I Human Reasoning

I The Suppression Task

I Abduction

I Summary

Steffen HolldoblerWeak Completion Semantics 1

Weak Completion Semantics

Steffen HolldoblerInternational Center for Computational LogicTechnische Universitat DresdenGermany

Human Reasoning – Two Examples

I Instructions on boarding card distributed at Amsterdam Schiphol Airport

. If it’s thirty minutes before your flight departure, make your way to the gateAs soon as the gate number is confirmed, make your way to the gate

I Notice in London Underground

. If there is an emergency then you press the alarm signal bottomThe driver will stop if any part of the train is in a station

I Observations

. Intended meaning differs from literal meaning

. Rigid adherence to classical logic is no help in modeling the examples

. There seems to be a reasoning process towards more plausible meanings

II The driver will stop the train in a stationif the driver is alerted to an emergencyand any part of the train is in the stationKowalski: Computational Logic and Human Life:How to be Artificially Intelligent. Cambridge University Press 2011

Steffen HolldoblerWeak Completion Semantics 2

The Suppression Task

I Byrne: Suppressing Valid Inferences with Conditionals. Cognition 31, 61-83: 1989

C1 If she has an essay to write, then she will go to the libraryC2 if she has a textbook to read, then she will go to the libraryC3 If the library is open, then she will go to the libraryE She has an essay to writeL She will go to the library

E ¬E L ¬LC1 96% 46% 71% 92%C1&C2 96% 4% 13% 96%C1&C3 38% 63% 54% 33%

(L) (¬L) (E) (¬E)

Steffen HolldoblerWeak Completion Semantics 3

The Suppression Task – Modus PonensI Stenning, van Lambalgen: Human Reasoning and Cognitive Science

MIT Press: 2008

I Programse ← > fact definition of e` ← e ∧ ¬ab1 rule definition of `

ab1 ← ⊥ assumption ab1 is assumed to be false

I Weakly completed programs & least models

e ↔ > true false` ↔ e ∧ ¬ab1 e ab1 Φ ↑ 1

ab1 ↔ ⊥ ` Φ ↑ 2

I Computing logical consequences with respect to least models

M = 〈{e, `}, {ab1}〉 |=wcs `

I Łukasiewicz: O logice trojwartosciowej. Ruch Filozoficzny 5, 169-171: 1920

I H., Kencana Ramli: Logic Programs under Three-Valued Łukasiewicz’s SemanticsLNCS 5649, 464-478: 2009

Steffen HolldoblerWeak Completion Semantics 4

The Suppression Task – Alternative Argument

I Programse ← > fact definition of e` ← e ∧ ¬ab1 rule definition of `

ab1 ← ⊥ assumption ab1 is assumed to be false` ← t ∧ ¬ab2 rule definition of `

ab2 ← ⊥ assumption ab2 is assumed to be false

I Weakly completed programs & least models

e ↔ > true false` ↔ (e ∧ ¬ab1) ∨ (t ∧ ¬ab2) e ab1

ab1 ↔ ⊥ ab2 Φ ↑ 1ab2 ↔ ⊥ ` Φ ↑ 2

I Computing logical consequences with respect to least models

M = 〈{e, `}, {ab1, ab2}〉 |=wcs `

Steffen HolldoblerWeak Completion Semantics 5

The Suppression Task – Additional Argument

I Programse ← > fact definition of e` ← e ∧ ¬ab1 rule definition of `

ab1 ← ⊥ assumption ab1 is assumed to be false` ← o ∧ ¬ab3 rule definition of `

ab3 ← ⊥ assumption ab3 is assumed to be falseab1 ← ¬o rule definition of ab1ab3 ← ¬e rule definition of ab3

I Weakly completed programs & least models

e ↔ > true false` ↔ (e ∧ ¬ab1) ∨ (o ∧ ¬ab3) e Φ ↑ 1

ab1 ↔ ⊥∨ ¬o ab3 Φ ↑ 2ab3 ↔ ⊥∨ ¬e

I Computing logical consequences with respect to least models

M = 〈{e}, {ab3}〉 6|=wcs `

Steffen HolldoblerWeak Completion Semantics 6

The Suppression Task – Denial of the Antecedent

I Programse ← ⊥` ← e ∧ ¬ab1

ab1 ← ⊥

I Weakly completed programs & least models

e ↔ ⊥ true false` ↔ e ∧ ¬ab1 e

ab1 ↔ ⊥ ab1`

I Computing logical consequences with respect to least models

M = 〈∅, {e, ab1, `}〉 |=wcs ¬`

Steffen HolldoblerWeak Completion Semantics 7

The Suppression Task – Affirmation of the Consequent

I Programs` ← >` ← e ∧ ¬ab1

ab1 ← ⊥

I Weakly completed programs & least models

` ↔ >∨ (e ∧ ¬ab1) true falseab1 ↔ ⊥ ` ab1

I Computing logical consequences with respect to least models

M = 〈{`}, {ab1}〉 6|=wcs e

. Byrne 1989 most humans conclude e!

