2-lecture_01.pdf
- 1. Dr.Nermeen Kame Cairo University Fall 2022 Introduction to Logic Propositional Logic
- 2. Syntax of Propositional Logic Propositional Constants Logical Operators Semantics Truth Assignments for propositional constants Meaning of logical operators Evaluation Truth Assignments to values of compound sentences Satisfaction Values of compound sentences to truth assignments Truth Tables Agenda
- 3. A propositional vocablary is a set/sequence of primitive symbols, called proposition constants. Given a propositional vocabulary, a propositional sentence is either (1) a member of the vocabulary or (2) a compound expression formed from members of the vocabulary and logical operators and parentheses. (Details to follow.) A propositional language is the set of all propositional sentences that can be formed from a propositional vocabulary. Propositional Languages
- 4. By convention (in this course), proposition constants are written as strings of alphanumeric characters beginning with a lower case letter. Examples: raining r32aining rAiNiNg rainingorsnowing Non-Examples: 324567 raining.or.snowing Proposition Constants
- 5. Negations: ¬raining The argument of a negation is called the target. Conjunctions: (raining Ù snowing) The arguments of a conjunction are called conjuncts. Disjunctions: (raining Ú snowing) The arguments of a disjunction are called disjuncts. Compound Sentences (part I)
- 6. Implications: (raining Þ cloudy) The left argument of an implication is the antecedent. The right argument is the consequent. Equivalences: (cloudy Û raining) Compound Sentences (part II)
- 7. ¬raining (raining Ù snowing) (raining Ú snowing) (raining Þ cloudy) (cloudy Û raining) ¬(raining Ù snowing) ((raining Ù snowing) Þ cloudy) (cloudy Þ (raining Ù snowing)) ((cloudy Ù wet) Û (raining Ú snowing)) (¬raining Þ (cloudy Þ snowing)) Nested Compound Sentences
- 8. Dropping Parentheses is good: (p Ù q) ® p Ù q But it can lead to ambiguities: ((p Ú q) Ù r) ® p Ú q Ù r (p Ú (q Ù r)) ® p Ú q Ù r Parentheses Removal
- 9. Parentheses can be dropped when the structure of an expression can be determined by precedence. ¬ Ù Ú Þ Û An operand surrounded by operators associates with operator of higher precedence. ¬p Ú q ® ((¬p) Ú q) p Ú q Ù r ® (p Ú (q Ù r)) p Ù q Þ r ® ((p Ù q) Þ r) p Þ q Û r ® ((p Þ q) Û r) Precedence
- 10. If surrounded by two occurrences of Ù or Ú, the operand associates with the operator to the left. p Ù q Ù r ® ((p Ù q) Ù r) p Ú q Ú r ® ((p Ú q) Ú r) If surrounded by two occurrences of Þ or Û, the operand associates with the operator to the right. p Þ q Þ r ® (p Þ (q Þ r)) p Û q Û r ® (p Û (q Û r)) Precedence (continued)
- 11. (a) All purple mushrooms are poisonous. (b) A mushroom is poisonous only if it is purple. (c) A mushroom is not poisonous unless it is purple. (d) No purple mushroom is poisonous. Natural Language Examples Consider a propositional language with three proposition constants—mushroom, purple, and poisonous—each indicating the property suggested by its spelling. Using these proposition constants, encode the following English sentences as Propositional Logic sentences.
- 12. Vocabulary: purple, mushroom, poisonous Natural Language Examples Consider a propositional language with three proposition constants—mushroom, purple, and poisonous—each indicating the property suggested by its spelling. Using these proposition constants, encode the following English sentences as Propositional Logic sentences.
