Download Logical Structures in Natural Language: Propositional Logic II

Logical Structures in Natural Language:
Propositional Logic II (Tableaux)
Raffaella Bernardi
Università degli Studi di Trento
e-mail: [email protected]
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What we have said last time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Remind: Propositional Logic: Basic Ideas . . . . . . . . . . . . . . . . . . . .
Remind: Language of Propositional Logic . . . . . . . . . . . . . . . . . . . .
Reminder: From English to Propositional Logic . . . . . . . . . . . . . . .
Reminder: Semantics: Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reminder: Interpretation Function . . . . . . . . . . . . . . . . . . . . . . . . . .
Reminder: Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reminder: Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tautologies and Contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of argumentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reminder: exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of key points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A formula: Tautology, Contradiction, Satisfiable, Falsifiable . . . . .
14.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An argumentation: Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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15.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NEW: Tableaux Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Tableaux: the calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Heuristics and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 Sets of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Done to be done and Home work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.
What we have said last time
• Logic
– Language: syntax, semantics.
– Reasoning
• Semantics
– Meaning of a sentence = Truth value
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1.
What we have said last time
• Logic
– Language: syntax, semantics.
– Reasoning
• Semantics
– Meaning of a sentence = Truth value
– Compositional meaning: truth-functional connectives
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What we have said last time
• Logic
– Language: syntax, semantics.
– Reasoning
• Semantics
– Meaning of a sentence = Truth value
– Compositional meaning: truth-functional connectives
– Interpretation Function: FORM → {true, f alse}
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1.
What we have said last time
• Logic
– Language: syntax, semantics.
– Reasoning
• Semantics
– Meaning of a sentence = Truth value
– Compositional meaning: truth-functional connectives
– Interpretation Function: FORM → {true, f alse}
• Reasoning: P remises |= α iff W (P remises) ⊆ W (α)
Today we look more into Propositional Logic (PL)
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2.
Remind: Propositional Logic: Basic Ideas
Statements:
The elementary building blocks of propositional logic are atomic statements that
cannot be decomposed any further: propositions.
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2.
Remind: Propositional Logic: Basic Ideas
Statements:
The elementary building blocks of propositional logic are atomic statements that
cannot be decomposed any further: propositions.
E.g.,
• “The box is red”
• “The proof of the pudding is in the eating”
• “It is raining”
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2.
Remind: Propositional Logic: Basic Ideas
Statements:
The elementary building blocks of propositional logic are atomic statements that
cannot be decomposed any further: propositions.
E.g.,
• “The box is red”
• “The proof of the pudding is in the eating”
• “It is raining”
and logical connectives “and”, “or”, “not”, by which we can build propositional
formulas.
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3.
Remind: Language of Propositional Logic
Alphabet The alphabet of PL consists of:
• A countable set of propositional symbols: p, q, r, . . .
• The logical connectives : ¬ (NOT), ∧ (AND), ∨ (OR), → (implication), ↔
(double implication).
• Parenthesis: (,) (they are used to disambiguate the language)
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3.
Remind: Language of Propositional Logic
Alphabet The alphabet of PL consists of:
• A countable set of propositional symbols: p, q, r, . . .
• The logical connectives : ¬ (NOT), ∧ (AND), ∨ (OR), → (implication), ↔
(double implication).
• Parenthesis: (,) (they are used to disambiguate the language)
Well formed formulas (wff) They are defined recursively
1. a propositional symbol is a wff:
2. if A is a wff then also ¬A is a wff
3. if A and B are wff then also (A ∧ B), (A ∨ B), (A → B) and (A → B) are wff
4. nothing else is a wff.
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4.
Reminder: From English to Propositional Logic
Eg. If you don’t sleep then you will be tired.
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4.
Reminder: From English to Propositional Logic
Eg. If you don’t sleep then you will be tired.
Keys: p = you sleep, q= you will be tired. Formula: ¬p → q.
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4.
Reminder: From English to Propositional Logic
Eg. If you don’t sleep then you will be tired.
Keys: p = you sleep, q= you will be tired. Formula: ¬p → q.
Exercise I:
1. If it rains while the sun shines, a rainbow will appear
2. Charles comes if Elsa does and the other way around
3. If I have lost if I cannot make a move, then I have lost.
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4.
Reminder: From English to Propositional Logic
Eg. If you don’t sleep then you will be tired.
Keys: p = you sleep, q= you will be tired. Formula: ¬p → q.
