Logical Structures in Natural Language:Propositional Logic II (Tableaux)

Raffaella BernardiUniversita degli Studi di Trento

e-mail: bernardi@disi.unitn.it

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Contents

1 What we have said last time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Remind: Propositional Logic: Basic Ideas . . . . . . . . . . . . . . . . . . . . 43 Remind: Language of Propositional Logic . . . . . . . . . . . . . . . . . . . . 54 Reminder: From English to Propositional Logic . . . . . . . . . . . . . . . 65 Reminder: Semantics: Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Reminder: Interpretation Function . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Reminder: Truth Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Reminder: Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Tautologies and Contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211 Example of argumentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 Reminder: exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513 Summary of key points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714 A formula: Tautology, Contradiction, Satisfiable, Falsifiable . . . . . 18

14.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915 An argumentation: Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

15.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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15.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216 Counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317 NEW: Tableaux Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

17.1 Tableaux: the calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2518 Heuristics and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

18.1 Sets of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2820 Done to be done and Home work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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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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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, false}

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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, false}

• Reasoning: Premises |= α iff W (Premises) ⊆ 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 thatcannot 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 thatcannot 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 thatcannot 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 propositionalformulas.

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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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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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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 theatomic formulas; for the complex formula they obey the following conditions. Giventhe 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 theatomic formulas; for the complex formula they obey the following conditions. Giventhe 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 theatomic formulas; for the complex formula they obey the following conditions. Giventhe 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 theatomic formulas; for the complex formula they obey the following conditions. Giventhe 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 theatomic formulas; for the complex formula they obey the following conditions. Giventhe 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

φ ¬φI1 T FI2 F T

(1)

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7. Reminder: Truth Tables

φ ¬φI1 T FI2 F T

(1)

φ ψ φ ∧ ψI1 T T TI2 T F FI3 F T FI4 F F F

(1)

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7. Reminder: Truth Tables

φ ¬φI1 T FI2 F T

(1)

φ ψ φ ∧ ψI1 T T TI2 T F FI3 F T FI4 F F F

(1)

φ ψ φ ∨ ψI1 T T TI2 T F TI3 F T TI4 F F F

(1)

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7. Reminder: Truth Tables

φ ¬φI1 T FI2 F T

(1)

φ ψ φ ∧ ψI1 T T TI2 T F FI3 F T FI4 F F F

(1)

φ ψ φ ∨ ψI1 T T TI2 T F TI3 F T TI4 F F F

(1)

φ ψ φ→ ψI1 T T TI2 T F FI3 F T TI4 F F 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 bymeans 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 bymeans 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 bymeans 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 ) = True

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10. Reasoning

P1, . . . , Pn |= C

a valid deductive argumentation is such that its conclusion cannot be false whenthe premises are true.

In other words, there is no interpretation for which the conclusion is false and thepremises are true.

W (Premise), the set of interpretations for which the premises are all true, andW (C) the set of interpretations for which the conclusion is true:

W (Premises) ⊆ W (C)

The premises entail α iff α is true for all the interpretations for which all the premisesare true.

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10. Reasoning

P1, . . . , Pn |= C

a valid deductive argumentation is such that its conclusion cannot be false whenthe premises are true.

In other words, there is no interpretation for which the conclusion is false and thepremises are true.

W (Premise), the set of interpretations for which the premises are all true, andW (C) the set of interpretations for which the conclusion is true:

W (Premises) ⊆ W (C)

The premises entail α iff α is true for all the interpretations for which all the premisesare true.

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11. Example of argumentations

Today is Monday or today is Thursday P v Q

Today is not Monday not P

================= =====

Today is Thursday Q

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11. Example of argumentations

Today is Monday or today is Thursday P v Q

Today is not Monday not P

================= =====

Today is Thursday Q

If today is Thursday, then today I’ve a lecture Q --> R

Today is Thursday Q

=============== =======

Today I’ve a lecture R

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11. Example of argumentations

Today is Monday or today is Thursday P v Q

Today is not Monday not P

================= =====

Today is Thursday Q

If today is Thursday, then today I’ve a lecture Q --> R

Today is Thursday Q

=============== =======

Today I’ve a lecture R

P ∨Q,¬P |= Q Q→ R,Q |= R

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11. Example of argumentations

