The Argument Mapping Tool of the Carneades Argumentation System DIAGRAMMING EVIDENCE: VISUALIZING CONNECTIONS IN SCIENCE AND HUMANITIES’ University of Windsor, April 25 & 26, 2014. Douglas Walton CRRAR The Advent of Defeasible Logic • Argumentation has long been used to study defeasible reasoning, and recently formal and computational argumentation systems have been built in artificial intelligence and multiagent computing that show how nonmonotonic reasoning can be modeled in argumentation-theoretic terms. • The basic model is a tree structure representing a connected sequence of argumentation both pro and con an ultimate conclusion. An evaluation shows the conclusion as accepted if the pro-arguments supporting it are not defeated by the con arguments against it. • Other devices may also be used, such as a priority ordering on rules, a system for assigning standards of proof to arguments, and defeasible argumentation schemes that provide tentative support for a conclusion in the absence of counterarguments or the posing of critical questions to rebut or undercut the argument. The DefLog System • • • • • Bart Verheij in 1999 began to build an automated argument assistant called ArguMed that helps a user to construct an argument diagram in order to analyze and evaluate a given argument (Verheij, 2003, 320). The development of this system was guided by his parallel research on the logical system called DefLog (Verjeij, 2003) that represents the underlying structure of the reasoning displayed by ArguMed. ArguMed (http://www.ai.rug.nl/~verheij/aaa/argumed3.htm) is available at no cost on the Internet. The ultimate conclusion, called the issue in ArguMed, appears in a text box at the top of the diagram, and the premises supporting it form arguments that can be linked together, and that lead by arrows to the conclusion. The user can input statements directly into the boxes shown on the screen in a way comparable to other argument mapping tools. An Example of an ArguMed Map The System ASPIC+ • ASPIC+ is built on a logical language containing a contrariness function, a set of strict and defeasible inference rules, and an ordering of this set of inference rules (Prakken, 2010). • Suppose that we have the goal of increasing productivity because we think that increasing productivity is good. And suppose we think that reducing taxes is the way to increase productivity. Then we should conclude that taxes should be reduced. • However we also accept the premises that reducing taxes increases inequality, that increasing inequality is bad, and that equality is more important than productivity. Then we should conclude that we should not reduce taxes. • The second argument defeats the first one, even though the conclusion of the first argument was acceptable before the second argument was considered. Abstract Argumentation Frameworks • ASPIC+ is built on the basis of abstract argumentation frameworks (Dung, 1995). An abstract argumentation framework is a structure used to determine which arguments can be accepted among a set of arguments in which some arguments defeat others. • The proponent starts with the argument he wants to prove and when the opponent has his turn, he must provide a defeating counterargument. • An abstract argumentation framework (AF) is defined as a pair (Args, Def ), where Args is a set of arguments and Def ⊆ Args × Args is a binary relation of defeat. Undercutters and Rebutters • ASPIC+ works by determining the success of rebutting and undercutting attacks that compare conflicts in arguments at the points where they conflict. • ASPIC+ is based on the system of (Pollock, 1995) where an important distinction was drawn between two kinds of refutations called rebutting defeaters and undercutting defeaters (often referred to as rebutters versus undercutters). • A rebutter gives a reason for denying a claim by arguing that the claim is false whereas an undercutter defeater attacks the inferential link between the claim and the reason supporting it (Pollock, 1995, 40). Famous Example • Pollock used a famous example (1995, 41) to illustrate his distinction. According to the example, if I see an object that looks red to me, that is a reason for thinking it is red, but if I know that the object is illuminated by a red light, that is an undercutter, because red objects look red in red light too. • This is not a reason for thinking the object is not red, but it is a reason for undercutting the argument that it is red. Despite the attacking argument, the object may be red, for all we know. • As an undercutter it acts like a critical question that casts an argument into doubt rather than strongly refuting it. • The use of three kinds of counterarguments that can be used to attack any given argument is highly characteristic of ASPIC+. The three kinds of counterarguments are undercutters, rebutters and premise attacks. The Carneades Argumentation System • Thomas F. Gordon (Fraunhofer Fokus) • There are many software packages to assist with argument diagramming (Scheuer et al., 2010), based on informal logic models of argument. • Carneades provides a number of software tools, including argument diagramming, based on a formal, computational model of argument. • Carneades is named after the Greek skeptical philosopher (Gordon and Walton, 2006) and is open source software, available at http://carneades.github.com/. Why Carneades is a Formal, Computational Model of Argument • computational, because the model consists of mathematical structure whose operations are all computable • formal, because there is a formal calculus for computable functions (lambda calculus) Argument Graphs • Definition (Argument Graph) An argument graph is a bipartite, directed, labeled graph, consisting of statement nodes and argument nodes connected by premise and conclusion edges. Formally, an argument graph is a structure ⟨, A, P, ⟩, where: • is a set of statement nodes, • is a set of argument nodes, • is a set of premises, and • is a set of conclusions. Argument Maps • In Carneades argument maps, statement nodes are shown as propositions in text boxes. • Argument nodes are displayed as circles, with a + or – sign inside the circle, to distinguish pro and con arguments, respectively. • Premises and conclusions are visualized as lines and arrows, respectively, connecting statement and argument nodes. Single Argument Linked Argument Convergent Argument Divergent Argument Serial Argument The Role of the Audience • Argument graphs are evaluated, relative to audiences, to determine the acceptability of statements in a stage. • Audiences are modelled as a set of assumptions and an assignment of weights to argument nodes. • Where L