Steffen HolldoblerWeak Completion Semantics 8

AbductionI Hartshorn et. al.: Collected Papers of C. Sanders Peirce. Harvard Univ. Press: 1931

I Programs & observations` ← e ∧ ¬ab1 `

ab1 ← ⊥

I Abducibles

e ← > e ← ⊥

I Weakly completed programs plus explanations & least models

` ↔ e ∧ ¬ab1 true falseab1 ↔ ⊥ e ab1

e ↔ > `

I Computing logical consequences with respect to least models

M = 〈{e, `}, {ab1}〉 |=wcs e

I H., Philipp, Wernhard: An Abductive Model for Human Reasoning. In: Proc. Tenth Int.Symposium on Logical Formalizations of Commonsense Reasoning: 2011

Steffen HolldoblerWeak Completion Semantics 9

Alternative Arguments and Affirmation of the ConsequentI Programs & observations

` ← e ∧ ¬ab1 `ab1 ← ⊥

` ← t ∧ ¬ab2ab2 ← ⊥

I Abducibles

e ← > t ← > e ← ⊥ t ← ⊥

I Weakly completed programs plus explanations & least models

` ↔ (e ∧ ¬ab1) ∨ (t ∧ ¬ab2) true false true falseab1 ↔ ⊥ e ab1 t ab1ab2 ↔ ⊥ ab2 ab2

e ↔ > or t ↔ > ` `

I Computing skeptical consequences with respect to both models

. e does not follow

I Dietz, H., Ragni: A Computational Logic Approach to the Suppression TaskProc. COGSCI, 1500-1505: 2012

Steffen HolldoblerWeak Completion Semantics 10

Usually Birds Fly

I Programs & observations

fly(X) ← bird(X) ∧ ¬ab(X) ¬fly(tweety)ab(X) ← ⊥

bird(tweety) ← >bird(jerry) ← >

I Abducibles∅

I Weakly completed programs plus explanations & least models

fly(tweety) ↔ bird(tweety) ∧ ¬ab(tweety) true falsefly(jerry) ↔ bird(jerry) ∧ ¬ab(jerry) bird(tweety) ab(tweety)

ab(tweety) ↔ ⊥ bird(jerry) ab(jerry)ab(jerry) ↔ ⊥ fly(tweety)

bird(tweety) ↔ > fly(jerry)bird(jerry) ↔ >

. We cannot explain the observation!

Steffen HolldoblerWeak Completion Semantics 11

Abducibles Revised

I Programs & observations

fly(X) ← bird(X) ∧ ¬ab(X) ¬fly(tweety)ab(X) ← ⊥

bird(tweety) ← >bird(jerry) ← >

I Abducibles

ab(tweety) ← > ab(jerry) ← >

I Weakly completed programs plus explanations & least models

fly(tweety) ↔ bird(tweety) ∧ ¬ab(tweety) true falsefly(jerry) ↔ bird(jerry) ∧ ¬ab(jerry) bird(tweety) ab(jerry)

ab(tweety) ↔ ⊥∨> bird(jerry)ab(jerry) ↔ ⊥ ab(tweety)

bird(tweety) ↔ > fly(jerry) fly(tweety)bird(jerry) ↔ >

I Dietz Saldanha, H., Pereira: Contextual Reasoning: Usually Birds can AbductivelyFly. In: Proc. LPNMR 2017 (to appear)

Steffen HolldoblerWeak Completion Semantics 12

Summary

I Usually,

. P denotes a program, i.e. a finite set of facts, rules and assumptions

. gP denotes the ground instance of P

. wcP denotes the weak completion of gP

. MP denotes the least model of wcP

. AP denotes the set of abducibles of gP

. IC denotes a finite set of integrity constraints of the form U← L1 ∧ . . . ∧ Ln

I Weak Completion Semantics is the approach

. to consider weakly completed programs

. to compute their least models

. and to reason with respect to these models

Steffen HolldoblerWeak Completion Semantics 13