Transcript Forms of Reasoning: Deduction, Abduction, Induction

Computational Logic
and Cognitive Science:
An Overview
Session 1: Logical Foundations
ICCL Summer School 2008
Technical University of Dresden
25th of August, 2008
Helmar Gust & Kai-Uwe Kühnberger
University of Osnabrück
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Who we are…
Helmar Gust
Kai-Uwe Kühnberger
Interests: Analogical
Reasoning, Logic
Programming, E-Learning
Systems, Neuro-Symbolic
Integration
Interests: Analogical
Reasoning, Ontologies,
Neuro-Symbolic
Integration
Where we work:
University of Osnabrück
Institute of Cognitive Science
Working Group: Artificial Intelligence
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Cognitive Science in Osnabrück
 Institute of Cognitive Science
 International Study Programs
 Bachelor Program
 Master Program
 Joined degree with
Trento/Rovereto
 PhD Program
 Doctorate Program
“Cognitive Science”
 Graduate School
“Adaptivity in Hybrid Cognitive Systems”
 Web: www.cogsci.uos.de
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Who are You?
 Prerequisites?


Logic?
 Propositional logic, FOL, models?
 Calculi, theorem proving?
 Non-classical logics: many-valued logic, non-monotonicity,
modal logic?
Topics in Cognitive Science?
 Rationality (bounded, unbounded, heuristics), human
reasoning?
 Cognitive models / architectures (symbolic, neural, hybrid)?
 Creativity?
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Overview of the Course
 First Session (Monday)

Foundations: Forms of reasoning, propositional and FOL, properties of
logical systems, Boolean algebras, normal forms
 Second Session (Tuesday)

Cognitive findings: Wason-selection task, theories of mind, creativity,
causality, types of reasoning, analogies
 Third Session (Thursday morning)

Non-classical types of reasoning: many-valued logics, fuzzy logics,
modal logics, probabilistic reasoning
 Fourth Session (Thursday afternoon)

Non-monotonicity
 Fifth Session (Friday)


Analogies, neuro-symbolic approaches
Wrap-up
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Forms of Reasoning:
Deduction, Abduction,
Induction
Theorem Proving,
Sherlock Holmes,
and All Swans are White...
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Basic Types of Inferences:
Deduction
 Deduction: Derive a conclusion from given axioms
(“knowledge”) and facts (“observations”).
 Example:
All humans are mortal.
Socrates is a human.
(axiom)
(fact/ premise)
Therefore, it follows that Socrates is mortal.
(conclusion)
 The conclusion can be derived by applying the modus ponens
inference rule (Aristotelian logic).
 Theorem proving is based on deductive reasoning techniques.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Basic Types of Inferences:
Induction
 Induction: Derive a general rule (axiom) from background
knowledge and observations.
 Example:
Socrates is a human
Socrates is mortal
(background knowledge)
(observation/ example)
Therefore, I hypothesize that all humans are mortal
(generalization)
 Remarks:

Induction means to infer generalized knowledge from example
observations: Induction is the inference mechanism for
(machine) learning.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Basic Types of Inferences:
Abduction
 Abduction: From a known axiom (theory) and some
observation, derive a premise.
 Example:
All humans are mortal
Socrates is mortal
(theory)
(observation)
Therefore, Socrates must have been a human (diagnosis)
 Remarks:

Abduction is typical for diagnostic and expert systems.




If one has the flue, one has moderate fewer.
Patient X has moderate fewer.
Therefore, he has the flue.
Strong relation to causation
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Deduction
 Deductive inferences are also called theorem proving or logical
inference.

Deduction is truth preserving: If the premises (axioms and
facts) are true, then the conclusion (theorem) is true.
 To perform deductive inferences on a machine, a calculus is
needed:

A calculus is a set of syntactical rewriting rules defined for
some (formal) language. These rules must be sound and
should be complete.
 We will focus on first-order logic (FOL).


