140 likes | 445 Vues
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
E N D
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 Qs 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 Morgans 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 Morgans Law will help
(AVB)?A^B , (A^B)?AVB
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