if Ψ Support(B) s.t. Claim(A) Ψ A is a classical direct undercut of B if φ Support(B) s.t. Claim(A) φ A is a classical canonical undercut of B if Claim(A) Support(B). A is a classical rebuttal of B if Claim(A) Claim(B). A is a classical defeating rebuttal of B if Claim(A) Claim(B). 21 / 134

22 Counterarguments based on classical logic Example (Attack relations) {a b, c}, (a b) c is a classical defeater of { a, b}, a b {a b, c}, (a b) c is a classical direct defeater of { a b}, a b { a b}, (a b) is a classical undercut of {a, b, c}, a b c { a b}, a is a classical direct undercut of {a, b, c}, a b c { a b}, (a b c) is a classical canonical undercut of {a, b, c}, a b c {a, a b}, b c is a classical rebuttal of { a b, c}, (b c) {a, a b}, b is a classical defeating rebuttal of { a b, c}, (b c) 22 / 134

23 Counterarguments based on classical logic Hierarchy of arrack relations Defeater Direct defeat Undercut Direct rebut Direct undercut Canonical undercut Rebut An arrow from D 1 to D 2 indicates that D 1 D / 134

24 Argument graphs Approaches to constructing logical argument graphs Descriptive graphs Here we assume that the structure of the argument graph is given, and the task is to identify the premises and claim of each argument. Therefore the input is an abstract argument graph, and the output is an instantiated argument graph. Generative graphs Here we assume that we start with a knowledgebase (i.e. a set of logical formula), and the task is to generate the arguments and counterarguments (and hence the attacks between arguments). Therefore, the input is a knowledgebase, and the output is an instantiated argument graph. 24 / 134

25 Argument graphs descriptive Abstract graph Instantiated graph Arguments Attacks generative Knowledgebase Approaches to constructing logical argument graphs 25 / 134

26 Argument graphs Example (Abstract graph and descriptive graph) The flight is low cost and luxury, therefore it is a good flight A flight cannot be both low cost and luxury A 1 = {lowcostfly, luxuryfly, lowcostfly luxuryfly goodfly}, goodfly A 2 = { (lowcostfly luxuryfly)}, lowcostfly luxuryfly 26 / 134

27 Argument graphs Example (Abstract graph) A 1 = Patient has hypertension so prescribe diuretics A 2 = Patient has hypertension so prescribe betablockers A 3 = Patient has emphysema which is a contraindication for betablockers 27 / 134

28 Argument graphs Example (Descriptive graph using classical logic with integrity constraint) bp(high) ok(diuretic) bp(high) ok(diuretic) give(diuretic) ok(diuretic) ok(betablocker) bp(high) ok(betablocker) bp(high) ok(betablocker) give(betablocker) ok(diuretic) ok(betablocker) give(diuretic) ok(betablocker) give(betablocker) ok(diuretic) symptom(emphysema), symptom(emphysema) ok(betablocker) ok(betablocker) 28 / 134

29 Argument graphs Example (Generative graph using simple logic) Let = {a, b, c, a c a, b c, a c b}. {a, c, a c a}, a {a, c, a c b}, b {b, b c}, c 29 / 134

30 Argument graphs Example (Generative graph using classical logic) Consider = {a, b, a a, b a, a b}, let the arguments be those that involves one or more rules. {c, b c}, b {b, b c}, c {b, a b}, a {a, a b}, b {c, c a}, a {a, c a}, c 30 / 134

31 Summary of deductive argumentation Specification of a deductive argumentation system Base logic Specification of language and of consequence relation (proof theoretic) or entailment relation (semantic) for defining arguments. Arguments Specification of conditions for premises and claims of arguments. Attack Specification of claim of attacker and its relationship with premises or claim of attackee. Generative graph Specification of which arguments and attacks (that can be generated from knowledgebase) are to appear in the graph. Most applications also need meta-level information about formulae (e.g. preferences, probability assignments, ethical value judgments, etc.) which are harnessed in the specification of a deductive argumentation system. 31 / 134

32 Comparison with ASPIC+ and ABA Controversial statement: The difference between ABA, ASPIC+, and deductive argumentation is not substantial. All provide structure to arguments with some/all premises and claim. All implicitly/explicitly use a base logic. All have notions of counterargument. All can instantiate abstract argument graphs. All can capture defeasible reasoning. All can be implemented. 32 / 134

33 Comparison with ASPIC+ and ABA There are some differences ASPIC+ has proof structure in argument. ASPIC+ use names to block application of rules. ASPIC+ distinguishes between strict and defeasible information. ASPIC+ and ABA do not put all the premises in the argument. ASPIC+ and ABA use rules for inference rules and for object-level information. ABA is tightly coupled with abstract argumentation. ASPIC+ and ABA use simple logic (or similar) for most examples. ASPIC+ is perhaps the most fully developed for rule-based applications. However, most of what can be done in practice in one approach can be done in the other approaches. 33 / 134

34 Properties Overview of properties of deductive argumentation systems Deductive argumentation offers numerous choices for the specification of base logic, argument, attack, and graph construction. Need to compare and contrast these choices Need to ensure choices are well-understood and well-behaved. We will consider the following kinds of property Attack properties concern classifications of attack relations Extension properties concern properties of the formulae in the support of arguments in extensions Structural properties concern the types of graph that can be constructed by given different choices for argument and attack relation. 34 / 134

35 Attack properties Some desirable properties of attack relations Name Property D0 if A A, B B then D(A, B) = D(A, B ) D1 if D(A, B) = then {Claim(A)} Support(B) D2 if D(A, B) = and Claim(C) Claim(A) then D(C, B) = D2i if D(A, B) = and Claim(C) Claim(A) then D(C, B) = D3 if D(A, B) = and support(b) = support(c) then D(A, C) = D3i if D(A, B) = and support(b) support(c) then D(A, C) = D4 if Arcs(G) = then MinIncon( ) = Postulates for an attack relation D. We denote an attack relation from A to B as holding when D(A, B) =. [Gorogiannis + Hunter 2011] 35 / 134

36 Attack properties Postulates satisfied by attack functions D D D DD D U D DU D CU D R D DR D0 Y Y Y Y Y Y Y D1 Y Y Y Y Y Y Y D2 Y Y Y Y Y Y Y D2i Y Y N N N N Y D3 Y Y Y Y Y N N D3i Y Y Y Y N N N D4 Y Y Y Y Y Y Y where D D is defeater, D DD is direct defeater, D U is undercut, D DU is direct undercut, D CU is cannonical undercut, D R is rebuttal, and D DR is defeating rebuttal. 36 / 134

