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Visualizing Arguments Using Carneades

Thomas F. GordonFraunhofer FOKUS, Berlin, Germanythomas.gordon@fokus.fraunhofer.de

Doug WaltonCentre for Research in Reasoning, Argumentation & Rhetoric (CRRAR),University of Windsor, Windsor, Canadadwalton@uwindsor.ca

October, 2009

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The Carneades Model of Argument

Carneades models – the structure and applicability of arguments– the acceptability of statements– burdens of proof (questioning, production and persuasion)– proof standards, such as preponderance of the evidece

Carneades is a – mathematical model: It consists of definitions of mathematical structures and

functions on these structures.– computational model: the functions of the model are all computable (i.e.

recursive, decidable)– semantic model: it defines arguments and (interesting) relations on arguments,

not a syntax for representing or encoding arguments. Carneades has been implemented using a functional programming language.

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Carneades (c. 213 - c. 128 B.C.)

Leader of the Academic Skeptics and head of Plato’s Academy

Developed a theory enabling human action despite the lack of perfect knowledge, based on “reasonable grounds” and “probable” inferences.

Carneades’ notion of probability corresponds closely to what is now called defeasible, plausible, presumptive or nonmonotonic reasoning.

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Linked Arguments

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Convergent Arguments

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Serial Arguments

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Pros and Cons

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Argumentation Schemes

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Premise Roles in Schemes

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Proof Burdens and Standards

Who? vs. How Much?– Who has the burden? (burden of proof)– How much burden is there? (proof standard)

Three kinds of proof burdens– Burden of Questioning– Burden of Production– Burden of Persuasion

Example proof standards, from the legal domain– Preponderance of Evidence– Beyond Reasonable Doubt

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Burden of Proof and Critical Questions

Two theories of what happens to the burden of proof when a critical question is asked:

■ Thesis: Shifting Burden Theory (SB)

• The respondent has the burden of questioning, but once the question is asked the burden shifts back to the proponent, who then has the burden of production and persuasion.

■ Antithesis: Backup Evidence Theory (BE)

• The respondent has not only the burden of questioning but also the burden of production and perhaps the burden of persuasion.

■ Synthesis: Depends on the Critical Question

• Which theory applies depends on the particular critical question.

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Example: Argumentation Scheme from Expert Opinion

■ The scheme

• Major Premise: E is an expert in subject domain S containing proposition A.

• Minor Premise: E asserts A.• Conclusion: A is true.

■ Critical questions which shift the burden back to the proponent

• How credible is E as an expert source?

• Is E’s assertion based on evidence?

■ Critical questions which require the respondent to provide backup evidence

• Is E personally reliable as a source?

• Is A consistent with what other experts assert?

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Modeling Critical Questions in Carneades

Critical questions are modeled as additional premises, of two kinds:Assumptions shift burden back to proponent after questioningExceptions place burden on respondent to provide backup evidence

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Examples of an Exception and an Assumption

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Argument Evaluation

Basic Idea– The acceptability and rejectability of a statement depends on its proof standard. – Whether or not a proof standard is satisfied by a statement depends on the

applicability of the arguments pro and con the statement. – The applicability of an argument depends on whether all its premises hold. (1)– Finally, whether a premise holds depends on the type of premise (ordinary,

negative, assumption, exception), its status (stated, at issue, accepted, rejected) and whether or not its statement is acceptable or rejectable.

These functions are mutually recursive.

(1) Somewhat unintuitively, exceptions must also hold for an argument to be defensible.

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Acceptability

Intuitively, a statement is acceptable iff the given arguments provide sufficient justification for deciding to accept the statement. That is, it would be possible to justify a decision to accept the statement.

For each statement P and its complement ¬P, there are four possibilities:1. P acceptable, ¬P not acceptable2. P not acceptable, ¬P acceptable3. P acceptable, ¬P acceptable4. P not acceptable, ¬P not acceptable

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Holding of Premises

An ordinary premise holds iff its statement has been assumed true, accepted or, if at issue, is acceptable (i.e. its proof standard is satisfied).

A negative premise holds iff its statement has been assumed false, rejected or, if at issue, the complement of its statement is acceptable.

An exception holds iff it would not hold if it were an ordinary premise, i.e. iff its statement is assumed false, rejected, or, if at issue, does not meet its proof standard.

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Argument Context

Arguments are evaluated in a context consisting of:– status: statement → {stated, issue, accepted, rejected}– proof-standard: statement → {SE, DV, PE, CCE, BRD}– weight: argument → 0.0-1.0

The status of statements is set by speech acts, during the dialogue. The context can change as the dialogue progresses.

The proof standard appropriate for some statement depends on the problem domain and dialogue type, perhaps even the stage of the dialogue.

Argument weights are used only by predonderance of the evidence (PE), clear and convincing evidence (CCE) and beyond reasonable doubt (BRD) standards.

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Proof Standards

Example proof standards– SE – Scintilla of Evidence– PE – Preponderance of the Evidence– CCE - Clear and Convincing Evidence– BRD – Beyond a Reasonable Doubt– DV – Dialectical Validity

The SE and DV standards do no use weights. The other standards do. The legal proof standards are ordered: SE < PE < CCE < BRD < DV. A statement

which satisfies one of these standards also satisfies all weaker standards. (Assumption: there is at least one pro argument which weighs more than alpha, the threshold weight for the CCE standard.)

Example: The PE standard is satisfied iff there is at least one applicable pro argument which is stronger than any applicable con argument.

Note: Arguments are not aggregated by summing weights, since arguments are not assumed to be independent.

Other proof standards can be defined, similarly.

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Using Exceptions

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Another Example: Pollock’s Undercutting Defeaters

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Using Argument Weights and Proof Standards

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Conclusions

Carneades is a formal, mathematical model of argument graphs which applies proof standards to determine the acceptability of arguments and the acceptability of statements in argument graphs, on an issue-by-issue basis.

The diagramming method for Carneades presented here captures more information than other popular methods (e.g. proof standards, schemes, critical questions, roles of premises).

Carneades is one of the few systems, like Verheij’s ArguMed, in which argument diagrams are views onto an underlying mathematical model of arguments

The mathematical model allows interesting properties to be computed and visualized automatically, such as argument applicability and statement acceptability.

Carneades subsumes all the features of the Beardsley/Freeman, Toulmin and Wigmore methods of argument diagramming, but on the foundation of a mathematical model of argument evaluation.