is a propositional language, an audience is a structure <assumptions, weight>, where assumptions ⊆ L is a consistent set of literals assumed to be acceptable by the audience and weight is a partial function mapping arguments to real numbers in the range 0.0...1.0, representing the relative weights assigned by the audience to the arguments. Acceptability Acceptability Acceptability Premise Attack Rebutter An Example of an Undercutter • Messerli (2012) found there was a close significant linear correlation between chocolate consumption per capita and the number of Nobel laureates in a country. • He found some theory-based biochemical evidence in the scientific observation that flavonoids present in cocoa are known to improve blood flow to the brain. • He used the form of argument called argument from correlation to cause. Argument from Correlation to Cause [Scheme in Carneades] id: correlation-to-cause strict: false direction: pro conclusion: Event E1 causes event E2. premises: Events E1 and E2 are correlated. assumptions: There exists a theory explaining how event E1 causes event E2. exceptions: Event E3 causes events E1 and E2. Undercutter Undercutter In, Out and Undecided • Statements which have been accepted or rejected by the audience are in or out, respectively. • The values of the remaining statement nodes are computed using proof standards and the weights assigned by the audience to the argument nodes. • This example shows how values can be propagated along an argumentation tree structure to prove an ultimate conclusion (the statement shown at the extreme left of the map). Propagation 1 Propagation 2 Propagation 3 Propagation 4 Propagation 5 Propagation 6 Propagation 7 Propagation 8 Standards and Schemes • Conflicts between pro and con arguments are resolved using proof standards, such as preponderance of the evidence and beyond reasonable doubt, inspired by the legal domain. • Carneades also formalizes argumentation schemes. • Schemes can be used to construct or reconstruct arguments, as well as to check whether arguments are “valid”, i.e. whether they properly instantiate the types of argument deemed normatively appropriate for the type of dialogue. Choosing a Scheme DMP • This form of argument can be used for evaluating defeasible inferences like the Tweety argument: If Tweety is a bird, Tweety flies; Tweety is a bird; therefore Tweety flies. • This form of argument was called defeasible modus ponens (DMP) by Walton (2002). • An example (Copi and Cohen, 1998, p. 363) also illustrates DMP: if he has a good lawyer then he will be acquitted; he has a good lawyer; therefore he will be acquitted. Form of DMP • Using a concept from defeasible logic called defeasible implication, or the defeasible conditional as it might be called, we can represent DMP is having the following form. Major Premise: A => B Minor Premise: A Conclusion: B • The first premise states the defeasible conditional, ‘If A is true then generally, but subject to exceptions, B is true’. • DMP is a scheme in DefLog, ASPIC+ and CAS. Argument Construction • Ballnat and Gordon (2010) provided a method of argument invention for Carneades, and Walton and Gordon (2012) have shown how the method can be applied in informal logic. • To use the system, an arguer first provides input on the premises he assumes the audience already accepts or not. • Then the system searches for minimal subsets of the remaining premises which together with the assumed premises are sufficient to (defeasibly) prove a given claim. • The arguer can then focus his attention on trying to construct arguments that support the premises in one of the minimal sets found. Menu for Finding an Argument Instantiating a Scheme Cycle in a Graph The RAS Triangle • Blair (2012, 87) wrote that when he and Ralph Johnson first wrote their textbook Logical Self-defense (first edition, 1977), they used the relevance-sufficiencyacceptability (RSA) triangle to determine whether an argument is a good one. • According to the RSA principle, an argument is a good one if its grounds (or premises) singly or in combination meet three criteria. 1. The premises have to be individually acceptable. 2. Taken together the premises have to be sufficient to support the claim that is the conclusion of the argument. 3. The argument needs to be relevant as a support for the conclusion. Sufficiency • Carneades is built around the idea of modelling sufficiency by using proof standards to aggregate pro and con arguments. • The conclusion of an argument is in (acceptable) if has been accepted by the audience or it satisfies the proof standard appropriate for the type of dialogue. • Proof standards have a legal flavor, and the notions of proof standards and burdens of proof modelled in Carneades are motivated by our interest in legal applications. • Several legal standards of proof exist, for example the preponderance of the evidence standard and the beyond reasonable doubt standard. Conclusions • Carneades has been used to illustrate some of the capabilities of current AI argumentation systems, but other systems such as DefLog and ASPIC+ are comparable. • These systems can be used to assist with making argument diagrams, but they can also be used to evaluate arguments, and to construct new arguments from a knowledge base to prove a conclusion. • They incorporate argumentation schemes and dialectical features of argumentation that can use standards of proof to model burden of proof shifts. • Argument invention is something that is fundamental to rhetoric. The notion of audience is basic for Carneades to evaluate and construct arguments. Some References Ballnat, S. and Gordon, T. F. (2010). Goal Selection in Argumentation Processes, Computational Models of Argument: Proceedings of COMMA 2010, ed. P. Baroni. Blair, J. A. (2001). Walton’s Argumentation Schemes for Presumptive Reasoning: A Critique and Development, Argumentation, 5, 365-379. Blair, J. A. (2012). Groundwork in the Theory of Argumentation. Dordrecht: Springer. Gordon, T. F., and Walton, D. (2009). Proof Burdens and Standards. In Argumentation in Artificial Intelligence, I. Rahwan and G. Simari, Eds. Springer-Verlag, Berlin, Germany, 2009, 239-260. Johnson, R. H. (2009). Some Reflections on the Informal Logic Initiatives, Studies in Logic, Grammar and Rhetoric, 16 (29), 17-46. Tindale, C. W. (1999). Acts of Arguing: A Rhetorical Model of Argument. Albany: State University of New York Press. Wadler, P. (2014). University of Edinburgh, Propositions as Types [preprint]: http://homepages.inf.ed.ac.uk/wadler/topics/recent.html Walton, D. and Gordon T. F. (2012). The Carneades Model of Argument Invention, Pragmatics & Cognition, 20(1), 1-31.