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Tutorial: Nonmonotonic Logic PhDs in Logic (2017) Christian Straßer May 2, 2017

Outline Defeasible Reasoning Scratching the Surface of Nonmonotonic Logic 1/52

Defeasible Reasoning

What is defeasible reasoning? 1/52

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abductive inference infer a (good!?) explanation α β and β thus α 2/52

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closed world assumption reasoning on the assumption that the given information is complete 3/52

default inferences Tweety is a bird. Thus,...? 4/52

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inductive generalisations 5/52

Domains of defeasible reasoning everyday reasoning expert reasoning (e.g. medical diagnosis) scientific reasoning 6/52

Commonalities tentative conclusions jumping to conclusions retraction possible if problems arise 7/52

Commonalities tentative conclusions jumping to conclusions retraction possible if problems arise Two tiers of defeasible reasoning 1. illative tier (support, concluding) 2. dialectic tier (retraction) 7/52

Pessimism in the 60ies Toulmin in The Uses of Argument, 1958 8/52

Toulmin Scheme (Toulmin, 1958) Premises Conclusion Backing Warrant Defeat 9/52

Nonmonotonic Logic to the Rescue Artificial Intelligence, Volume 13, Issues 1 2, Pages 1-174,(April 1980), Special Issue on Non-Monotonic Logic 10/52

Nonmonotonic Logic to the Rescue Artificial Intelligence, Volume 13, Issues 1 2, Pages 1-174,(April 1980), Special Issue on Non-Monotonic Logic capture defeasible reasoning in a mathematically precise way reproduce the success of CL in the domain of mathematical reasoning in the less sterile/idealized domain of defeasible reasoning where incompleteness 10/52

Nowadays cooperations between formal and informal logicians e.g., Douglas Walton and formal argumentation (e.g., Gordon, Prakken and Walton, Artificial Intelligence, 2007) shift of normative standards in cognitive science, Stenning and Van Lambalgen, 2008, MIT Press Pfeifer, Studia Logica, 2014 11/52

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Monotonicity If A 1,..., A n B then A 1,..., A n, A n+1 B. 14/52

Monotonicity If A 1,..., A n B then A 1,..., A n, A n+1 B. Premises/Input A 1,..., A n, Logic Conclusions/Output B 1,..., B m, B 14/52

Monotonicity If A 1,..., A n B then A 1,..., A n, A n+1 B. Premises/Input A 1,..., A n, A Logic Conclusions/Output B 1,..., B m, B 14/52

Cautious Monotonicity If A 1,..., A n B and A 1,..., A n C, then A 1,..., A n, B C. 15/52

Cautious Monotonicity If A 1,..., A n B and A 1,..., A n C, then A 1,..., A n, B C. Premises/Input A 1,..., A n Logic Conclusions/Output B, C 15/52

Rational Monotonicity If it is not the case that A 1,..., A n B, and moreover A 1,..., A n C, then A 1,..., A n, B C. 16/52

Rational Monotonicity If it is not the case that A 1,..., A n B, and moreover A 1,..., A n C, then A 1,..., A n, B C. Premises/Input A 1,..., A n, Logic Conclusions/Output C B My beliefs are robust/cumulative under adding consistent information. 16/52

Rational Monotonicity If it is not the case that A 1,..., A n B, and moreover A 1,..., A n C, then A 1,..., A n, B C. Premises/Input A 1,..., A n, B Logic Conclusions/Output C B My beliefs are robust/cumulative under adding consistent information. 16/52

The nonmononotonic Zoo See: Straßer, C., & Antonelli, G. A. (2014). Non-monotonic logic. In E. N. Zalta (Eds.), The Stanford Encyclopedia of Philosophy inheritance networks default logic logic programming autoepistemic logic circumscription preferential semantics maximal consistent subset approaches adaptive logics formal argumentation etc. 17/52

Scratching the Surface of Nonmonotonic Logic

In the following we will work with defeasible theories consisting of 1. non-defeasible information F including strict rules 2. defeasible information D including defeasible rules How exactly these defeasible theories are expressed in terms of a formal language depends on the underlying formalism. 18/52