37 Attack properties Conclusions on attack properties Since classical logic offers a variety of different options for defining a counterargument (i.e. an attack relation), it is helpful to characterize the options in terms of postulates. Furthermore, these postulates can be used or adapted for classifying attack relations for a variety of base logics. With a few further postulates we can get characterization results for the attack relation (i.e. for each attack relation there is a combination of posulates that is equivalent to the attack relation) Further postulates on attack include Martinez, Garcia + Simari (IJCAI 07) has a postulate on the attack relation similar to D3 Amgoud+Besnard (SUM 2009) has a postulate on the attack relation similar to D1 Amgoud+Besnard (SUM 2009) have a number of further postulates and properties for classical logic instantiations 37 / 134

38 Extension properties Overview of extension properties Given an instantiated argument graph, we can look at the deductive arguments in the extension. Are the claims of the arguments in the extension consistent? Are the premises of the arguments in the extension consistent? What is the relationships between the conflicts in the knowledge and the conflicts between the arguments? We investigate these kinds of issue in terms of postulates. [Gorogiannis + Hunter 2011] 38 / 134

39 Extension properties Sceptical versus credulous extensions Let Extensions σ(g) be the set of extensions obtained according to σ {pr, gr, st, id} Sceptical σ (G) = S Credulous σ(g) = S Extension σ(g) S Extension σ(g) S Example A 4 Extension pr(g) = {{A 1, A 4}, {A 2, A 4}} A 3 A 1 A 2 Sceptical pr (G) = {A 4} Credulous pr(g) = {A 1, A 2, A 4} 39 / 134

40 Extension properties Free formulae Let Free( ) be the set of formulae not in any minimal inconsistent subset of. Free( ) = {α α Γ} Γ MinIncon( ) where MinIncon( ) = {Γ Γ and for all Γ Γ, Γ } Example Let = {a b, a, b c, c, d, d b, e, e d} MinIncon( ) = {{a b, a, b c, c}{a b, a, b c, d, d b}} Hence, Free( ) = {e, e d} 40 / 134

41 Extension properties Free and non-free arguments FreeArguments(G) = {A Nodes(G) Support(A) Free( )} NonFreeArguments(G) = {A Nodes(G) Support(A) Free( )} Example Let = {a b, a, b c, c, d, d b, e, e d} So Free( ) = {e, e d}. An example of a free arguments is {e}, e f Some examples of non-free arguments are {a b, a}, b { b c, c}, b 41 / 134

42 Extension properties Non-free postulate For a descriptive or generative argument graph G based on classical logic arguments, the non-free postulate is as follows, where σ {pr, gr, st, id}. s.t. A Nodes(G) Support(A) and NonFreeArguments(G) and Credulous σ(g) Nodes(G) Failure means that if Support(A), then A is credulously accepted. So for any argument that can be formed from a knowledgebase, there is a preferred extension that contains that argument. 42 / 134

43 Extension properties Consistency postulates For a descriptive or generative argument graph G based on classical logic arguments, the consistency postulates are defined as follows, where σ {pr, gr, st, id}. (CN1) Support(A) A sceptical σ (G) (CN2) A S Support(A), for all S Extension σ(g) (CN1 ) A Sceptical σ (G) Claim(A) (CN2 ) A S Claim(A), for all S Extension σ(g) 43 / 134

44 Results on consistency postulates For grounded and ideal semantics Attack CN1 CN1 CN2 CN2 Direct undercut Direct defeat Canonical undercut Rebut For preferred, semi-stable, stable, and complete semantics Attack CN1 CN1 CN2 CN2 Direct undercut Direct defeat Canonical undercut Rebut 44 / 134

45 Examples w.r.t. consistency postulates CN1 Postulate A Sceptical σ(g) Support(A) = Let = {a b, a c}. For the reviewed semantics for rebut, the following are arguments in G. {a b}, b { a c}, c Clearly {a b, a c}. Hence, CN1 postulate fails for rebut. 45 / 134

46 Examples w.r.t. consistency postulates CN2 Postulate A S Support(A) =, for all S Extensionsσ(G) Let = {a, a b, b} and let D be direct undercut. {b}, b {a, a b}, b { a b}, ( a b) {a, b}, ( a b) {a}, a {b, a b}, a Any credulous extension (e.g. red nodes) satisfies CN2 and CN2. 46 / 134

47 Examples w.r.t. consistency postulates CN2 Postulate A S Support(A) =, for all S Extensionsσ(G) Consider = {a, b, a b} with canonical undercuts. {a, b}, ( a b) { a b}, (a b) {a, a b}, b {b}, (a ( a b)) {b, a b}, a {a}, (b ( a b)) Any credulous extension (e.g. red nodes) violates CN2 and CN2. 47 / 134

48 Examples w.r.t. consistency postulates Consider = {a, b, a b} with undercuts. {b}, (a ( a b)) {b}, b {a, a b)}, b { a b}, (a b) {a, b}, ( a b) {a}, a {b, a b)}, a {a}, (b ( a b)) The stable extension (red nodes) violates CN2 and CN2. 48 / 134

49 Extension properties Related work Cayrol (IJCAI 95) was the first to systematically consider instantiation of Dung s proposal with classical logic arguments. Cayrol (IJCAI 95) has a result similar to satisfaction of CR with direct undercuts. Amgoud+Besnard (SUM 2009) have a number of further postulates and properties for classical logic instantiations. Caminada+Amgoud (AIJ 2007) postulates for defeasible logic instantiations. Amgoud + Vesic (IJAR 2014) have investigate relationships between extensions and maximally consistent subsets of the knowledgebase. 49 / 134

50 Extension properties Conclusions on extension properties Postulates on extensions can be used for any logic-based argumentation system For classical logic, results for the non-free postulate suggest that sceptical semantics are too sceptical credulous semantics are too credulous results for CN2 & CN2 suggest that exhaustive constructing argument graphs G is too simplistic need to better understand how meta-level information is used to select arguments 50 / 134

51 Extension properties More on shortcomings of exhaustive construction of argument graph There is also a need to be aposite for an audience Consider an article in a current affairs magazine: Only a small subset of all possible arguments, that either the writer or the reader could construct from their own knowledgebases, are used. A journalist regards some arguments as having higher impact and/or more believable for the intended audience and/or... than others, and so makes a selection. This need for appositeness is reflected in law, medicine, science, politics, advertising, management,..., and just ordinary every-day life. This need for appositeness creates challenges for developing nomative forms of argumentation for decision making with appropriate extension properties. 51 / 134

