Introduction to Structural Argumentation

Transcription

1 Introduction to Structural Argumentation Anthony Hunter Department of Computer Science, University College London, UK July 8, / 28

2 Approaches to structured argumentation 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. 2 / 28

3 Overview of structured argumentation Descriptive graphs Generative graphs Counterarguments Arguments Base logic 3 / 28

4 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. 4 / 28

5 Arguments Definition for deductive argument 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. Examples {report(rain), report(rain) carry(umbrella)}, carry(umbrella) {study(sid, logic), study(sid, logic)}, KingOfFrance(Sid) 5 / 28

6 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) 6 / 28

7 Arguments The minimality constraint An argument Φ, α satisfies the minimality constraint when there is no Ψ Φ such that Ψ 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) 7 / 28

8 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. 8 / 28

9 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. 9 / 28

10 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) 10 / 28

11 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) Examples A 1 = {efficientmetro, efficientmetro usemetro}, usemetro A 2 = {strikemetro, strikemetro efficientmetro}, efficientmetro A 3 = {govdeficit, govdeficit cutgovspending}, cutgovspending A 4 = {weakeconomy, weakeconomy cutgovspending}, cutgovspending 11 / 28

12 Counterarguments based on simple logic Example of 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. 12 / 28

13 Counterarguments based on classical logic Counterarguments 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 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 13 / 28

14 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 14 / 28

15 Counterarguments based on classical logic Example 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 15 / 28

16 Counterarguments based on classical logic Example with 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) Another example with 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) 16 / 28

17 Counterarguments based on classical logic Further kinds of classical counterargument Let A and B be two classical arguments. We define the following types of classical attack. A is a classical defeater of B if Claim(A) Support(B). A is a classical direct defeater of B if A is a classical undercut of B if φ Support(B) s.t. Claim(A) φ 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). 17 / 28

18 Counterarguments based on classical logic Examples of attack functions/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) 18 / 28

19 Counterarguments based on classical logic An arrow from D 1 to D 2 indicates that D 1 D 2. Defeater Direct defeat Undercut Direct rebut Direct undercut Canonical undercut Rebut 19 / 28

20 Argument graphs Approaches to generating 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. 20 / 28

21 Argument graphs Example of 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 21 / 28

22 Argument graphs Example of 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 22 / 28

23 Argument graphs Example of 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) 23 / 28

24 Argument graphs Example of 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 24 / 28

25 Argument graphs Example of 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 25 / 28

26 Need for meta-level information Normally, meta-level information is also needed for logical argumentation. Examples of meta-level information Preferences over formulae to give a preference over arguments [see for example Amgoud and Cayrol 2002]. Preference for premises that are based on more reliable sources Preference for claims that meet more important goals Use probability theory to quantify uncertainty of each argument (e.g. probability that premises are true, or probability that the argument comes from a reliable source, etc). Use meta-level argumentation to reason about the quality of arguments (e.g. argumentation about whether proponents for arguments are qualified to argue about a topic). 26 / 28

27 Probabilistic logic Drug X is an effective treatment for improving survival However, strokes are a serious side-effect p = problematic treatment, s = effective treatment for improving survival, r = strokes are a serious side-effect. A 1 = { p, p s}, s A 2 = {r, r p}, p Model p s r P m 1 false true true 0.35 m 2 false true false 0.35 m 3 true true true 0.12 m 4 true false true 0.18 P(A 1) = P(m 1) + P(m 2) = 0.7 P(A 2) = P(m 3) + P(m 4) = / 28

28 Conclusions Argumentation is an important cognitive process for dealing with incomplete and inconsistent information. Computational models of argument provide a range of insights into argumentation. Abstract argumentation captures the dialectical nature of argumentation. Logical argumentation captures the internal structure of arguments and attacks. Dialogical argumentation captures protocols and strategies for multiple agents to argue together. Argumentation technology offers promising solutions for a range of applications. Many interesting and important research questions remain (e.g. using richer formalisms from computational linguistics such as discourse representation theory) 28 / 28