Default logic A B: B follows defeasibly from A. 19/52

Default logic A B: B follows defeasibly from A. Example: (slightly enhanced) Nixon Diamond facts: {Nixon Republican, Nixon Quaker} defeasible information: {Republican pacifist, Quaker pacifist, Quaker pro-syndicate, Republican pro-gun} pro-gun Nixon is a is a Republican Quaker Pacifist Pacifist pro-syndicate 19/52

pro-gun is a Republican Pacifist Nixon is a Quaker Pacifist pro-syndicate Idea: Apply iteratively modus ponens to defaults while preserving consistency. This way build step-wise an extension (sets of beliefs that are obtained in this way). This may lead to several choices. 20/52

Basic Idea: Extension 1 pro-gun is a Republican Pacifist Nixon is a Quaker Pacifist pro-syndicate 21/52

Basic Idea: Extension 2 pro-gun is a Republican Pacifist Nixon is a Quaker Pacifist pro-syndicate 22/52

Basic Idea: Extension 2 pro-gun is a Republican Pacifist Nixon is a Quaker Pacifist pro-syndicate There are two extensions: Cn({pacifist, Nixon, Republican, Quaker, pro-syndicate, pro-gun}) Cn({ pacifist, Nixon, Republican, Quaker, pro-syndicate, pro-gun}). 22/52

Default consequences Question What to derive from a defeasible theory? Answer(s) build all extensions of the theory Sceptical reasoner: derive what is in the intersection of all extensions e.g., do not derive pacifist from the Nixon diamond Credulous reasoner: derive what is in the union of all extensions e.g., derive both pacifist and pacifist from the Nixon diamond mapping out rational choices 23/52

Floating Conclusions Nixon Republican Quaker Hawk Dove political where Hawk Dove and Dove Hawk What are the extensions? Does political follow? 24/52

Floating Conclusions: Extension 1 Nixon Republican Quaker Hawk Dove political {Nixon, Republican, Hawk, Quaker, political, Dove} 25/52

Floating Conclusions: Extension 2 Nixon Republican Quaker Hawk Dove political {Nixon, Republican, Hawk, Quaker, political, Dove} 26/52

Floating Conclusions: Extension 3 Nixon Republican Quaker Hawk Dove political {Nixon, Republican, Hawk, Quaker, political, Dove} 27/52

Reasoning by Cases no handling of disjunctive facts out-of-the-box for instance: Σ = {Republican Democrat, Republican political, Democrat political}.? Republican Republican Democrat? political Democrat since the default is not triggered by the fact, MP cannot be applied 28/52

Extension-based Approaches: Default Logic (Reiter) idea: split the factual part of the knowledge base (Gelfond, Lifschitz, Przymusinska, 1991) Republican Republican Base 1 Republican Democrat political Base 2 Democrat Democrat 29/52

Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb rhb 30/52

Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb rhb 2. If somebody writes legibly then usually the right hand is not broken. wl rhb 3. He writes legibly. wl 30/52

Different Idea Meta-rule for defaults: If A B and C B then A C B. OR 31/52

A Problematic Example for OR Suppose we have Σ = {p q r, q s, s v, r u, u v, p}. q s v p q r v r u v 32/52

A Problematic Example for OR q s v p q r v r u v 33/52

A Problematic Example for OR q s v p q r s u v r u v by (OR): from s v and u v 33/52

A Problematic Example for OR q s v p q r s u v r u v by (OR): from s v and u v by (Right-Weakening), from q s and r u 33/52

A Problematic Example for OR q s v p q r s u v r u v by (OR): from s v and u v by (Right-Weakening), from q s and r u by (OR): from q s u and r s u 33/52

A Problematic Example for OR t s! q s v p q r s u v r u v Suppose now we also have t and t s. the possible defeater has no effect on the generalized path 34/52

A Problematic Example for OR t s! q s v p q r s u v t! r r u v Suppose now we also have t and t r. the additional possible defeater has no effect on the generalized path 35/52

Greedy Reasoning (Conclusion maximizing) Best-candidate job new-car new-car Can we derive job? 36/52

Problem with Cautious Monotonicity Birthday party: we know Will comes and that (Will Diane Eli Peter): Will Diane Eli Peter 37/52

Problem with Cautious Monotonicity Birthday party: we know Will comes and that (Will Diane Eli Peter): Will Diane Eli Peter If we add Eli to the facts: we also have the following extension: Will Diane Eli Peter 37/52

