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Logical Structures in Natural Language: Propositional Logic II (Tableaux) Raffaella Bernardi Università degli Studi di Trento e-mail: [email protected] Contents First Last Prev Next J Contents 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents First Last Prev 3 4 5 6 7 8 9 10 11 12 13 15 17 18 19 20 21 Next J 16 17 18 19 20 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents First Last Prev 22 23 24 25 26 27 28 29 Next J 1. What we have said last time • Logic – Language: syntax, semantics. – Reasoning • Semantics – Meaning of a sentence = Truth value Contents First Last Prev Next J 1. What we have said last time • Logic – Language: syntax, semantics. – Reasoning • Semantics – Meaning of a sentence = Truth value – Compositional meaning: truth-functional connectives Contents First Last Prev Next J 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} Contents First Last Prev Next J 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) Contents First Last Prev Next J 2. Remind: Propositional Logic: Basic Ideas Statements: The elementary building blocks of propositional logic are atomic statements that cannot be decomposed any further: propositions. Contents First Last Prev Next J 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” Contents First Last Prev Next J 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. Contents First Last Prev Next J 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) Contents First Last Prev Next J 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. Contents First Last Prev Next J 4. Reminder: From English to Propositional Logic Eg. If you don’t sleep then you will be tired. Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last Prev Next J 5. Reminder: Semantics: Intuition • Atomic propositions can be true T or false F. Contents First Last Prev Next J 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). Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last Prev Next J 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) Contents First Last Prev Next J 7. Reminder: Truth Tables Contents First Last Prev Next J 7. Reminder: Truth Tables I1 I2 φ T F ¬φ F T (1) Contents First Last Prev Next J 7. Reminder: Truth Tables I1 I2 φ T F ¬φ F T (1) I1 I2 I3 I4 φ T T F F Contents ψ T F T F φ∧ψ T F F F (1) First Last Prev Next J 7. 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) Contents Prev Next J 7. 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) First Last Contents Prev Next J 8. Reminder: Model A model consists of two pieces of information: Contents First Last Prev Next J 8. Reminder: Model A model consists of two pieces of information: • which collection of atomic propositions we are talking about (domain, D), Contents First Last Prev Next J 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). Contents First Last Prev Next J 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). Contents First Last Prev Next J 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). Contents First Last Prev Next J 9. Tautologies and Contradictions Build the truth table of p ∧ ¬p. Contents First Last Prev Next J 9. Tautologies and Contradictions Build the truth table of p ∧ ¬p. It’s a contradiction: always false. Contents First Last Prev Next J 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). Contents First Last Prev Next J 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 Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J 11. Example of argumentations Today is Monday or today is Thursday Today is not Monday ================= Today is Thursday P v Q not P ===== Q Contents First Last Prev Next J 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 Contents First Q --> R Q ======= R Last Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last Prev Next J 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) Contents First Last Prev Next J 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 } Contents First Last Prev Next J 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) Contents First Last Prev Next J 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 ) Contents First Last Prev Next J 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 Contents First Last Prev Next J 14. A formula: Tautology, Contradiction, Satisfiable, Falsifiable Recall, a formula P is: Contents First Last Prev Next J 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) Contents First Last Prev Next J 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. Contents First Last Prev Next J 14.1. Example I1 I2 P T F ¬P F T ¬P ∨ P T T ¬P ∨ P is a tautology. Contents First Last Prev Next J 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) Contents First Last Prev Next J 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) Contents First Last Prev Next J 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 } Contents First Last Prev Next J 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. Contents First Last Prev Next J 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) Contents First Last Prev Next J 17. NEW: Tableaux Calculus • The Tableaux Calculus is a decision procedure solving the problem of satisfiability. Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J 17.1. Tableaux: the calculus Contents First Last Prev Next J 17.1. Tableaux: the calculus A∧B A B Contents First Last Prev Next J 17.1. Tableaux: the calculus A∧B A B A∨B A B Contents First Last Prev Next J 17.1. Tableaux: the calculus A∧B A B A∨B A A→B ¬A B Contents First Last B Prev Next J 17.1. Tableaux: the calculus A∧B A B A∨B A A→B ¬A B B A↔B A∧B ¬A ∧ ¬B Contents First Last Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last ¬B Prev Next J 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 Contents First Last Prev Next J 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 Contents First Last Prev Next J 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 Contents ¬B ¬(A ↔ B) A ∧ ¬B First Last ¬A ∧ B Prev Next J 18. Heuristics and Exercises Apply non-branching rules before branching rules. Contents First Last Prev Next J 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) Contents First Last Prev Next J 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)} Contents First Last Prev Next J 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? Contents First Last Prev Next J 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. Contents First Last Prev Next J 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? Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J 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? Contents First Last Prev Next J 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. Contents First Last Prev Next J 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. Contents First Last Prev Next J