Today is Monday or today is Thursday P v Q

Today is not Monday not P

================= =====

Today is Thursday Q

If today is Thursday, then today I’ve a lecture Q --> R

Today is Thursday Q

=============== =======

Today I’ve a lecture R

P ∨Q,¬P |= Q Q→ R,Q |= R

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Try to build truth tables to verify: P ∨Q,¬P |= Q

P Q P ∨Q ¬P QI1 T T T F TI2 T F T F FI3 F T T T TI4 F F F T F

W (Premesse) ⊆ W (Q)

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Try to build truth tables to verify: P ∨Q,¬P |= Q

P Q P ∨Q ¬P QI1 T T T F TI2 T F T F FI3 F T T T TI4 F F F T F

W (Premesse) ⊆ 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 satis-fiable, 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 arevalid

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, Satisfi-

able, Falsifiable

Recall, a formula P is:

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14. A formula: Tautology, Contradiction, Satisfi-

able, Falsifiable

Recall, a formula P is:

• tautology if for all the interpretations I, I(P ) = True (it’s always true)

• contradiction if for all the interpretations I, I(P ) = False (is always false)

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14. A formula: Tautology, Contradiction, Satisfi-

able, Falsifiable

Recall, a formula P is:

• tautology if for all the interpretations I, I(P ) = True (it’s always true)

• contradiction if for all the interpretations I, I(P ) = False (is always false)

A formula P is:

• satisfiabiliy if there is at least an interpretation I such that I(P ) = True

• falsifiable if there is at least an interpretation I such that I(P ) = False

A formula that is false in all interpretation is also called unsatisfiable.

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14.1. Example

P ¬P ¬P ∨ PI1 T F TI2 F 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 whenthe premises are true.

In other words, there is no interpretation for which the conclusion is false and thepremises are true.

W (Premise), the set of interpretations for which the premises are all true, andW (C) the set of interpretations for which the conclusion is true:

W (Premises) ⊆ W (C)

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15.1. Example

P Q P ∨Q ¬P QI1 T T T F TI2 T F T F FI3 F T T T TI4 F F F T F

W (Premesse) ⊆ W (Q)

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15.1. Example

P Q P ∨Q ¬P QI1 T T T F TI2 T F T F FI3 F T T T TI4 F F F T F

W (Premesse) ⊆ 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 thenthe air pressure did not remain constant.

If Paul lives in Dublin, he lives in Ireland. Paul lives in Ireland. ThereforePaul 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 theargument.

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16. Counter-example

Counterexample an interpretation in which the reasoning does not hold. In otherwords, 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 theargumentation 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 thenthe air pressure did not remain constant.

If Paul lives in Dublin, he lives in Ireland. Paul lives in Ireland. ThereforePaul 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 satisfia-bility.

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17. NEW: Tableaux Calculus

• The Tableaux Calculus is a decision procedure solving the problem of satisfia-bility.

• If a formula is satisfiable, the procedure will constructively exhibit an interpre-tation 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 satisfia-bility.

• If a formula is satisfiable, the procedure will constructively exhibit an interpre-tation 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 ∧BAB

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

A↔ B��

A ∧B ¬A ∧ ¬B

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

A↔ B��

A ∧B ¬A ∧ ¬B

¬¬AA

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

A↔ B��

A ∧B ¬A ∧ ¬B

¬¬AA

¬(A ∧B)��

¬A ¬B

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

A↔ B��

A ∧B ¬A ∧ ¬B

¬¬AA

¬(A ∧B)��

¬A ¬B

¬(A ∨B)¬A¬B

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

A↔ B��

A ∧B ¬A ∧ ¬B

¬¬AA

¬(A ∧B)��

¬A ¬B

¬(A ∨B)¬A¬B

¬(A→ B)A¬B

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17.1. Tableaux: the calculus

A ∧BAB

A ∨B��

A B

A→ B��

¬A B

A↔ B��

A ∧B ¬A ∧ ¬B

¬¬AA

¬(A ∧B)��

¬A ¬B

¬(A ∨B)¬A¬B

¬(A→ B)A¬B

¬(A↔ B)��

A ∧ ¬B ¬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 oftableaux 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 oftruth tables first and then by tableaux method; in other words, you are asked tocheck whether there is at least an interpretation in which all the formulas in the setare 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 atautology?

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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 atautology?

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 atautology?

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 atautology?

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 atautology?

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 atautology?

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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