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Conjunctions, Disjunctions, and Negations

Conjunctions, Disjunctions, and Negations. Symbolic Logic 2/12/2001. Outline. Test Homework & Schedule Truth Tables Negation Conjunction Disjunction Ambiguity & Parentheses. Test. Handed back Wednesday Solutions posted after all have taken it. See Sara if you have questions.

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Conjunctions, Disjunctions, and Negations

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  1. Conjunctions, Disjunctions, and Negations Symbolic Logic 2/12/2001

  2. Outline • Test • Homework & Schedule • Truth Tables • Negation • Conjunction • Disjunction • Ambiguity & Parentheses

  3. Test • Handed back Wednesday • Solutions posted after all have taken it. • See Sara if you have questions. • Note on Caps & letters: • Caps (A, B, C…) are reserved for Predicates. • Letters like (a, b, c…) are usually for constants. • Letters like (x, y, z…) are usually for variables.

  4. Homeworks & Schedule • New Procedures for homeworks. • Homework will be Due in class every Monday. • You will hand in a disk and/or paper. • What is due is the problems for the sections on the schedule for the preceding week. • An updated schedule with tests & assignments is online.

  5. More on Homework • Problems will be sampled. That is, not every problem of every assignment will be graded. You are expected to do all the assignments. • You will need two disks for your homework so that you can keep working while the current assignment is being graded. • Your second disk will also serve as a backup.

  6. Truth Tables • We use a truth table to define logical constants. • A truth table lists all possible combinations of truth values for a particular constant. • The left side of the table contains the truth values of the components of a sentence. • The right side contains the truth value of the sentence.

  7. Negation • Here’s the truth table for negation (~) P ~P TRUE FALSE FALSE TRUE • This table lists all possible combinations of truth values for the sentence P.

  8. Negation Continued • Negation simply reverses the truth value of whatever it is negating. • For example, if we have the true sentence, Home(John) the negation of this sentence is ~Home(John) and its truth value will be false.

  9. Truth table for Home(John) Home(John) ~Home(John) TRUE FALSE FALSE TRUE

  10. Conjunction • The ordinary language definition of conjunction is AND. • Words like “but” and “moreover” also do much the same thing in English as “^” in FOL. • ^ implies that a compound sentence will be truth IFF both sides are true, false otherwise.

  11. Truth table for ^ P Q P^Q T T T T F F F T F F F F

  12. Example of ^ Happy(John) Sad(Mary) Happy(John)^Sad(Mary) T T T T F F F T F F F F

  13. Disjunction • Logical OR (v) • The resulting sentence is true if either or both sentences are true. • This differs from the ordinary English meaning of OR

  14. Disjunction P Q PvQ T T T T F T F T T F F F

  15. v vs. XOR • The ordinary English use of “or” ordinarily means XOR, which is true IFF only one of the sentences is true. P Q P xor Q T T F T F T F T T F F F

  16. Exercises

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