52 Structural properties: Motivation Utility assumption for Dung s abstract argumentation For any directed graph, a user can produce a textual description for each argument that would be an appropriate reflection of the nodes and arcs in the graph. So every graph has a use in modelling real-world scenarios of argumentation. A 1 A 2 A 3 B 1 B 2 B 3 Can a logical-based argumentation system produce all graphs? For each abstract argument graph G, can we specify a knowledgebase K such that the system induces an instantiated argument graph G such that G is isomorphic to G. 52 / 134

53 Structural properties Deductive argumentation system with exhaustive generation of graphs A deductive argumentation system is a tuple (Arguments i, Attacks i ) where Arguments i ( ) (respectively Attacks i ( )) gives the arguments (respectively attacks) that can be formed from. Structural properties A system (Arguments i, Attacks i ) constructively covers Φ iff for all G X, there is a and there is an A Arguments i ( ), such that (Arguments i ( ), Attacks i ( )) = G. A system (Arguments i, Attacks i ) is constructively covered by Φ iff for all and for all A Arguments i ( ), if (Arguments i ( ), Attacks i ( )) = G then G Φ. A system (Arguments i, Attacks i ) is constructively complete for Φ iff (Arguments i, Attacks i ) constructively covers Φ and is constructively covered by Φ [See Hunter + Woltran 2013] 53 / 134

54 Structural properties Argumentation based on simple logic Let (Arguments i, Attacks i ) be the argument system based on simple logic arguments and simple logic undercuts. (Arguments i, Attacks i ) is constructively complete for all graphs. Showing constructive completeness for simple logic For each node x in the graph, create a rule where the atom α n+1 is a unique identifier for this rule α 1... α n α n+1 For each attack on the node x, there is a condition α k in the above rule where α k is the unique identified for the attacker. The knowledgebase is the set of these rules plus the set of atoms formed from the conditions in these rules. Each argument contains exactly one of these rules. 54 / 134

55 Structural properties From = {a, b, c, a c a, a c b, b c}, we can construct the following exhaustive argument graph which is isomorphic to the above directed graph. {b, b c}, c {a, c, a c a}, a {a, c, a c b}, b 55 / 134

56 Structural properties There is no knowledgebase such that we can instantiate the following abstract graph with classical logic arguments and undercuts. From = {a, a}, we can only produce a more complicated graph below. A 1 = {a}, a A 2 = { a}, a A 3 = { a},... A 4 = {a}, / 134

57 Structural properties Preference-based argumentation frameworks PAF introduces a preference relation over arguments that in effect causes an attack to be ignored when the attacked argument is preferred over the attacker. From this, we need to define a defeat relation D as follows, and then (A, D) is used as the argument graph, instead of (A, R) D = {(A i, A j ) R (A j, A i ) } Example For the left graph, if A 2 A 1 and A 2 A 3, then (A, D) is the right graph. A 1 A 2 A 3 A 1 A 2 A 3 [Amgoud and Cayrol 2002] 57 / 134

58 Structural properties Example (Using preference-based argumentation) p 1 is the fact that the patient has glaucoma. p 2 is the assumption that it is ok to give PGA treatment. p 3 is the assumption that it is ok to give BB treatment. p 6 is an integrity constraint that ensures that only one treatment is given. p 1 = glaucoma p 2 = ok(pga) p 3 = ok(bb) p 4 = glaucoma ok(pga) give(pga) p 5 = glaucoma ok(bb) give(bb) p 6 = ok(pga) ok(bb) There are numerous arguments that can be constructed from this set of formulae such as the following. A 1 = {p 1, p 2, p 4, p 6}, give(pga) ok(bb) A 2 = {p 1, p 3, p 5, p 6}, give(bb) ok(pga) A 3 = {p 2, p 3}, ok(pga) ok(bb) A 4 = {p 6}, ok(pga) ok(bb) A 5 = {p 1, p 2, p 4}, give(pga) A 6 = {p 1, p 3, p 5}, give(bb) A 7 = {p 2, p 6}, ok(bb) A 8 = {p 3, p 6}, ok(pga) 58 / 134

59 Structural properties Example (Using preference-based argumentation (continued)) Let Arguments c ( ) be the set of all classical arguments that can be constructed from {p 1,..., p 6}. Let the preference relation be such that A i A 1 and A i A 2 for all i such that i 1 and i 2. Following is the subgraph involving A 1 and A 2 plus any arguments that defeat them. A 1 = {p 1, p 2, p 4, p 6}, give(pga) ok(bb) A 2 = {p 1, p 3, p 5, p 6}, give(bb) ok(pga) By taking this focal graph, we have ignored arguments such as A 3 to A 8 which do not affect the dialectical status of A 1 or A 2 given this preference relation. 59 / 134

60 Structural properties Some results 1 For simple logic arguments, with simple rebuts and simple undercuts, the exhaustive graphs are constructively complete for the set of all graphs. 2 For classical logic arguments, with any choice of attack, the exhaustive graphs are not constructively complete for the set of all graphs. For classical logic arguments, with any choice of attack, it is an open question as to which set of graphs for which the exhaustive graphs are complete. For classical logic arguments, with direct rebuttal, the exhaustive connective graphs are constructively complete for the set of bipartite graphs. 3 Every graph can be regarded as an argument graph, and we should be able to construct every argument graph from a logical knowledgebase using a generative mechanism. 4 To meet this requirement, we need generative mechanisms that harness meta-level information. 60 / 134

61 Defeasible reasoning Monotonicity An important property for many logics i including classical logic. i α {β} i α Example (Failure of monotonicity for defeasible reasoning) Consider the following set of formulae. Therefore, bird(x) flying-thing(x) ostrich(x) flying-thing(x) ostrich(x) bird(x) {bird(tweety)} i flying-thing(tweety) {bird(tweety), ostrich(tweety)} i flying-thing(tweety) 61 / 134

62 Defeasible reasoning Default logic (by Ray Reiter in 1980) A default theory is a set of first-order formulae and a set of default rules of the following form, where α, β and γ are classical formulae, α : β γ The inference rules are those of classical logic plus a special mechanism to deal with default rules If α is inferred, and β cannot be inferred, then infer γ. All classical inferences from the classical information in a default theory is derivable (if there is an extension). 62 / 134