Exercise I:
1. If it rains while the sun shines, a rainbow will appear
2. Charles comes if Elsa does and the other way around
3. If I have lost if I cannot make a move, then I have lost.
1. (rain ∧ sun) → rainbow
2. elsa ↔ charles
3. (¬move → lost) → lost
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4.
Reminder: From English to Propositional Logic
Eg. If you don’t sleep then you will be tired.
Keys: p = you sleep, q= you will be tired. Formula: ¬p → q.
Exercise I:
1. If it rains while the sun shines, a rainbow will appear
2. Charles comes if Elsa does and the other way around
3. If I have lost if I cannot make a move, then I have lost.
1. (rain ∧ sun) → rainbow
2. elsa ↔ charles
3. (¬move → lost) → lost
Use: http://www.earlham.edu/~peters/courses/log/transtip.htm
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5.
Reminder: Semantics: Intuition
• Atomic propositions can be true T or false F.
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5.
Reminder: Semantics: Intuition
• Atomic propositions can be true T or false F.
• The truth value of formulas is determined by the truth values of the atoms
(truth value assignment or interpretation).
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Reminder: Semantics: Intuition
• Atomic propositions can be true T or false F.
• The truth value of formulas is determined by the truth values of the atoms
(truth value assignment or interpretation).
Example: (a ∨ b) ∧ c: If a and b are false and c is true, then the formula is not true.
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5.
Reminder: Semantics: Intuition
• Atomic propositions can be true T or false F.
• The truth value of formulas is determined by the truth values of the atoms
(truth value assignment or interpretation).
Example: (a ∨ b) ∧ c: If a and b are false and c is true, then the formula is not true.
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6.
Reminder: Interpretation Function
The interpretation function, denoted by I, can assign true (T) or false (F) to the
atomic formulas; for the complex formula they obey the following conditions. Given
the formulas P, Q of L:
a. I(¬P ) = T iff I(P ) = F
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6.
Reminder: Interpretation Function
The interpretation function, denoted by I, can assign true (T) or false (F) to the
atomic formulas; for the complex formula they obey the following conditions. Given
the formulas P, Q of L:
a. I(¬P ) = T iff I(P ) = F
b. I(P ∧ Q) = T iff I(P ) = T e I(Q) = T
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6.
Reminder: Interpretation Function
The interpretation function, denoted by I, can assign true (T) or false (F) to the
atomic formulas; for the complex formula they obey the following conditions. Given
the formulas P, Q of L:
a. I(¬P ) = T iff I(P ) = F
b. I(P ∧ Q) = T iff I(P ) = T e I(Q) = T
c. I(P ∨ Q) = F iff I(P ) = F e I(Q) = F
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6.
Reminder: Interpretation Function
The interpretation function, denoted by I, can assign true (T) or false (F) to the
atomic formulas; for the complex formula they obey the following conditions. Given
the formulas P, Q of L:
a. I(¬P ) = T iff I(P ) = F
b. I(P ∧ Q) = T iff I(P ) = T e I(Q) = T
c. I(P ∨ Q) = F iff I(P ) = F e I(Q) = F
d. I(P → Q) = F iff I(P ) = T e I(Q) = F
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6.
Reminder: Interpretation Function
The interpretation function, denoted by I, can assign true (T) or false (F) to the
atomic formulas; for the complex formula they obey the following conditions. Given
the formulas P, Q of L:
a. I(¬P ) = T iff I(P ) = F
b. I(P ∧ Q) = T iff I(P ) = T e I(Q) = T
c. I(P ∨ Q) = F iff I(P ) = F e I(Q) = F
d. I(P → Q) = F iff I(P ) = T e I(Q) = F
e. I(P ↔ Q) = F iff I(P ) = I(Q)
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7.
Reminder: Truth Tables
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7.
Reminder: Truth Tables
I1
I2
φ
T
F
¬φ
F
T
(1)
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7.
Reminder: Truth Tables
I1
I2
φ
T
F
¬φ
F
T
(1)
I1
I2
I3
I4
φ
T
T
F
F
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ψ
T
F
T
F
φ∧ψ
T
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Reminder: Truth Tables
I1
I2
I1
I2
I3
I4
φ
T
T
F
F
φ
T
F
ψ
T
F
T
F
¬φ
F
T
(1)
I1
I2
I3
I4
φ
T
T
F
F
ψ
T
F
T
F
φ∧ψ
T
F
F
F
(1)
First
Last
φ∨ψ
T
T
T
F
(1)
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Reminder: Truth Tables
I1
I2
I1
I2
I3
I4
φ
T
T
F
F
φ
T
F
ψ
T
F
T
F
¬φ
F
T
(1)
φ∨ψ
T
T
T
F
(1)
I1
I2
I3
I4
φ
T
T
F
F
ψ
T
F
T
F
φ∧ψ
T
F
F
F
(1)
I1
I2
I3
I4
φ
T
T
F
F
ψ
T
F
T
F
φ→ψ
T
F
T
T
(1)
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8.
