Law Logic Summerschool 2012 Session 3.2.1 (Part 1): Formalising argumentation

1. Law Logic Summerschool 2012Session 3.2.1 (Part 1):Formalising argumentation Henry Prakken July 18, 2012

2. Argument schemes: general form • But also critical questions • Negative answers are counterarguments Premise 1, … , Premise n Therefore (presumably), conclusion

3. Logical account of argument schemes • Argument schemes are inference rules • Critical questions are pointers to counterarguments • Some point to premise attacks • Some point to rebutting attacks • Some point to undercutting attacks • If all critical questions point to premise attack, then the inference rule is deductive, otherwise it is defeasible

4. Evaluating arguments Can be indirect • Does each step instantiate an acceptable argument scheme? • Have all its counterarguments been refuted? • Are its premises acceptable? • If defeasible: what about attacks on inference or conclusion? • Argument schemes help in identifying sources of doubt in an argument.

5. Contents • Formalisation of argumentation • Labelling semantics • Argument games • Positions

6. We should lower taxes Lower taxes increase productivity Increased productivity is good

7. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad

8. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity USA lowered taxes but productivity decreased

9. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … USA lowered taxes but productivity decreased

10. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Prof. P has political ambitions

11. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Prof. P has political ambitions

12. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good

13. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good

14. Dung A B E D C

15. 1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. Grounded semantics minimisesIn labelling Preferred semantics maximisesIn labelling Dung A B E D C P.M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and n–person games. Artificial Intelligence, 77:321–357, 1995.

16. Dialectical status of arguments • Justified: survive conflict with all counterarguments • In in all labellings • Overruled: defeated by justified argument • Out in all labellings • Defensible: neither justified not overruled

17. Justifying your argument: refute every attack Argument for claim Proponent: Proponent of argument must strictly defeat opponents arguments, Opponent may weakly defeat proponent’s arguments

18. Dialectical status: an argument game Argument for claim Proponent: Defeater 1 Defeater 2 Opponent:

19. Dialectical status: an argument game Argument for claim Proponent: Defeater 1 Defeater 2 Opponent: Strict defeater 1.1 Strict defeater 2.1 Proponent:

20. Dialectical status: an argument game Argument for claim Proponent: Defeater 1 Defeater 2 Opponent: Strict defeater 1.1 Strict defeater 2.1 Proponent: Defeater 2.1.1 Defeater 2.1.2 Opponent:

21. Dialectical status: an argument game Argument for claim Proponent: Defeater 1 Defeater 2 Opponent: The initial argument is justified in grounded semantics iff it can be successfully defended in an argument game, that is, if each branch end with an argument by proponent Strict defeater 1.1 Strict defeater 2.1 Proponent: Defeater 2.1.1 Defeater 2.1.2 Opponent: Strict Defeater 2.1.1.1 Strict defeater 2.1.1.2 Proponent:

22. Coherent positions • Let us call any set of arguments a position • A position P is coherent iff: • P is conflict-free (no arguments in P defeat each other) • Pdefends all its members (every defeater of an argument in P is defeated by an argument in P ) • Theorem: If a position P is coherent then there exists a labelling in which all arguments in P are labelled In.

23. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

24. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

25. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

26. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

27. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

28. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

29. S defends A if all defeaters of A are defeated by a member of S S is coherent if it is conflict-free and defends all its members A B E D C Coherent?

30. A B We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective E People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good C D

31. Coherentism vs. foundationalism • Coherentism ≈ beliefs are justified by their mutual relations of coherence • Foundationalism ≈ beliefs are justified by founding them on ultimate justified beliefs • Argumentation-based inference combines the two approaches: • Arguments must be built on acceptable premises • Arguments must be part of a coherent position

32. Formalising structured argumentation Arguments: Trees where Nodes are formulas of a logical language L Links are applications of inference rules Rs = Strict rules (all deductively valid inferences over L ) Rd= Defeasible rules (presumptive argument schemes) Attack: on premises On defeasible inferences On conclusions of defeasible inference Defeat: attack + preferences

33. What goes beyond mechanisation • The choice of premises • The choice of preferences • The selection of argument schemes