- 13. Vocabulary: purple, mushroom, poisonous Purple mushrooms are poisonous. mushroom Ù purple Þ poisonous mushroom Þ (purple Þ poisonous) Natural Language Examples
- 14. Vocabulary: purple, mushroom, poisonous A mushroom is poisonous only if it is purple. mushroom Þ (¬purple Þ ¬poisonous) mushroom Þ (poisonous Þ purple) mushroom Ù poisonous Þ purple Natural Language Examples
- 15. Vocabulary: purple, mushroom, poisonous A mushroom is not poisonous unless it is purple. mushroom Þ (¬purple Þ ¬poisonous) mushroom Þ (poisonous Þ purple) mushroom Ù poisonous Þ purple Natural Language Examples
- 16. Vocabulary: purple, mushroom, poisonous No purple mushroom is poisonous ¬(mushroom Ù poisonous Ù purple) mushroom Ù poisonous Þ ¬purple Natural Language Examples
- 17. A propositional interpretation is an association between the propositional constants in a propositional language and the values T or F. (Later, written as 1 and 0.) We sometimes view an interpretation as a Boolean vector of values for the items in the signature of the language (when the signature is ordered). i = TFT Propositional Interpretation
- 18. A sentential interpretation is an association between the sentences in a propositional language and the truth values T or F. pi = T (p Ú q)i = T qi = F (¬q Ú r)i = T ri = T ((p Ú q) Ù (¬q Ú r))i = T A propositional interpretation defines a sentential interpretation by application of operator semantics. Sentential Interpretation
- 19. Negation: For example, if the interpretation of p is F, then the interpretation of ¬p is T. For example, if the interpretation of (pÙq) is T, then the interpretation of ¬(pÙq) is F. Operator Semantics
- 20. Conjunction: Disjunction: NB: The type of disjunction here is called inclusive or, which says that a disjunction is true if and only if at least one of its disjuncts is true. This contrasts with exclusive or, which says that a disjunction is true if and only if an odd number of its disjuncts is true. Operator Semantics (continued)
- 21. Implication: NB: The semantics of implication here is called material implication. Any implication is true if the antecedent is false, whether or not there is a connection to the consequent. Operator Semantics (continued)
- 23. Interpretation i: Compound Sentence (p Ú q) Ù (¬q Ú r) Evaluation (T Ú F) Ù (¬F Ú T) (T Ú F) Ù (T Ú T) T Ù T T
- 24. A truth table is a table of all possible interpretations for the propositional constants in a language. One column per constant. One row per interpretation. For a language with n constants, there are 2n interpretations. Truth Tables
- 26. Method to find all propositional interpretations that satisfy a given set of sentences: (1)Form a truth table for the propositional constants. (2) For each sentence in the set and each row in the truth table, check whether the row satisfies the sentence. If not, cross out the row. (3) Any row remaining satisfies all sentences in the set. (Note that there might be more than one.) Satisfaction
- 28. qÞr p ÞqÙr Satisfaction Example (continued)
- 29. qÞr p ÞqÙr ¬r Satisfaction Example (concluded)
- 30. EXAMPLE–NATURAL LANGUAGE The following examples concern three properties of people, and we assign a different proposition constant to each of these properties. We use the constant c to mean that a person is cool. We use the constant f to mean that a person is funny. And we use the constant p to mean that a person is popular.
- 31. EXAMPLE–NATURAL LANGUAGE consider the English sentence - If a person is cool or funny, then he is popular. - A person is popular only if he is either cool or funny - A person is popular if and only if he is either cool or funny. - There is no one who is both cool and funny.
- 32. EXAMPLE–NATURAL LANGUAGE Suppose we were to imagine a person who is cool and funny and popular, i.e., the proposition constants c and f and p are all true. - Evaluate the sentence - Evaluate the sentence
- 33. EXAMPLE–NATURAL LANGUAGE Suppose we were to imagine a person who is cool and funny and popular, i.e., the proposition constants c and f and p are all true. - Evaluate the sentence
- 34. Exercise 1 Consider a truth assignment in which p is true, q is false, r is true. Use this truth assignment to evaluate the following sentences.
- 35. Exercise 2 Consider the sentences shown below. There are three proposition constants here, meaning that there are eight possible truth assignments. How many of these assignments satisfy all of these sentences?