63 Defeasible reasoning Default logic (by Ray Reiter in 1980) The operator Γ indicates the conclusions (i.e. the extension) to be associated with a set of classical formulae E. Let (D, W ) be a default theory, where D is a set of default rules and W is a set of classical formulae. Let Cn return the set of classical consequences of a set of formulae. Γ(E) is the smallest set of classical formulae such that the following three conditions are satisfied. 1 W Γ(E) 2 Γ(E) = Cn(Γ(E)) 3 For each default in α : β/γ D, the following holds: if α Γ(E) and β E then γ Γ(E) 63 / 134

64 Defeasible reasoning Attempt E Γ(E) Extension? 1 bird(tweety) bird(tweety) E Γ(E) fly(tweety) 2 fly(tweety) bird(tweety) E Γ(E) fly(tweety) 3 bird(tweety) bird(tweety) E = Γ(E) fly(tweety) fly(tweety) 4 bird(tweety), bird(tweety) Γ(E) E fly(tweety) 5 bird(tweety), bird(tweety) Γ(E) E & E Γ(E) fly(tweety) A non-exhaustive number of attempts are made for determining an extension. In each attempt, a guess is made for E, and then Γ(E) is calculated. 64 / 134

65 Defeasible reasoning Example (Default logic) Let D be the following set of defaults: bird(x) : fly(x) fly(x) penguin(x) : bird(x) bird(x) penguin(x) : fly(x) fly(x) and W is {bird(tweety)}. For (D, W ), we obtain one extension Cn({bird(Tweety), fly(tweety)}) Now, consider W = {bird(tweety), penguin(nigel)}. For (D, W ), we obtain one extension Cn({bird(Tweety), fly(tweety), penguin(nigel), bird(nigel), fly(nigel)}) Default logic suppresses inconsistency Note how potential inconsistency is suppressed by the definition of default logic. 65 / 134

66 Defeasible reasoning Argumentation is non-monotonic Adding formulae to the knowledgebase can cause arguments to be withdrawn. Example Let = {a}, and so the following is a generated argument graph. Hence, {A 1} is the grounded extension of the graph. A 1 = {a}, a A 4 = {a},... If we add a to, we get the following generated argument graph. Hence, A 1 is no longer in the grounded extension. A 1 = {a}, a A 2 = { a}, a A 3 = { a},... A 4 = {a}, / 134

67 Defeasible reasoning Non-monotonic logics versus argumentation systems Both are non-monotonic processes However Non-monotonic logics suppress inconsistency: The aim is to provide a consistent set of inferences. Argumentation systems highlight inconsistency via the use of arguments and counterarguments. 67 / 134

68 Defeasible reasoning Overview Argumentation is a non-monotonic process. This reflects the fact that argumentation involves uncertain information, and so new information can cause a change in the conclusions drawn. However, the base logic does not need to be non-monotonic. Indeed, most proposals for structured argumentation use a monotonic base logic (e.g. simple logic, or classical logic). Nonetheless, there are issues in capturing defeasible reasoning in argumentation. Choice of base logic Modelling of defeasible knowledge And there are insights and tools to be harnessed for research in non-monontonic logics 68 / 134

69 Defeasible reasoning Example (Defeasible reasoning using simple logic with normality predicate) The first argument A 1 captures the general rule that if workday holds, then usemetro(sid) holds. A 1 = {workday, normal, workday normal usemetro(sid)}, usemetro(sid) A 2 = {workathome(sid), workathome(sid) normal}, normal Example (Defeasible reasoning using simple logic augmented with rule names) The first argument A 1 captures the general rule that if workday holds, then usemetro(sid) holds. A 1 = {workday, r1 : workday usemetro(sid)}, usemetro(sid) A 2 = {workathome(sid), workathome(sid) r1}, r1 69 / 134

70 Defeasible reasoning Going beyond simple logic Simple logic is monotonic. Simple logic is very weak it only has modus ponens. There is a range of conditional logics that are monotonic that are richer than classical logic that capture interesting aspects of defeasible reasoning There is also a range of non-monotonic logics that may useful in argumentation (e.g. default logic). 70 / 134

71 Defeasible reasoning Proof rules for system P (REF ) A A (CUT ) A B A B C A C (LLE) A B A C B C (AND) A B A C A B C (CM) A B A C A B C (RW ) A B C A C B (OR) A C B C A B C (LOOP) A0 A1 A1 A2... A k 1 A k A k A 0 A 0 A k [See Kraus Lehmann and Magidor 1990] 71 / 134

72 Defeasible reasoning Example From the following statements penguin bird penguin fly bird fly we get the following inferences penguin bird fly fly penguin bird penguin bird penguin fly bird penguin penguin 72 / 134

73 Defeasible reasoning Example From the following statements, teenager poor teenager student poor employed student employed we get the following inferences teenager (poor student) but we don t get the following. poor student student poor 73 / 134

74 Defeasible reasoning Conditional logic argument For a set of conditional statements Φ and a set of propositional formulae Ψ, if Φ P α β and Ψ α, then a preferential argument is Φ Ψ, α β Example For Φ = {penguin bird, penguin fly} and Ψ = {penguin, bird}, the following is a preferential argument. Φ Ψ, penguin bird fly 74 / 134

75 Defeasible reasoning Some preferential attack relations Let A 1 = Φ Ψ, α β and A 2 = Φ Ψ, γ δ. A 2 is a rebuttal of A 1 iff δ β and γ α A 2 is a direct rebuttal of A 1 iff δ β and γ α A 2 is a undercut of A 1 iff δ α A 2 is a canonical undercut of A 1 iff δ α A 2 is a direct undercut of A 1 iff there is σ Ψ such that δ σ 75 / 134

76 Defeasible reasoning Example For A 1 and A 2 below, A 2 is a direct rebuttal of A 1, but not vice versa, A 1 = Φ 1 Ψ 1, bird fly A 2 = Φ 2 Ψ 2, penguin bird fly where Φ 1 = {bird fly}, Ψ 1 = {bird}, Φ 2 = {penguin bird, penguin fly}, Ψ 2 = {penguin, bird}. Example For A 1 and A 2 below, A 2 is a direct rebuttal of A 1, but not vice versa, A 1 = Φ 1 Ψ 1, matchisstruck matchlights A 2 = Φ 2 Ψ 2, matchisstruck matchiswet matchlights 76 / 134

