# Introduction to Logic

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```							                                     Introduction to Logic

Overview:                          Literature:
What is Logic?                 Handouts: (J. Kelly)
Propositional Logic              The Essence of Logic:
Predicate Logic                  chapter 1 (not 1.6)
chapter 6

1

What is Logic?

  λογικη (logikè) = relative to logos (reason)
  Definitions of ‘logic’
  The art of reasoning
  The branch of philosophy that analyzes inference
  Reasoned and reasonable judgment; "it made a certain kind of
logic"
  The principles that guide reasoning within a given field or
situation: "economic logic requires it"; "by the logic of war"
  A system of reasoning

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Aristotoles (384–322 B.C.)

  The instrument (the "organon") by means of
which we come to know anything.
  Formal rules for correct reasoning

  Syllogism:

All man are mortal
Socrates is a man
Socrates is mortal

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Gottfried Leibniz (1646-1716)

  Human reasoning can be reduced to calculation
  Characteristica universalis
  universal language to describe all scientific concepts
  Calculus ratiocinator
  method to reason with that language
  Salva veritate
  two expressions can be interchanged without changing the
truth-value of the statements in which they occur

“when there are disputes among persons, we can simply
say: Let us calculate [calculemus]!”

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Criticism to logic

  “… logic is not a model of reasoning at all but is more a
means of constraining the reasoning process; we [people]
don’t solve problems using logic, we just use it to explain
our solutions.”
[Marvin Minsky]

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The Importance of Logic

  High-level language for expressing knowledge
  High expressive power
  Well-understood formal semantics
  Precise notion of logical consequence
  Proof systems that can automatically derive statements
syntactically from a set of premises
  Logic can provide explanations for answers
  By tracing a proof that leads to a logical consequence

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Uses of Logic

  Argumentation
  Inference (systems)
  Cognitive psychology & AI
  descriptive use of logic
  Normative systems
  prescriptive use of logic
  Knowledge representation

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Warning!

  Meta-language
  Different levels
  Abstraction
  Difference to everyday use of concepts
  Temporal, causal, … relations

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Propositional Logic and Predicate Logic

  Propositional Logic
  The study of statements and their connectivity structure.
  Predicate Logic
  The study of individuals and their properties.

  Propositional logic more abstract and hence less detailed
than predicate logic.
  Propositional/predicate logic are unique in the sense that
sound and complete proof systems do exist.
  Not for more expressive logics (higher-order logics)

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Reasoning = Reach valid conclusions

All man are mortal                    All man have blue skin
Socrates is a man                       Socrates is a man
Socrates is mortal                   Socrates has blue skin

  Reasoning is valid (deduction, syllogism)
  Nothing is said about the actual truth of premisses!
  The concern of logic is that…
  … from true premises one can never reach a false
conclusion!

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Propositional Logic

  Truth Tables
  Logical equivalence
  Tautology

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Basic elements

  Logical connectives
  ∧       and
  ∨       or
  ¬       not
  →       implication, if … then …
  ↔       bi-implication, iff (if and only if)

  Connectives are used to link atoms
  Atoms = facts = statements
  Propositions can be constructed based on atoms and
connectives

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Truth values

  A proposition has a truth value:
  true (T) or false (F)
  v(A) represents the truth value of A
  T and F are often represented as 1 resp. 0

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Truth tables

  Truth tables give the operational definition of logical
connectives
  List ALL possible cases
  Example, for negation :

A             ¬A
T             F
F             T

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Truth table for AND (∧)

A         B            A∧B
F         F             F
F         T                F
T         F                F
T         T                T

V( (1+1 = 2) ∧ (the sun is a star) ) = T

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Truth table for OR (∨)

A         B            A∨B
F         F             F
F         T                T
T         F                T
T         T                T

V( (1+1 = 2) ∨ (the sun is a planet)) = T

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Truth table for implication (→)

A      B           A→B
F       F           T
F       T               T
T       F               F
T       T               T

V((the sun is a planet) → (1+1 = 3)) = T
(dog is animal) → (dog breaths) ) = T
V(
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Truth table for iff (↔)

A         B            A↔B
F          F            T
F          T                F
T          F                F
T          T                T

V(   (the sun is a star) ↔ (1+1 = 2) )           =T

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Remarks on implication

  Implication in propositional logic is different from its use in
natural language
  Temporal issues
  Causal relations
  Implication in propositional logic refers to the truth values
of the atoms!