 Syntax of FOL.
 Semantics of FOL.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Propositional Logic
and First-Order Logic
Some rather Abstract Stuff…
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Propositional Logic
 Formulas:

Given is a countable set of atomic propositions AtProp = {p,q,r,...}.
The set of well-formed formulas Form of propositional logic is the
smallest class such that it holds:




p  AtProp: p  Form
,   Form:     Form
,   Form:     Form
  Form:
  Form
 Semantics:



A formula  is valid if  is true for all possible assignments of the
atomic propositions occurring in 
A formula  is satisfiable if  is true for some assignment of the
atomic propositions occurring in 
Models of propositional logic are specified by Boolean algebras
(A model is a distribution of truth-values over AtProp making  true)
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Propositional Logic
 Hilbert-style calculus
 Axioms:








 p  (q  p)
 [p  (q  r)]  [(p  q)  (p  r)]
 (p  q)  (q  p)
pqp
and
 (p  q)  q
 (r  p)  ((r  q)  (r  p  q))
 p  (p  q) and
 q  (p  q)
 (p  r)  ((q  r)  (p  q  r))
Rules:
Modus Ponens: If expressions  and    are provable then 
is also provable.
Remark: There are other possible axiomatizations of propositional
logic.


Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Propositional Logic
 Other calculi:

Gentzen-type calculus
http://en.wikipedia.org/wiki/Sequent_calculus

Tableaux-calculus
http://en.wikipedia.org/wiki/Method_of_analytic_tableaux
 Propositional logic is relatively weak: no temporal or
modal statements, no rules can be expressed
 Therefore a stronger system is needed
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
First-Order Logic
 Syntactically well-formed first-order formulas for a signature
 = {c1,...,cn,f1,...,fm,R1,...,Rl} are inductively defined.

The set of Terms is the smallest class such that:



A variable x  Var is a term, a constant ci  {c1,...,cn} is a term.
 Var is a countable set of variables.
If fi is a function symbol of arity r and t1,...,tr are terms, then fi(t1,...,tr) is a term.
The set of Formulas is the smallest class such that:



If Rj is a predicate symbol of arity r and t1,...,tr are terms, then Rj(t1,...,tr) is a
formula (atomic formula or literal).
For all formulas  and :   ,   , ,   ,    are formulas.
If x  Var and  is a formula, then x and x are formulas.
 Notice that “term” and “formula” are rather different concepts.

Terms are used to define formulas and not vice versa.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
First-order Logic
 Semantics (meaning) of FOL formulas.

Expressions of FOL are interpreted using an interpretation function
I:   ()

I(ci)  
I(fi) : arity(fi)  
I(Ri) : arity(Ri)  {true, false}
 is the called the universe or the domain

A pair  = <,I> is called a structure.



Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
First-order Logic
 Semantics (meaning) of FOL formulas.

Recursive definition for interpreting terms and evaluating truth values
of formulas:








For c  {c1,...,cn}: [[ci]] = I(ci)
[[fi(t1,...,tr)]] = I(fI)([[t1]],...,[[tr]])
[[R(t1,...,tr)]] = true
iff
[[  ]] = true
iff
[[  ]] = true
iff
[[]] = true
iff
[[x (x)]] = true
iff
[[x (x)]] = true
iff
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
<[[t1]],...,[[tr]]>  I(R)
[[]] = true and [[]] = true
[[]] = true or [[]] = true
[[]] = false
for all d  : [[(x)]]x=d = true
there exists d  : [[(x)]]x=d = true
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
First-order Logic
 Semantics

Model



If the interpretation of a formula  with respect to a structure  = <,I>
results in the truth value true,  is called a model for  (formal:
  )
Validity
 If every structure  = <,I> is a model for  we call  valid ( )
Satisfiability


If there exists a model  = <,I> for  we call  satisfiable
Example:

xy (R(x)  R(y)  R(x)  R(y))
[valid]
 „If x and y are rich then either x is rich or y is rich“
 „If x and y are even then either x is even or y is even“
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
First-order Logic
 Semantics

An example:

 x (N(x)  P(x,c))




[satisfiable]
„There is a natural number that is smaller than 17.“
„There exists someone who is a student and likes logic.“
Notice that there are models which make the statement false
Logical consequence

A formula  is a logical consequence (or a logical entailment)
of A = {A1,...,An}, if each model for A is also a model for .


