Aussagenlogik

Aussagenlogik
Schnelldurchlauf
Michael Leuschel
Softwaretechnik und Programmiersprachen
05/11/2012
Lecture 3
Teil 1:
Sprache (Syntax)
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Bestandteile
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Atomare Aussagen (atomic propositions)
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Entweder wahr oder falsch (Wahrheitswert, truth
value auf Englisch; true oder false)
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zwei-wertig
Wahrheitswert einer komplexen Aussage:
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bestimmt von Wahrheitswerten der Bestandteile
it rains and it is cold
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it rains
(compositionality)
it is cold
Alphabet der Aussagenlogik
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Connectives
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Atomare Aussagen
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Und (Konjunktion): ∧
Oder (Disjunktion): ∨
Wenn (Implikation): →
Not (Negation):
¬
Äquivalenz ⟺, Exklusives Oder, . . .
Bezeichner:
p
q
(
)
Klammerung
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Klammern
rains ...
Well Formed Formulas (wff’s)
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Atomic propositions are wff’s
If α and β are wff’s then so are:
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(¬ α)
(α ∧ β)
(α ∨ β)
(α → β)
No other expressions are wff’s
Note: α, β : meta-variables (outside of
propositional logic)
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Precedence
1. Negation ¬ binds highest
2. Then come ∧ and ∨
3. Finally comes →
Instead of
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(((¬ p) ∧ (¬ q)) → ((¬ p) ∨ (¬ q)))
we can write
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(¬ p ∧ ¬ q) → (¬ p ∨ ¬ q)
¬p∧¬q → ¬p∨¬q
Teil 2:
Semantik der Aussagenlogik
Interpretations, Models, Logical
Consequence
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Wahrheitswert einer Formel
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Zuweisung: wff ↝ Wahrheitswert
Junktoren:
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feste Bedeutung
Bsp: true ∧ false ergibt false
Wahrheitstabellen (truth tables)
Atomare Aussagen:
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egal (wahr oder falsch)
(we study form, not content !)
Wahrheitstabellen
α
β
¬α
α∧ β
α∨ β
α→ β α⇔ β
true
true
false
true
true
true
true
true
false
false
false
true
false
false
false
true
true
false
true
true
false
false
false
true
false
false
true
true
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Interpretations and Models
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An interpretation v is a mapping from atomic
propositions to {true,false}
Example: v(p) = true, v(q) = false
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Extension to wff’s by using truth tables
Example: v(p ∨ q) = ?
n  1. Determine v for subparts:
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v(p) = true, v(q) = false
2. Apply truth table: true ∨ false = true
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v(p ∨ q) = true
For v’(p) = false, v’(q) = false: v’(p ∨ q) = false
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Interpretation v is a model for α iff v(α) = true
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Exercises:
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v(p) = true,v(q) = false
v’(p) = false,v’(q) = false
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v(p ∧ (q ∨ (¬p) )) =
v(p → (p → q )) =
v’(p → (p → q )) =
v(p ∨ (¬p)) =
v’(p ∨ (¬p)) =
v(p ∧ (¬p)) =
v’(p ∧ (¬p)) =
Exercises: Solutions
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v(p) = true,v(q) = false
v’(p) = false,v’(q) = false
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v(p ∧ (q ∨ (¬p) )) false
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v(p)
v(q ∨ (¬p) )
true
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v(p ∧ (q ∨ (¬p) )) = false
v(p → (p → q )) = false
v’(p → (p → q )) = true
v(p ∨ (¬p)) = true
v’(p ∨ (¬p)) = true
false
v(q)
false
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v(¬p) false
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v(p)
true
v(p ∧ (¬p)) = false
v’(p ∧ (¬p)) = false
Satisfaction, Tautology,
Contradiction
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A wff α is called
satisfiable (erfüllbar) iff it has a model
n  unsatisfiable or a contradiction
(unerfüllbar, Widerspruch) iff it has no model
n  a tautology iff all interpretations are models
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p ∧ (¬p)
p ∧ (q ∨ (¬p) )
p ∨ (¬p)
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Consequence and
Equivalence
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Two wff’s α and β are logically equivalent,
denoted by α ≡ β, iff they have the same
models (i.e., same truth table)
(p ∧ q) ≡ (q ∧ p)
(p → q) ≡ (¬p ∨ q)
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A wff α is called a logical consequence of β,
denoted by β ⇒ α, iff every model of β is also
a model of α
p ⇒ (q ∨ p)
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often
instead of ⇒
Consequence for Sets
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An interpretation v is a model for a set S of
wff’s iff it is a model for every element of S
v(p)=true,
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v is a model of {(p ∨ q), (¬p → ¬q)}
A set A of wff’s is called a logical
consequence of another set B,
denoted by B ⇒ A,
iff every model of B is also a model of Α
{p} ⇒ {(p ∨ q), (¬p → ¬q)}
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Beispiel
Ja
{p → q , ¬q} ⇒ {¬p}
?
auch Modell von {¬p} v(p)=true,
v(q)= true
v(p)=true,
v(q)=false
v(p)= false,
v(q)= true
4 mögliche Interpretationen
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v(p)= false,
v(q)= false
einziges Modell von
{p → q , ¬q}
Ein anderes Puzzle
Ein Zettel sagt die
Wahrheit der
andere nicht.
?
In dieser Zelle befindet
sich eine Prinzessin und
ein Tiger in der anderen
Zelle
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?
?
?
In einer der Zellen
befindet sich eine
Prinzessin und in der
anderen ein Tiger
Beware: Logic vs Language
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Commutativity: p ∧ q ≡ q ∧ p
But:
He got scared and he killed the intruder.
n  He killed the intruder and he got scared.
(causal and temporal meaning attached to
and)
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Implication
If 1 = 2 then I am Winnie the Poh.
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What (definitely) to know for
the exam:
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Language of Propositional Logic
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wff’s
Semantics of Propositional Logic
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Truth tables, Interpretations, Models,
Satisfiability,Tautology, Contradiction
Logical Equivalence and Consequence