  E.g. compare:
  If the moon is made of cheese then it is tasty
  If the moon is made of cheese then 2 x 2 = 5
  Both are true, but the first sounds more ‘logical’!!

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  Tautology
  A logical expression that has truth value T in all cases

  A logical expression that has truth value F in all cases

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Example: tautology

(A∧B)→ (C∨(¬B→¬C))
A   B   C   A∧B   ¬B     ¬C   ¬B→¬C    C∨(¬B→¬C)       (A∧B)→ (C∨(¬B→¬C))

F   F   F    F     T      T     T            T                 T

F   F   T    F     T      F     F            T                 T

F   T   F    F     F      T     T            T                 T

F   T   T    F     F      F     T            T                 T

T   F   F    F     T      T     T            T                 T

T   F   T    F     T      F     F            T                 T

T   T   F    T     F      T     T            T                 T

T   T   T    T     F      F     T            T                 T

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Example: tautology

(A∧B)→ (C∨(¬B→¬C))
A   B   C   A∧B   ¬B     ¬C   ¬B→¬C    C∨(¬B→¬C)       (A∧B)→ (C∨(¬B→¬C))

F   F   F    F     T      T     T            T                 T

F   F   T    F     T      F     F            T                 T

F   T   F    F     F      T     T            T                 T

F   T   T    F     F      F     T            T                 T

T   F   F    F     T      T     T            T                 T

T   F   T    F     T      F     F            T                 T

T   T   F    T     F      T     T            T                 T

T   T   T    T     F      F     T            T                 T

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A        ¬A           A∧¬A

F        T                F

T        F                F

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Logical equivalence

  Two logical expressions are logically equivalent if
they have the same truth table
  That is, each truth assignment of the atoms results in
the same truth value for the expression

  Notation: A ≡ B
  A and B are logically equivalent

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Example

A   B   A∧B                A   B   ¬A      ¬B       ¬A∨¬B   ¬(¬A∨¬B)

F   F    F                 F   F    T       T         T        F
F   T    F
F   T    T       F         T        F
T   F    F
T   F    F       T         T        F
T   T    T
T   T    F       F         F        T

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Example

A   B   A∧B                A   B   ¬A      ¬B       ¬A∨¬B   ¬(¬A∨¬B)

F   F    F                 F   F    T       T         T        F
F   T    F
F   T    T       F         T        F
T   F    F
T   F    F       T         T        F
T   T    T
T   T    F       F         F        T

Conclusion:

A ∧ B ≡ ¬(¬A ∨ ¬B)

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Equivalence laws

  A ∧ F ≡ F
  A ∧ T ≡ A
  A ∨ F ≡ A
  A ∨ T ≡ T
  A ∧ A ≡ A
  A ∨ A ≡ A

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Equivalence laws

  A ∧ F ≡ F                     A ∧ B ≡ B ∧ A
  A ∧ T ≡ A                     A ∨ B ≡ B ∨ A
  A ∨ F ≡ A                     A ∧ (A ∨ B) ≡ A
  A ∨ T ≡ T                     A ∨ (A ∧ B) ≡ A
  A ∧ A ≡ A                     A ∨ (¬A ∧ B) ≡ A ∨ B
  A ∨ A ≡ A                     A ∧ (¬A ∨ B) ≡ A ∧ B
  A ∧ ¬A ≡ F                    (A ∧ B) ∨ (A ∧ ¬B) ≡ A
  A ∨ ¬A ≡ T                    A → B ≡ ¬A ∨ B
  ¬¬A ≡ A                       A → B ≡ ¬(A ∧ ¬B)

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Equivalence laws (general)

  Distributivity
  A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C)
  A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C)

  De Morgan Laws
  ¬(A ∧ B) ≡ ¬A ∨ ¬B
  ¬(A ∨ B) ≡ ¬A ∧ ¬B

  Logical equivalence satisfies the laws of a Boolean algebra.