We write A  
Notice: A   can mean that A is a model for  or that  is a logical
consequence of A
 Therefore people usually use different alphabets or fonts to make this
difference visible
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Theories
 The theory Th(A) of a set of formulas A:
Th(A) := { | A  }
 Theories are closed under semantic entailment
 The operator:
Th : A  Th(A)
is a so called closure operator:



X  Th(X)
X  Y  Th(X)  Th(Y)
Th(Th(X)) = Th(X)
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
extensive / inductive
monotone
idempotent
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
First-order Logic
 Semantic equivalences

Two formulas  and  are semantically equivalent (we write   ) if for all
interpretations of  and  it holds:  is a model for  iff  is a model for .


The following statements are equivalent (based on the deduction theorem):




A few examples:
 
 
   (  )  (  )  (  )
G is a logical consequence of {A1,...,An}
A1  ...  An  G is valid
 Every structure is a model for this expression.
A1  ...  An  G is not satisfiable.
 There is no structure making this expression true
This can be used in the resolution calculus: If an expression
A1  ...  An  G is not satisfiable, then false
can be derived syntactically.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Repetition: Semantic
Equivalences

Here is a list of semantic equivalences










(  )  (  ), (  )  (  )
(  )      (  ), (  )      (  )
(  (  ))  , (  (  ))  
(  (  ))  (  )  (  )
(  (  ))  (  )  (  )
  
(  )  (  ), (  )  (  )
(  )  , (  )  
(  )  , (  )  
(commutativity)
(associativity)
(absorption)
(distributivity)
(distributivity)
(double negation)
(deMorgan)
Here are some more semantic equivalences







(  )  , (  )  
    
    
x  x, x  x
(x   )  x (  ), (x   )  x (  )
x(  )  (x  x)
Etc.
(idempotency)
(tautology)
(contradiction)
(quantifiers)
ICCL Summer School 2008
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
Technical University of Dresden, August 25th – August 29th, 2008
Properties of Logical Systems
 Soundness

A calculus is sound, if only such conclusions can be derived which
also hold in the model

In other words: Everything that can be derived is semantically true
 Completeness

A calculus is complete, if all conclusions can be derived which hold
in the models

In other words: Everything that is semantically true can syntactically be derived
 Decidability

A calculus is decidable if there is an algorithm that calculates
effectively for every formula whether such a formula is a theorem or
not
 Usually people are interested in completeness results and decidability
results
 We say a logic is sound/complete/decidable if there exists a calculus with
these properties
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Some Properties of Classical
Logic
 Propositional Logic:
 Sound and Complete, i.e. everything that can be proven is
valid and everything that is valid can be proven
 Decidable, i.e. there is an algorithm that decides for every
input whether this input is a theorem or not
 First-order logic:
 Complete (Gödel 1930)
 Undecidable, i.e. no algorithm exists that decides
for every input whether this input is a theorem or
not (Church 1936)
 More
precisely FOL is semi-decidable
 Models
 The classical model for FOL are Boolean algebras
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Boolean Algebras
 P  [[P]] 

if arity is 1 (or [[P]]  ...  if arity > 1)
  x1,...,xn: P(x1,...,xn)  Q(x1,...,xn)  [[P]]  [[Q]]
 We can draw Venn diagrams:

P
Q
 Regions (e.g. arbitrary subsets) of the n-dimensional real space
can be interpreted as a Boolean algebra
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Boolean Algebras
 The power set () has the following properties:






It is a partially ordered set with order 
A  B is the largest set X with X  A and X  B
A  B is the smallest set X with A  X and B  X
comp(A) is the largest set X with A  X = 
 is the largest set in (), such that X   for all X ()
 is the smallest set in (), such that   X for all X ()
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Boolean Algebras
 The concept of a lattice

Definition: A partial order  = <D,> is called a lattice if for each
two elements x,y  D it holds: sup(x,y) exists and inf(x,y) exists


sup(x,y) is the least upper bound of elements x and y
inf(x,y) is the greatest lower bound of x and y
 The concept of a Boolean Algebra

Definition: A Boolean algebra is a tuple  = <D,,,,> (or
alternatively <D,,,,,>) such that



<D,> = <D,,> is a distributive lattice
 is the top and  the bottom element
 is a complement operation
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Lindenbaum Algebras
 The Linbebaum algebra for propositional logic with atomic propositions
p and q
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Normal Forms
 If there are a lot of different representations of the same statement

Are there simple ones?
 Are there “normal forms”?
 Different normal forms for FOL

Negation normal form


Prenex normal form


Only conjunctions of disjunctions
Disjunctive normal form


No embedded Quantifiers
Conjunctive normal form


Only negations of atomic formulas
Only disjunctions of conjunctions
Gentzen normal form

Only implications where the condition is an atomic conjunction and the conclusion is
an atomic disjunction
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Normal Forms
 If there are a lot of different representations of the same statement

Are there simple ones?
 Are there “normal forms”?
 Different normal forms for FOL

Negation normal form




p(cx) ¬q(cx,y)
Only conjunctions of disjunctions
Disjunctive normal form


xy:(p(x) :¬q(x,y))
No embedded Quantifiers
Conjunctive normal form

x:(p(x) y:¬q(x,y))
Only negations of atomic formulas
Prenex normal form

¬(x:(p(x) y:q(x,y)))
Only disjunctions of conjunctions
Gentzen normal form

Only implications where the condition is an atomic conjunction and the conclusion is
an atomic disjunction
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
q(cx,y)  p(cx)
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Clause Form
 Conjunctive normal form.




We know: Every formula of propositional logic can be rewritten
as a conjunction of disjunctions of atomic propositions.
Similarly every formula of predicate logic can be rewritten as a
conjunction of disjunctions of literals (modulo the quantifiers).
A formula is in clause form if it is rewritten as a set of
disjunctions of (possibly negative) literals.
Example: {{p(cx) },{¬q(cx,y)}}
 Theorem: Every FOL formula F can be transformed into clause
form F’ such that
F is satisfiable iff F’ is satisfiable
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
What is the ‘meaning’ of these
Axioms?


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Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
x: C(x,x)
x,y: C(x,y)  C(y,x)
x,y: P(x,y)  z: (C(z,x)  C(z,y))
x,y: O(x,y)  z: (P(z,x)  P(z,y))
x,y: DC(x,y)  C(x,y)
x,y: EC(x,y)  C(x,y)  O(x,y)
x,y: PO(x,y)  O(x,y)  P(x,y)  P(y,x)
x,y: EQ(x,y)  P(x,y)  P(y,x)
x,y: PP(x,y)  P(x,y)  P(y,x)
x,y: TPP(x,y)  PP(x,y)  z(EC(z,x)  EC(z,y))
x,y: TPPI(x,y)  PP(y,x)  z(EC(z,y)  EC(z,x))
x,y: NTPP(x,y)  PP(x,y)  z(EC(z,x)  EC(z,y))
x,y: NTPPI(x,y)  PP(y,x)  z(EC(z,y)  EC(z,x))
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Is This a Theorem?
 x,y,z: NTPP(x,y)  NTPP(y,z)  NTPP(x,z)
 Easy to see if we look at models!
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Relations of Regions of the
RCC-8
(a canonical model: n-dimensional closed discs)
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Thank you very much!!
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008