77 Defeasible reasoning MP system of conditional logic (Chellas 1975) Conditional logics can be used for hypothetical statements of the form If α were true, then β would be true. RCEA RCEC CC CM c α β c (α γ) (β γ) c α β c (γ α) (γ β) c ((α β) (α γ)) (α (β γ)) c (α (β γ)) ((α β) (α γ)) CN c (α ) MP c (α β) (α β) Informally, α β is valid when β is true in the possible worlds where α is true. 77 / 134

78 Defeasible reasoning Example (Using MP system of conditional logic) Let = {matchisstruck matchlights, matchisstruck}. From this knowledgebase, we get the following argument. {matchisstruck matchlights, matchisstruck}, matchlights Note, from, we cannot get the following argument. {matchisstruck matchiswet matchlights, matchisstruck}, matchlights [See Besnard et al 2013] 78 / 134

79 Defeasible reasoning matchtonight matchtonight JohnGoesToTheStadium JohnGoesToTheStadium matchtonight JohnHasEnoughMoney JohnGoesToTheStadiumt matchtonight JohnHasEnoughMoney JohnGoesToTheStadium A descriptive graph using MP system of conditional logic. The first argument says John will go to the stadium because there is a match tonight. The second corrects the first argument by correcting the circumstances under which John will go to watch the match at the stadium. For more details of definition of counterargument, see Besnard et al / 134

80 Defeasible reasoning: What is a defeasible rule? birds fly If we take all the normal situations (or observations, or worlds or days), and in the majority of these birds fly, then (*) may be equal to the following ( ) birds normally fly Or if we take the set of all birds (or all the birds you have seen, or read about, or watched on TV) and the majority of this set fly, then (*) may be equal to the following most birds fly Or perhaps if we take the set of all birds, we know the majority have the capability to fly, then (*) could equate with the following. though this in turn raises the question of what it is to know something has a capability most birds have the capability to fly Or if we take an idealized notion of a bird that we in a society are happy to agree upon, then we could equate (*) with the following a prototypical bird flies 80 / 134

81 Defeasible reasoning: What is a defeasible rule? birds lay eggs ( ) We may take (**) as meaning the following but on a given day, (or given situation, or observation) a given bird bird will probably not lay an egg. birds normally lay eggs Or we could say (**) means the following but half the bird population is male and therefore don t lay eggs. most birds have the capability lay eggs Or perhaps we could say (**) means the following where most species of bird reproduce by laying eggs However, the above involves a lot of missing information to be filled in, and in general it is challenging to know how to formalize a defeasible rule. 81 / 134

82 Defeasible reasoning The choice of proof rules depends on the interpretation of defeasible rules Example (Failure of contraposition from Brewka) Men usually do not have beards But, this does not imply that if someone does have a beard, then that person is not a man. Example (Another failure of contraposition) If I buy a lottery ticket, then I will normally not win a prize. But, this does not imply that if I do win a prize, then I did not buy a ticket. 82 / 134

83 Defeasible reasoning Dichotomy of situations for whether contrapositive reasoning is appropriate Epistemic Defeasible rules describe how certain facts hold in relation to each other in the world. So the world exists independently of the rules. Contrapositive reasoning may be appropriate. Constitutive Defeasible rules (in part) describe how the world is constructed (e.g. regulations). So the world does not exist independently of the rules. Contrapositive reasoning is not appropriate. [Caminada 2008] 83 / 134

84 Defeasible reasoning Note that these examples are identical syntactically. Example (Epistemic situation) O A - goods ordered three months ago will probably have arrived now. A C - arrived good will probably have a customs document. L C - goods listed as unfulfilled will probably lack customs document. From these rules, it would be reasonable to infer A from L. Example (Constitutive situation) S M - snoring in the library is form of misbehaviour. M R - misbehaviour in the library can result in removal from the library. P R - professors cannot be removed from the library. From these rules, it would not be reasonable to infer M from P. [Caminada 2008] 84 / 134

85 Defeasible reasoning Appropriateness of contraposition also depends on the kind of epistemic information and how we model it bird 1 Rule 1: bird flyingbird 2 Rule 2: flyingbird bird 3 Rule 3: penguin flyingbird 4 Rule 4: flyingbird bird 5 From 1: flyingbird bird 6 But step 5 conflicts with Rules 3+4 flyingbird penguin 85 / 134

86 Defeasible reasoning Brewka s preferred subtheories A default theory Γ is a tuple (Γ 1,..., Γ n), where each Γ i is a set of formulae in a classical first-order language. The formulae in Γ i are more preferred to those in Γ i+1 A preferred subtheory is a set Σ = Σ 1 Σ n such that for i = 1...n, Σ 1 Σ i is a maximal (under set inclusion) consistent subset of Γ 1,..., Γ i. Example Γ i Σ i 1 {a, a b, a b} {a, a b} 2 { a c, c} { c} 3 {c, b, d} {d} [Brewka 1989] 86 / 134

87 Defeasible reasoning Relationship between preferred subtheories and deductive arguments Let = Γ 1 Γ n. If Σ is a preferred subtheory of Γ, then Args(Σ) is a stable extension of Args( ) with direct undercut and. If E is a stable extension of Args( ) with direct undercut and, then Support(A) is a preferred subtheory of Γ. A E where A B iff α Support(A) s.t. β Support(B), α β. Example For Γ = ({a, b}{ a b}), the nodes in the stable extension are yellow {b}, b {a, a b}, b { a b}, ( a b) {a, b}, ( a b) {a}, a {b, a b}, a [Adapated from Modgil and Prakken 2013] 87 / 134

88 Defeasible reasoning Conclusions Defeasible reasoning is a central feature of argumentation Structured argumentation may benefit from logics developed for defeasible reasoning Computational models of argument may benefit from insights into defeasible reasoning developed by research into non-monontonic logics. We need better ways of deciding what kinds of defeasible rules we have for an application, and for deciding which proof rules are appropriate. 88 / 134

89 Meta-level argumentation Argument schema An argument schema is an informal specification of a general pattern of reasoning used in argumentation, and defined in terms of Types of premise Conclusion Critical questions that when answered may bring conclusion into doubt A variety of argument schema have been proposed by Walton and by others They are popular for capturing the complexities of applications They can easily be adapted for specific applications They can be formalized (e.g. Prakken, Hunter, etc). [See Walton 2006] 89 / 134