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Inference

  How to reason (deduce new facts) with propositional logic

  Modus Ponens
A→B
A
∴B

  IfA → B and A hold (are true) then it can be safely
concluded that B also is true
  E.g. if the rule is (smoke → fire) and you see smoke
then you can conclude that there is fire

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Predicate logic

  Objects, predicates, functions
  Quantifiers
  1st order language
  Substitutions, interpretations

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Syllogistic reasoning

  Syllogistic reasoning as

All man are mortal
Socrates is a man
∴Socrates is mortal
cannot be expressed in propositional logic

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Objects, predicates, quantifiers

  Needed:
  Objects, such as Socrates
  Predicates, such as mortal
  Quantifiers, such as all

  Formally:                           or:
  ∀x : M(x) → S(x)                      ∀x : man(x) → mortal(x)
  M(s)                                  man(socrates)
∴ S(s)                                   ∴mortal(socrates)

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Quantifiers

  Universal quantifier
  Notation: ∀
  “for all”
  Ex. “(∀x)M(x)” or “∀x : M(x)”

  Existential quantifier
  Notation: ∃
  “exists, there is”
  Ex. “(∃x)M(x)” of “∃x : M(x)”

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Often used formulas

  All A are B:
  For all x: if x is A then x is B
  (∀x)(A(x) → B(x))

  Some A are B:
  There is x: x is A and x is B
  (∃x)(A(x) ∧ B(x))

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Warning

  Order of quantifiers is essential!
  Big difference:
  (∀x)(∃y)A(x,y)
•  For all x there is a y such that A(x,y)
•  For all men x there is a woman y such that mother(x,y)

  (∃y)(∀x)A(x,y)
•  There is a y such that for all x, A(x,y)
•  There is a woman y such that for all man x, mother(x,y)

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Quantifiers in finite domains

  When the domain is finite,
say {a1, …, an}, the universal and existential quantifiers
are a abbreviation for the finite conjunction resp.
disjunction:
  (∀x)A(x) = A(a1) ∧ … ∧ A(an)
  (∃x)A(x) = A(a1) ∨ … ∨ A(an)

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Relation between ∀ and ∃

  (∃x)A(x) ↔ ¬(∀x)¬A(x)

  (∀x)A(x) ↔ ¬(∃x)¬A(x)

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Inference

  How to reason (deduce new facts) with predicate logic

  Modus ponens

  Universal Elimination (Syllogism)
(∀x) P(x) → Q(x)
P(a)
∴ Q(a)
where x ∈ X, a ∈ X
  E.g (all kids like ice-cream) and (Bob is a kid) hold, then
deduce (Bob likes ice-cream)
  Exercise: translate into predicate logic!

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Examples of Translations

  Old men are wise
  (∀x)[(old(x) ∧ man(x)) → wise(x)]

  When a man loves a woman, they are happy
  (∀x)(∃y)[(man(x) ∧ woman(y) ∧ loves(x,y)) → (happy(x) ∧ happy(y))]

  Everyone loves Piet
  (∀x)[loves(x,piet)]

  Some farmers beat the donkey
  (∃x)[farmer(x) ∧ beats(x,donkey)]

  Some farmers beat their donkeys
  (∃x)(∀y)[(farmer(x) ∧ donkey(y) ∧ owns(x,y)) → beats(x,y)]

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Next module:

Rule Based Systems
(22 September 2007)

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