Reminder: Model
A model consists of two pieces of information:
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8.
Reminder: Model
A model consists of two pieces of information:
• which collection of atomic propositions we are talking about (domain, D),
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8.
Reminder: Model
A model consists of two pieces of information:
• which collection of atomic propositions we are talking about (domain, D),
• and for each formula which is the appropriate semantic value, this is done by
means of a function called interpretation function (I).
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8.
Reminder: Model
A model consists of two pieces of information:
• which collection of atomic propositions we are talking about (domain, D),
• and for each formula which is the appropriate semantic value, this is done by
means of a function called interpretation function (I).
Thus a model M is a pair: (D, I).
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8.
Reminder: Model
A model consists of two pieces of information:
• which collection of atomic propositions we are talking about (domain, D),
• and for each formula which is the appropriate semantic value, this is done by
means of a function called interpretation function (I).
Thus a model M is a pair: (D, I).
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9.
Tautologies and Contradictions
Build the truth table of p ∧ ¬p.
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9.
Tautologies and Contradictions
Build the truth table of p ∧ ¬p.
It’s a contradiction: always false.
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9.
Tautologies and Contradictions
Build the truth table of p ∧ ¬p.
It’s a contradiction: always false.
Build the truth table of (p → q) ∨ (q → p).
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9.
Tautologies and Contradictions
Build the truth table of p ∧ ¬p.
It’s a contradiction: always false.
Build the truth table of (p → q) ∨ (q → p).
It’s a tautology: always true.
A formula P is:
• satisfiabiliy if there is at least an interpretation I such that I(P ) = T rue
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10.
Reasoning
P1 , . . . , Pn |= C
a valid deductive argumentation is such that its conclusion cannot be false when
the premises are true.
In other words, there is no interpretation for which the conclusion is false and the
premises are true.
W (P remise), the set of interpretations for which the premises are all true, and
W (C) the set of interpretations for which the conclusion is true:
W (P remises) ⊆ W (C)
The premises entail α iff α is true for all the interpretations for which all the premises
are true.
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Reasoning
P1 , . . . , Pn |= C
a valid deductive argumentation is such that its conclusion cannot be false when
the premises are true.
In other words, there is no interpretation for which the conclusion is false and the
premises are true.
W (P remise), the set of interpretations for which the premises are all true, and
W (C) the set of interpretations for which the conclusion is true:
W (P remises) ⊆ W (C)
The premises entail α iff α is true for all the interpretations for which all the premises
are true.
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11.
Example of argumentations
Today is Monday or today is Thursday
Today is not Monday
=================
Today is Thursday
P v Q
not P
=====
Q
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11.
Example of argumentations
Today is Monday or today is Thursday
Today is not Monday
=================
Today is Thursday
P v Q
not P
=====
Q
If today is Thursday, then today I’ve a lecture
Today is Thursday
===============
Today I’ve a lecture
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Q --> R
Q
=======
R
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11.
Example of argumentations
Today is Monday or today is Thursday
Today is not Monday
=================
Today is Thursday
P v Q
not P
=====
Q
If today is Thursday, then today I’ve a lecture
Today is Thursday
===============
Today I’ve a lecture
P ∨ Q, ¬P |= Q
Q --> R
Q
=======
R
Q → R, Q |= R
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11.
Example of argumentations
Today is Monday or today is Thursday
Today is not Monday
=================
Today is Thursday
P v Q
not P
=====
Q
If today is Thursday, then today I’ve a lecture
Today is Thursday
===============
Today I’ve a lecture
P ∨ Q, ¬P |= Q
Q --> R
Q
=======
R
Q → R, Q |= R
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Try to build truth tables to verify: P ∨ Q, ¬P |= Q
I1
I2
I3
I4
P
T
T
F
F
Q
T
F
T
F
P ∨Q
T
T
T
F
¬P
F
F
T
T
Q
T
F
T
F
W (P remesse) ⊆ W (Q)
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Try to build truth tables to verify: P ∨ Q, ¬P |= Q
I1
I2
I3
I4
P
T
T
F
F
Q
T
F
T
F
P ∨Q
T
T
T
F
¬P
F
F
T
T
Q
T
F
T
F
W (P remesse) ⊆ W (Q)
{I3 } ⊆ {I1 , I3 }
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12.
