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Lecture 1-Formal Logic

A Classic Example . All men are mortal, Socrates is a man, so Socrates is mortal. True or False? Prove it. A more complex example. blood on the glove matched the murder victims' DNA; a trail led from the crime scene to a man's bedroom; the gloves, allowing for shrinkage, fit; no one el

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Lecture 1-Formal Logic

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    1. Lecture 1-Formal Logic May 12th, 2003

    2. A Classic Example All men are mortal, Socrates is a man, so Socrates is mortal. True or False? Prove it

    3. A more complex example blood on the glove matched the murder victims' DNA; a trail led from the crime scene to a man's bedroom; the gloves, allowing for shrinkage, fit; no one else had entered the bedroom; the man is therefore guilty of the murder.

    4. Definitions Statement – A sentence that is either true or false. Logical connective – binary connective conjunction -- (A^B) disjunction -- (A U B) unary connective negation -- A’ Implication -- A?B antecedent (A) consequent (B) Equivalence A?B (A?B ^ B?A)

    5. Well-formed formula (wff) Well-formed formula (wff) An expression that is a legitimate string. 1. connectives within parentheses, innermost parentheses first. 2. ‘ 3. ^,V 4. ? 5. ? order of precedence is top to bottom.

    6. Truth table

    7. Logic connectives in English word It is a warm-up for your GRE logic section A is a sufficient condition for B ? B is a necessary condition of A ??? A only if B ??? Answer: A?B ?? What is the Negation ?? Peter is tall and thin

    8. Tautologies Tautology: A wff whose truth value is always true. Contradiction: A wff whose truth value is always false. Equivalent wffs: P,Q { P?Q } P is Q’s dual and vice versa Some Tautological equivalences (P8) Verify the equivalence (truth table).

    9. Some Tautological equivalences Commutative Properties AVB? BVA, A^B?B^A Associative Properties (AVB)VC?AV(BVC),(A^B)^C?A^(B^C) Distributive Properties AV(B^C)?(AVB)^(AVC),A^(BVC)?(A^B)V(A^C) Identity properties AV0=A, A^1?A Complement properties AVA’=1, A^A’=0

    10. De Morgan’s Law Some information about De Morgan if you are interested (not required for test). ?? Find the negation of the statement?? “I am not on my knees, and never will be again” De Morgan’s Law will help (AVB)’?A’^B’ , (A^B)’?A’VB’

    11. Logical Connectives in the Real World Web Search Engine Programming language

    12. How to test the tautology? Truth table Algorithm (TautTest) Formal logic --derivation rules (will be covered later)

    13. Algorithm An algorithm is a set of instructions that can be mechanically executed in a finite amount of time in order to solve some problem. Psudocode: The compromise form to describe algorithms Algorithm tautologyTest (handout).

    14. Assignments I Due date (May 19th, next Monday) Excercises1.1 2,5,6(c,d), 7(a,c), 14(e,h), 17(e), 26(c), use Algorithm TautTest to prove the proposition (A^B’)’^(AVB)?B

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