Another approach: adaptive logics / default assumptions Model defeasible rules as strict rules with defeasible assumptions: Will Diane becomes Will π 1 Diane Diane Eli becomes Diane π 2 Eli Eli Peter becomes Eli π 3 Peter. where π 1, π 2, π 3 are normality assumptions which are assumed to be true as much as possible. π 1 π 2 π 3 Will Diane Eli Peter 38/52

Another approach: adaptive logics / default assumptions π 1 π 2 π 3 Will Diane Eli Peter since we know that not all of them are at the party, it cannot be that all π 1, π 2 and π 3 are true. 39/52

Another approach: adaptive logics / default assumptions π 1 π 2 π 3 Will Diane Eli Peter since we know that not all of them are at the party, it cannot be that all π 1, π 2 and π 3 are true. We have the following options: {π 1, π 2 } {π 1, π 3 } {π 2, π 3 } {π 1 } {π 2 } {π 3 } 39/52

Contraposition π 1 π 2 π 3 Will Diane Eli Peter Note that in this approach defeasible rules are contrapositable! 40/52

Contraposition π 1 π 2 π 3 Will Diane Eli Peter Note that in this approach defeasible rules are contrapositable! π 1 π 2 π 3 Will Diane Eli Peter 40/52

Consequence Relation Given a set of premises Σ and a set of normality assumptions Π we define: Definition Ξ MCS(Σ, Π) iff 1. Ξ Π 2. Ξ Σ is consistent 3. for every Ξ Π, if Ξ Ξ then Ξ Σ is inconsistent. Definition Σ Π A iff A Ξ MCS(Σ,Π) Cn(Ξ Σ). 41/52

Preferential Semantics Let M(Σ) be the set of all models of Σ. We order them as follows: M Π M iff Π(M ) Π(M). Define: Σ Π A iff M = A for all M min Π (Σ) 42/52

Preferential Semantics Let M(Σ) be the set of all models of Σ. We order them as follows: M Π M iff Π(M ) Π(M). Define: Σ Π A iff M = A for all M min Π (Σ) Sometimes the perspective is inverted: instead of assuming normality assumptions to be true, we assume abnormality assumptions to be false. So, now Π contains abnormalities! 42/52

An Application: Inconsistency-adaptive logics in CLuN all connectives have the same truth-tables as in classical logic, just the table for negation is indeterministic: A A 0 1 1 [0/1] 43/52

An Application: Inconsistency-adaptive logics in CLuN all connectives have the same truth-tables as in classical logic, just the table for negation is indeterministic: A A 0 1 1 [0/1] note that A, A CLuN B. The logic is paraconsistent. 43/52

Example Let Σ = { p q, p, r, r, r s}. We have the following model types: model p p q q r r s s M 1 Yes M 2 Yes Yes M 3 Yes Yes M 4 Yes Yes M 5 Yes Yes Yes M 6 Yes Yes Yes M 7 Yes Yes Yes M 8 Yes Yes Yes Yes Note that M 1 is minimally abnormal! (i.e., M 1 M i where 1 i 8). 44/52

Other examples Σ = {p q, p r, q, r, q r}. Does p follow? What are the minimally abnormal models? 45/52

Shoham / Kraus-Lehmann-Magidor General perspective: impose a partial order on a given class of all models S read: M M as M is more normal than M (S, ) is called a preferential structure (given smoothness) define Σ S A iff for all -minimal models M in S of Σ, M = A. 46/52

Shoham / Kraus-Lehmann-Magidor Each such S fulfills the following properties: Reflexivity: A A Cut: If A B C and A B then A C Cautious monotonicity: If A B and A C then A B C. Left Logical Equivalence: If A B then A C iff B C. Right Weakening: If A B and C A then C B. OR: If A C and B C then A B C. is called a preferential consequence relation if it fulfills these properties. 47/52

Shoham / Kraus-Lehmann-Magidor Theorem is a preferential consequence relation iff = S for some preferential structure S. 48/52

Cautious Monotonicity Blackboard. 49/52

No Rational Monotonicity Blackboard. 50/52

Other properties Where S, is a preferential structure. Do we get: Relevance: If bird S fly, penguin S fly then penguin lives-in-alaska S fly? 51/52

Other properties Where S, is a preferential structure. Do we get: Relevance: If bird S fly, penguin S fly then penguin lives-in-alaska S fly? Drowning: If bird S fly, bird S wings, penguin S fly then penguin S wings. 51/52

Thank you! 52/52