Reminder: exercises
Build the truth tables for the following formulas and decide whether they are satisfiable, or a tautology or a contradiction.
• (¬A → B) ∧ (¬A ∨ B)
• P → (Q ∨ ¬R)
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Build the truth tables for the following entailments and decide whether they are
valid
1. P ∨ Q |= Q
2. P → Q, Q → R |= P → R
3. P → Q, Q |= P
4. P → Q |= ¬(Q → P )
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13.
Summary of key points.
• Tomorrow bring the solutions for the exercises.
• Today key concepts
– Syntax of PL: atomic vs. complex formulas
– Semantics of PL: truth tables
– Formalization of simple arguments
– Interpretation function
– Domain
– Model
– Entailment
– Satisfiability
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14.
A formula: Tautology, Contradiction, Satisfiable, Falsifiable
Recall, a formula P is:
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14.
A formula: Tautology, Contradiction, Satisfiable, Falsifiable
Recall, a formula P is:
• tautology if for all the interpretations I, I(P ) = T rue (it’s always true)
• contradiction if for all the interpretations I, I(P ) = F alse (is always false)
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14.
A formula: Tautology, Contradiction, Satisfiable, Falsifiable
Recall, a formula P is:
• tautology if for all the interpretations I, I(P ) = T rue (it’s always true)
• contradiction if for all the interpretations I, I(P ) = F alse (is always false)
A formula P is:
• satisfiabiliy if there is at least an interpretation I such that I(P ) = T rue
• falsifiable if there is at least an interpretation I such that I(P ) = F alse
A formula that is false in all interpretation is also called unsatisfiable.
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14.1.
Example
I1
I2
P
T
F
¬P
F
T
¬P ∨ P
T
T
¬P ∨ P is a tautology.
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15.
An argumentation: Validity
{P1 , . . . , Pn } |= C
a valid deductive argumentation is such that its conclusion cannot be false when
the premises are true.
In other words, there is no interpretation for which the conclusion is false and the
premises are true.
W (P remise), the set of interpretations for which the premises are all true, and
W (C) the set of interpretations for which the conclusion is true:
W (P remises) ⊆ W (C)
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15.1.
Example
I1
I2
I3
I4
P
T
T
F
F
Q
T
F
T
F
P ∨Q
T
T
T
F
¬P
F
F
T
T
Q
T
F
T
F
W (P remesse) ⊆ W (Q)
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15.1.
Example
I1
I2
I3
I4
P
T
T
F
F
Q
T
F
T
F
P ∨Q
T
T
T
F
¬P
F
F
T
T
Q
T
F
T
F
W (P remesse) ⊆ W (Q)
{I3 } ⊆ {I1 , I3 }
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15.2.
Exercises
Check whether the following arguments are valid:
If the temperature and air pressure remained constant, there was no rain.
The temperature did remain constant. Therefore, if there was rain then
the air pressure did not remain constant.
If Paul lives in Dublin, he lives in Ireland. Paul lives in Ireland. Therefore
Paul lives in Dublin.
(i) Give the keys of your formalization using PL; (ii) represent the argument formally,
and (iii) Apply the truth table method to prove or disprove the validity of the
argument.
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16.
Counter-example
Counterexample an interpretation in which the reasoning does not hold. In other
words, an interpretation such that the premises are true and the conclusion is false.
Exercise: together Take the previous exercise and build a counter-example if the
argumentation is not valid
If the temperature and air pressure remained constant, there was no rain.
The temperature did remain constant. Therefore, if there was rain then
the air pressure did not remain constant.
If Paul lives in Dublin, he lives in Ireland. Paul lives in Ireland. Therefore
Paul lives in Dublin.
Exercises: alone See printed paper (pl3)
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17.
NEW: Tableaux Calculus
• The Tableaux Calculus is a decision procedure solving the problem of satisfiability.
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17.
NEW: Tableaux Calculus
• The Tableaux Calculus is a decision procedure solving the problem of satisfiability.
• If a formula is satisfiable, the procedure will constructively exhibit an interpretation in which the formula is true.
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17.
NEW: Tableaux Calculus
• The Tableaux Calculus is a decision procedure solving the problem of satisfiability.
