A Formal Model of Legal Proof Standards

1. A Formal Model of Legal Proof Standards • Thomas F. Gordon, Fraunhofer FOKUS, Berlin • Douglas Walton, University of Windsor, Windsor, Ontario

2. Proof in Mathematics and Law • Proof in Mathematics • Strong notion • Requires the construction and presentation of a deductively valid argument, proving a theorem. • Proof in Law • Weaker notion • Requires the presentation and weighing of evidence for and against some factual claim • At issue is whether the claim fact is sufficient “probable”, given the evidence, to meet the required proof standard. • Which proof standard applies depends on the procedural context. Examples: • Civil cases: preponderance of the evidence • Criminal cases: beyond reasonable doubt

3. Example: California Civil Procedure

4. Different Kinds of Proof Burdens • Burden of Claiming • Burden of Questioning • Burden of Production • Burden of Persuasion • Tactical Burden of Proof

5. Some Legal Proof Standards • Scintilla of Evidence • “Any material evidence that, if true, would tend to establish issue in mind of reasonable juror.” (Black’s Law Dictionary) • Preponderance of the Evidence • “Evidence which is of greater weight or more convincing than the evidence which is offered in opposition to it; that is, evidence which as a whole shows that the fact is more probable than not.” (Black’s) • Clear and Convincing Evidence • “That measure or degree of proof which will produce in mind of trier of fact a firm belief or conviction as to allegations sought to be established; it is intermediate, being more than preponderance, but not to extend of such certainty as is required beyond reasonable doubt as in criminal cases.” (Black’s) • Beyond Reasonable Doubt • “In evidence means fully satisfied, entirely convinced, satisfied to a moral certainty.” (Black’s)

6. The Carneades Formal Model of Argument • Primary Reference • Gordon, T. F., Prakken, H., and Walton, D. The Carneades model of argument and burden of proof. Artificial Intelligence 171, 10-11 (2007), 875–896. • The formalization of the model, but not its basic ideas, has been refined in the meantime. Latest version used in this talk • Defined concepts • argument • dialogue (argumentation process) • argument evaluation structure • acceptability of statements • applicability of arguments • proof standards

7. Argument • Let L be a propositional language • An argument is a tuple ⟨P,E,c⟩, where • P is a set of premises • E is a set of exceptions • c is the conclusion • The premises, exceptions and conclusion are all literals, i.e. atomic propositions (p)or negated atomic propositions (¬p), in L • An argument need not be deductively valid. • Arguments are dialectically valid if they are instantiations of argumentation schemes allowed by the procedural rules (protocol) of the dialogue type,

8. Example Argument from Expert Witness Testimony

9. Dialogue • A dialogue is a tuple ⟨O,A,C>, where O, A, C are the opening, argumentation and closing stages, respectively, of the dialogue, where each stage is a sequence of states. • An argument state is ⟨Γ,σ⟩ pair, where • Γ is a set of arguments and • σ is a function mapping each statement in the language L to its dialectical status, “claimed” or “questioned”, in the dialogue. • The status of statements in a state is only relevant for assessing whether the burdens of claiming and questioning have been met. • When we are only interested in evaluating the arguments of a single state, the model of a state can be simplified, to be the set of arguments, Γ, without the status function.

10. Audience • The audience is the agent or party which evaluates the arguments of a stage in a dialogue. • The audience can be • the proponent, in single-agent argumentation processes • the respondent, in two-party dialogues. • a third party, such as a judge or jury in legal cases • An audience is modeled as a tuple a ⟨Θ,f ⟩, where • Θis the set of arguments of the stage, and • f is a function mapping arguments to their weights, real numbers in the range 0.0 to 1.0.

11. Argument Evaluation Structure • Intuitively, an argument evaluation structure provides information needed to evaluate the arguments in a stage of dialogue. • An argument evaluation structure is tuple ⟨Γ,A,g⟩, where • Γis the set of arguments of a stage • A is an audience ⟨Θ,f ⟩, and • g is a function mapping literal propositions to their proof standard in the dialogue.

12. Acceptability • Let S = ⟨Γ,A,g⟩ be an argument evaluation structure. • An literal p is acceptable in S if and only if g(p)(p,Γ,A) is true. • That is, in English, only if the proposition p satisfies its proof standard given the arguments in the stage.

13. Derivability (In and Out) • Let S = ⟨Γ,A,g⟩ be an argument evaluation structure, where A is an the audience ⟨Θ,f ⟩. • A literal p is in S, denoted (Γ,Θ) ⊢f,g p, if and only if: • p ∈ Θ or • (¬p ∉ Θ and p is acceptable is S) • Otherwise p is out, denoted (Γ,Θ) ⊬f,g p

14. Argument Applicability • Let S = ⟨Γ,A,g⟩ be an argument evaluation structure. • An argument ⟨P,E,c⟩ is applicable in an argument evaluation structure if and only if: • the argument is a member of Γ, the arguments of the stage, • every literal p ∈ P is in, and • every literal p ∈ E is out.

15. Models of Some Legal Proof Standards • Scintilla of Evidence (SE) • There is at least one applicable argument • Preponderance of Evidence (PE) • SE is satisfied, and • the maximum weight assigned to an applicable pro argument is greater than the maximum weight of an applicable con argument. • Clear and Convincing Evidence • PE is satisfied • the maximum weight of applicable pro arguments exceeds some threshold α, and • the difference between the maximum weight of the applicable pro arguments and the maximum weight of the applicable con arguments exceeds some threshold β. • Beyond Reasonable Doubt (BD) • CE is satisfied and • the maximum weight of the applicable con arguments is less than some threshold γ.

16. Scintilla of Evidence Example The defendant killed the victim.

17. Preponderance of Evidence Example The defendant killed the victim.

18. Clear and Convincing Evidence Example The defendant killed the victim.

19. Beyond Reasonable Doubt Example The defendant killed the victim.

20. Prakken’s Three Principles of Accrual • Combining several arguments pro or con some proposition can not only strengthen one’s position, but also weaken it. • Once several arguments have been accrued, the individual arguments, considered separately, should have no impact on the acceptability of the proposition at issue, and • Finally, any argument which is ‘flawed’ may not take part in the aggregation process. The models of proof standards presented in the section are designed to respect these principals.

21. Accrual Example The defendant killed the victim. The defendant killed the victim.

22. Conclusions • Proof burdens and standards are relevant only in dialogical models of argumentation. • Relational models of argument evaluation, such as Dung argumentation frameworks are not intended to, and do not, model proofs. • While constructing a proof is hard and ill-defined problem, checking a proof should be well-defined and easy (tractable). Intuitively, part of the burden of proof is tasking responsibility for the hard work. • [Relation to probabilistic conceptions of proof standards???]