90 Meta-level argumentation Scheme for appeal to expert opinion Major premise Source E is an expert in subject domain S containing proposition P. Minor premise E asserts that proposition A in domain S is true (false). Conclusion A may plausibly be taken to be true (false). Critical questions : Expertise: How credible is E as an expert source? Field: Is E an expert in the field that A is in? Opinion: What did E assert that implies A? Trustworthiness: Is E personally reliable as a source? Consistency: Is A consistent with what other experts assert? Evidence: Is E s assertion based on evidence? [See Walton 2006] 90 / 134

91 Meta-level argumentation Scheme for appeal to popular opinion General acceptance premise A is generally accepted as true. Presumption premise If A is generally accepted as true, there exists a presumption in favour of A. Conclusion There exists a presumption in favour of A. Critical questions : Evidence: What evidence, such as poll or appeal to common knowledge, supports the claim that A is generally accepted as true? Alternatives: Even if A is generally accepted as true, are there any reasons for doubting it is true? [See Walton 2006] 91 / 134

92 Meta-level argumentation Scheme for argument from negative consequences Premise If A is brought about, bad consequences will plausible occur. Conclusion A should not be brought about. Critical questions : Probability: How probable are the consequences? Evidence: What is the evidence? Alternatives: Are there positive consequences that should be taken into account? [See Walton 2006] 92 / 134

93 Meta-level argumentation Further argument schema from Walton The slippery slope argument Argument from commitment The circumstantial ad hominem Argument from position to know Argument from analogy Argument from sign Argument from example Argument from popular practice Argument from precedent Argument from consequences Argument from waste Argument from bias Ethotic argument Argument from cause to effect Argument from established rule Argument from gradualism Argument from verbal classification Argument from correlation to cause Plausible argument from ignorance The direct ad hominem argument Argument from an exceptional case Argument from inconsistent commitment Argument from positive consequences Circumstantial argument against the person Argument from vagueness of verbal classification Argument from arbitrariness of a verbal classification Argument from evidence to a hypothesis Argument from falsification of a hypothesis 93 / 134

94 Meta-level argumentation Meta-level argumentation Meta-level argumentation is argumentation about argumentation. Use logic as a meta-language for meta-level argumentation. Properties and constraints can be specified in the meta-language. Notions such as acceptability of an argument can be the subject of argumentation For example, arguments and counterarguments in the meta-level for an object-level argument being acceptable. Hence, argument schema can be captured in meta-level argumentation 94 / 134

95 Meta-level argumentation Example (Use meta-language of Wooldridge, McBurney + Parsons (2006)) For example, suppose {b, b c}, c is an argument then the following is a meta-level formula. {b, b c}, c Hence, the following are meta-level arguments. { {b, b c}, c }, {b, b c}, c { {b}, b c, a 7 = {b}, b c }, a 7 Following is a meta-level axiom for preference-based reasoning where a 1 and a 2 are object level arguments, and q is an individual. Defeats(a 1, a 2, q) (Attacks(a 1, a 2) Val(a 1, v 1) Val(a 2, v 2) (v 2 > q v 1)) 95 / 134

96 Meta-level argumentation Besnard+Doutre 2004 encoding of argument semantics uses a meta-language Conflictfreeness, admissibility, and extensions under different semantics, are specified by axioms in classical logic Conflictfree is a constraint ensuring no attacker and attackee are in a set of arguments X Conflictfree(X, G) = [a i ( a j )] a i X a j s.t. (a i,a j ) Attacks(G) Example (Conflictfree) a 4 a 3 a 1 a 2 For X 1 = {a 1, a 3}, [a 1 ( a 2 a 3)] [a 3 ( a 4)] For X 2 = {a 1, a 4}, [a 1 ( a 2 a 3)] [a 4] Unsatisfiable Satisfiable 96 / 134

97 Meta-level argumentation Besnard+Doutre 2004 encoding of argument semantics uses a meta-language The following constraint ensures that X is a stable extension Stable(G) = [a i ( a j )] a i Nodes(G) a j s.t. (a i,a j ) Attacks(G) Example (Stable) a 4 a 3 a 1 a 2 Hence, Stable(G) is [a 1 ( a 2 a 3)] [a 2 ( a 1)] [a 3 ( a 4)] [a 4 ( )] The models of this constraint are {{a 1, a 4}, {a 2, a 4}} 97 / 134

98 Meta-level argumentation Some axioms for meta-argumentation (M 1) argument(x) acceptable(x) (M 2) acceptable(x) warranted(x) (M 3) acceptable(y) undercut(y, x) acceptable(x) Example Let (F 1) be argument( {b, b c}, c ), {F 1, M 1}, acceptable( {b, b c}, c ) [see Hunter 2008] 98 / 134

99 Meta-level argumentation Simulating object-level argumentation Object level Meta-level { x.p(x) x.q(x), x. p(x)}, x.q(x) {F 2, M 1, M 2}, warranted(a 1) { x.( p(x) r(x))}, (...) {F 3, F 5, M 1, M 3}, (...) { x. r(x)}, (...) {F 4, F 6, M 1, M 3}, (...) (F 2) argument(a 1) (M 1) argument(x) acceptable(x) (F 3) argument(a 2) (M 2) acceptable(x) warranted(x) (F 4) argument(a 3) (M 3) acceptable(y) undercut(y, x) (F 5) undercut(a 2, a 1) acceptable(x) (F 6) undercut(a 3, a 2) 99 / 134

100 Meta-level argumentation Logical Formalization of Appropriateness In order to formalise appropriateness we drop the axiom and replace it with the axiom where (M 1) argument(x) acceptable(x) (M 4) assert(x, y) appropriate(x, y) acceptable(y) assert is for x asserts y appropriate is for x is an appropriate proponent for y So now our core axioms for appropriateness are (M 2) acceptable(x) warranted(x) (M 3) acceptable(y) undercut(y, x) acceptable(x) (M 4) assert(x, y) appropriate(x, y) acceptable(y) 100 / 134

101 Meta-level argumentation Logical Formalization of Appropriateness Our core axioms for appropriateness (M 2) acceptable(x) warranted(x) (M 3) acceptable(y) undercut(y, x) acceptable(x) (M 4) assert(x, y) appropriate(x, y) acceptable(y) Further axioms for appropriateness may include (M 5) liar(x) appropriate(x, y) (M 6) celebrity(x) appropriate(x, y) (M 6 ) celebrity(x) topic(y, showbiz) appropriate(x, y) Example Let (F 6 ) be liar(pinnocchio), {F 6, M 5 }, appropriate(pinocchio, a) Let (F 8 ) be celebrity(harrisonford), and (F 9 ) be topic(a, showbiz), {F 8, F 9, M 6 }, appropriate(harrisonford, a) 101 / 134