• If a formula is satisfiable, the procedure will constructively exhibit an interpretation in which the formula is true.
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17.1.
Tableaux: the calculus
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17.1.
Tableaux: the calculus
A∧B
A
B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A→B
¬A
B
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B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A→B
¬A
B
B
A↔B
A∧B
¬A ∧ ¬B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A↔B
A∧B
¬A ∧ ¬B
A→B
¬A
B
B
¬¬A
A
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A↔B
A∧B
¬A ∧ ¬B
A→B
¬A
B
B
¬(A ∧ B)
¬¬A
A
¬A
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¬B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A↔B
A∧B
¬A ∧ ¬B
A→B
¬A
B
B
¬(A ∧ B)
¬¬A
A
¬A
¬B
¬(A ∨ B)
¬A
¬B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A↔B
A∧B
A→B
¬A ∧ ¬B
¬(A ∨ B)
¬A
¬B
¬A
B
B
¬(A ∧ B)
¬¬A
A
¬A
¬B
¬(A → B)
A
¬B
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17.1.
Tableaux: the calculus
A∧B
A
B
A∨B
A
A↔B
A∧B
A→B
¬A ∧ ¬B
¬(A ∨ B)
¬A
¬B
¬A
B
B
¬(A ∧ B)
¬¬A
A
¬A
¬(A → B)
A
¬B
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¬B
¬(A ↔ B)
A ∧ ¬B
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¬A ∧ B
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18.
Heuristics and Exercises
Apply non-branching rules before branching rules.
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18.
Heuristics and Exercises
Apply non-branching rules before branching rules.
Exercises Take the exercises done so far using truth tables and prove by means of
tableaux whether the formula is satisfiable.
• A ∧ (B ∧ ¬A)
• (A → B) → ¬B
• A → (B → A)
• (B → A) → A
• (¬A → B) ∧ (¬A ∨ B)
• A → (B ∨ ¬C)
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18.1.
Sets of formulas
Determine whether the following sets of logical forms are satisfiable by means of
truth tables first and then by tableaux method; in other words, you are asked to
check whether there is at least an interpretation in which all the formulas in the set
are true.
{¬B → B, ¬(A → B), ¬A ∨ ¬B}
{¬A ∨ B, ¬(B ∧ ¬C), C → D, ¬(¬A ∨ D)}
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
Are you sure you cannot give a stronger answer, i.e. are you sure ψ is not a
tautology?
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
Are you sure you cannot give a stronger answer, i.e. are you sure ψ is not a
tautology?
In order to check whether ψ is a tautology you have to look at ¬ψ.
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
Are you sure you cannot give a stronger answer, i.e. are you sure ψ is not a
tautology?
In order to check whether ψ is a tautology you have to look at ¬ψ.
If ¬ψ is unsatisfiable then ψ is also a tautology.
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
Are you sure you cannot give a stronger answer, i.e. are you sure ψ is not a
tautology?
In order to check whether ψ is a tautology you have to look at ¬ψ.
If ¬ψ is unsatisfiable then ψ is also a tautology.
• If all branches close: ψ is unsatisfiable.
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
Are you sure you cannot give a stronger answer, i.e. are you sure ψ is not a
tautology?
In order to check whether ψ is a tautology you have to look at ¬ψ.
If ¬ψ is unsatisfiable then ψ is also a tautology.
• If all branches close: ψ is unsatisfiable.
Can you make a stronger claim?
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19.
Formula
You are asked to prove whether ψ is a tautology by means of tableaux.
• If all branches of your tableaux are open, what do you conclude?
ψ is satisfiable.
Are you sure you cannot give a stronger answer, i.e. are you sure ψ is not a
tautology?
In order to check whether ψ is a tautology you have to look at ¬ψ.
If ¬ψ is unsatisfiable then ψ is also a tautology.
• If all branches close: ψ is unsatisfiable.
Can you make a stronger claim?
No this is already a strong result, there is no need to look at ¬ψ.
More on this next time.
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20.
Done to be done and Home work
Today we have looked at:
• Recalled: Prove whether a formula is satisfiable by means of Truth Tables
• Recalled: Prove whether an entailment is valid by means of Truth Tables.
• Prove whether a formula is satisfiable by means of Tableaux.
Next time we will look at how to prove whether
• a set of formulas is satisfiable by means of Tableaux.
• a formula is a tautology by means of Tableaux
• an entailment is valid by means of Tableaux.
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