102 Meta-level argumentation Logical Formalization of Expert Argumentation Our core axioms for expert argumentation (M 2) acceptable(x) warranted(x) (M 3) acceptable(y) undercut(y, x) acceptable(x) (M 4) assert(x, y) appropriate(x, y) acceptable(y) (M 7) topic(z, w) role(x, y) scope(y, w) appropriate(x, z) (M 8) assert(x 2, y 2) acceptable(y 2) competing(y 1, y 2) outrank(x 2, x 1) appropriate(x 1, y 1) where topic(z, w) argument z is on topic w role(x, y) proponent x has role y scope(y, w) role y covers topic w competing(y 1, y 2) argument y 1 competes with argument y 2 outranks(x 1, x 2) proponent x 1 outranks proponent x / 134

103 Meta-level argumentation Example (Expert argumentation) (M 2) acceptable(x) warranted(x) (M 4) assert(x, y) appropriate(x, y) acceptable(y) (M 7) topic(z, w) role(x, y) scope(y, w) appropriate(x, z) (H 1) assert(drjones, diagnosis 1) (H 2) topic(diagnosis 1, infarction) (H 3) role(drjones, GP) (H 4) scope(gp, infarction) appropriate(drjones, diagnosis 1) warranted(diagnosis 1) 103 / 134

104 Meta-level argumentation Example (Expert argumentation) (M 4) assert(x, y) appropriate(x, y) acceptable(y) (M 7) topic(z, w) role(x, y) scope(y, w) appropriate(x, z) (M 8) assert(x 2, y 2) acceptable(y 2) competing(y 1, y 2) outrank(x 2, x 1) appropriate(x 1, y 1) (H 5) assert(drsmith, diagnosis 2) (H 6) topic(diagnosis 2, infarction) (H 7) role(drsmith, cardiologist) (H 8) scope(cardiologist, infarction) (H 9) competing(drjones, DrSmith) (H 10) outrank(drsmith, DrJones) appropriate(drsmith, diagnosis 2) appropriate(drjones, diagnosis 1) warranted(diagnosis 2) 104 / 134

105 Representing argument schema Criteria for good expert argumentation We consider four key criteria for delineating good expert argumentation. Qualified proponent The expert is suitably qualified in the field of the argument being proposed. Confident proponent The expert offers sufficient confidence in the argument being proposed. Best argument The argument by the expert is better than any competing argument by any expert. Safe argument No counterargument to the expert s argument has been overlooked. These adapt the conditions and some of the critical questions of Walton s scheme for appeal to expert opinion. 105 / 134

106 Meta-level argumentation Criteria for good expert argumentation Argument a is a good expert argument iff there is a proponent p and an undefeated argument Φ, α such that Φ appropriate(a, p) Φ assert(a, p) Qualified proponent Confident proponent and there is no undefeated argument Ψ, β such that Ψ appropriate(a, p) Ψ acceptable(a, p) Best argument Safe argument 106 / 134

107 Meta-level argumentation Meta-level (yellow) and object-level arguments (orange) can be arranged in a bimodal argument graph where the object-level arguments and attacks are supported (dotted line) by meta-level arguments. {...}, appropriate(p 1, a 1 {...}, attack(a 2, a 1) {...}, acceptable(a 1) {...}, acceptable(a 2) a 1 a 2 {...}, attack(a 1, a 2) [Muller + Hunter 2013] 107 / 134

108 Meta-level argumentation Advantages of meta-level argumentation Meta-level argumentation provides a separation of concerns Object level is for the issue being analyzed. Meta-level is for the quality of participants, evidence, etc. Meta-level logics can be used directly in deductive argumentation. Meta-level argumentation can be used for formalize argument schema. Bimodal argument graphs can be used to arrange and evaluate meta-level and object-level arguments 108 / 134

109 From natural language arguments to formalized arguments Overview Natural language is complex and so formalization calls for expressive logics such as classical logic. Computational linguistics offers mechanisms to translate fragments of natural language into logic. However, most arguments in natural language are enthymemes, and this raises further complications. 109 / 134

110 Understanding arguments in free text Red denotes outer reason-claim coupling, and blue denotes inner reason-claim coupling. Note, outer reason-claim coupling has two reasons for the claim. claim Heathrow needs more capacity \claim reason Heathrow runs at close to 100% capacity. With demand for air travel predicted to double in a generation, Heathrow will not be able to cope without a third runway, say those in favour of the plan. \reason reason reason Because the airport is over-stretched, any problems which arise cause knock-on delays. \reason claim Heathrow, the argument goes, needs extra capacity if it is to reach the levels of service found at competitors elsewhere in Europe, or it will be overtaken by its rivals. \claim \reason / 134

111 Understanding arguments in free text Beyond argument mining Argument mining only allows us to find parts of the text that pertain to premises and/or claim. Argument mining does not give us the logical structure of the arguments. If we are to use logical reasoning with arguments arising in natural language, we need natural language understanding. 111 / 134

112 Understanding arguments in free text S poison (ethel,the-cat ) NP V [+PAS] NP the-cat poison ethel NP N pr The cat was poisoned by Ethel 112 / 134

113 Understanding arguments in free text S rain snow S rain S snow If it didn t rain then it snowed 113 / 134

114 Understanding arguments in free text S (λq[λp[ x[q(x) P(x)]]](student ))(like (jo )) NP λq[λp[ x[q(x) P(x)]]](student ) VP like (jo ) Det N V t[+fin] NP λq[λp[ x[q(x) P(x)]]] student like jo every student liked N pr Jane Jane 114 / 134

115 Understanding arguments in free text Deep understanding within sentence Part of speech tagging Syntactic parsing Semantic parsing Identifying the predicate-argument structure Identifying the structure of the logical formulae Deep understanding across sentences Resolving pronouns Resolving ellipses Co-ordinating phrases Identifying implicit information Identifying rhetorical structure 115 / 134

116 Modelling enthymemes in logical argumentation A husband is clearing up breakfast as his wife is preparing to go to work. Husband thinks The weather report predicts rain and if the weather report predicts rain, then you should take an umbrella, so you should take an umbrella (intended argument) Husband speaks The weather report predicts rain, so you should take an umbrella (enthymeme) Wife thinks The weather report predicts rain and if the weather report predicts rain, then you should take an umbrella, so you should take an umbrella (received argument) Since if the weather report predicts rain, then you should take an umbrella is common knowledge, it is not communicated. 116 / 134

117 Modelling enthymemes in logical argumentation Enthymeme Let Ψ, α be an argument. Φ, α is an enthymeme for Ψ, α iff Φ Ψ. Example Let α be you need an umbrella today, and β be the weather report predicts rain. Intended argument is {β, β α}, α Enthymeme is {β}, α. So β α is treated as common knowledge. [Hunter 2007] 117 / 134

118 Modelling enthymemes in logical argumentation Assumption that there is a set of background knowledge Encode by deletion of premises that are in background knowledge Intended argument Enthymeme Decode by abduction of premises from background knowledge Abduction Potentially many decodings can be abduced for an enthymemes. Meta-information (preferences, probabilities, etc) can help to choose the best decoding. Risk that wrong argument is chosen for the decoding. 118 / 134

119 Modelling enthymemes in logical argumentation Approximate arguments An approximate argument is a pair Φ, α where Φ L and α L. If Φ α, then Φ, α is valid. If Φ, then Φ, α is consistent. If Φ α, and there is no Φ Φ s.t. Φ α, then Φ, α is minimal. If Φ α, and Φ, then Φ, α is expansive. If Φ α, and Φ α, then Φ, α is a precursor. An enthymeme is a precursor Therefore, if Φ, α is a precursor, then there exists some Ψ L such that Φ Ψ α and Φ Ψ, and hence Φ Ψ, α is expansive. 119 / 134

120 Modelling enthymemes in logical argumentation Example (Approximate arguments) Let = {α, α β, γ, β, β, γ, β γ}, Valid Consistent Minimal Expansive Precursor Argument {α, α β, γ, β}, β {γ, γ}, β {α, α β, γ}, β {α, α β, γ, γ}, β {α, α β}, β { α β}, β { α β, β γ, γ}, β 120 / 134

121 Modelling enthymemes in logical argumentation Real arguments Each agent i has the following two types of resource. Perbase A personal knowledgebase i that when i is proponent, is used for the support of the intended argument. Cobase A function µ i,j : L [0, 1] that represents what an agent i believes is common knowledge for i and j. For α L, the higher the value of µ i,j (α), the more that i regards α as being common knowledge for i and j. Note, we do not assume that µ i,j = µ j,i for any agents. Example When µ i,j (β α) = 1 then the premise β α is superfluous in any real argument consigned by proponent i to recipient j. 121 / 134

122 Modelling enthymemes in logical argumentation Encoding arguments For an argument Φ, α, and for a threshold τ [0, 1], the encodation of Φ, α from a proponent i for a recipient j, is the approximate argument Ψ, α where Ψ = {φ Φ µ i,j (φ) τ} Example When µ i,j (β α) = 1, and µ i,j (β) = 0.5, and τ = 0.7. the encodation of {β, β α}, α is {β}, α Example When µ i,j (β α) = 1, and µ i,j (β) = 1, and τ = 0.7, the encodation of {β, β α}, α is {}, α 122 / 134

123 Modelling enthymemes in logical argumentation Decoding arguments For an encodation Ψ, α from a proponent i for a recipient j, a decodation is of the form Ψ Ψ, α such that 1 Ψ L 2 Ψ Ψ, α is expansive 3 there is no Ψ s.t. Ψ > j,i Ψ and Ψ Ψ, α is expansive. Example Let µ j,i be defined by the following and for all other φ, µ j,i (φ) = 0. µ j,i (α β) = 1, µ j,i (α ɛ) = 1, µ j,i (ɛ β) = 1 So for the enthymeme {α}, β, the decodations are {α, α β}, β {α, α ɛ, ɛ β}, β 123 / 134

124 Modelling enthymemes in logical argumentation Background knowledge The RNLI is a charity providing most lifeboats for the UK. Birmingham is in the centre of England and so it is the furthest point from the sea. Conversation a fund-raiser and a potential donor Would you like to make a donation to the RNLI? No, I always spend my holidays in Birmingham 124 / 134

125 Modelling enthymemes in logical argumentation E 1 and E 3 are by a proponent for expanding Heathrow airport with a third runway, and E 2 is by an opponent, and so it appears that E 3 attacks E 2 and E 2 attacks E 1. E 1 = We should build a third runway at Heathrow because everyone will benefit from the increased capacity. E 2 = It is not true that everyone will benefit in the community. E 3 = Local residents won t have problems with traffic because we will increase public transport to the airport. Heathrow airport in the suburbs of London. 125 / 134

126 Modelling enthymemes in logical argumentation h = we should build a third runway at Heathrow, e = everyone will benefit from the increased capacity, t = local residents will have problems from increased traffic, s = we will increase public transport to the airport. {e, e h}, h {}, e {s, s t}, t 126 / 134

127 Modelling enthymemes in logical argumentation E 1 and E 3 are by a proponent for expanding Heathrow airport with a third runway, and E 2 is by an opponent, and so it appears that E 3 attacks E 2 and E 2 attacks E 1. E 1 = We should build a third runway at Heathrow because everyone will benefit from the increased capacity. E 2 = It is not true that everyone will benefit in the community. E 3 = Local residents won t have problems with traffic because we will increase public transport to the airport. Let us suppose that with the common knowledge we have available, we judge that E 2 decodes to either E 2 or E 2. E 2 = It is not true that everyone will benefit in the community. There are local residents who will suffer from increased noise from the increased number of aircraft. E 2 = It is not true that everyone will benefit in the community. There are local residents who will have problems from increased traffic on the roads to the airport. Given these decodings,e 1 is attacked by both interpretations of E 2 whereas only one interpretation of E 2 is attacked by E / 134

128 Modelling enthymemes in logical argumentation h = we should build a third runway at Heathrow, e = everyone will benefit from the increased capacity, t = some local residents will have problems from increased traffic, n = some local residents will suffer from noise from the increased number of aircraft. s = we will increase public transport to the airport. G 1 {e, e h}, h {e, e h}, h G 2 {n, n e}, e {t, t e}, e {s, s t}, t {s, s t}, t 128 / 134

129 Modelling enthymemes in logical argumentation Enthymemes where the claim is implicit An enthymeme has some/all support being implicit An enthymeme may have some/all of its claim being implicit Conversation late at night Would you like a coffee? Coffee will keep me awake. [Black and Hunter 2012] 129 / 134