LOGIC ANTHONY KAPOLKA FYF 101-9/3/2010
LIBERALLY EDUCATED PEOPLE......RESPECT RIGOR NOT SO MUCH FOR ITS OWN SAKE BUT AS A WAY OF SEEKING TRUTH.
LOGIC PUZZLE COOPER IS MURDERED. 3 SUSPECTS: SMITH, JONES, & WILLIAMS SMITH SAYS: COOPER IS A FRIEND OF JONES & WILLIAM DISLIKES HIM. JONES SAYS: I DON T KNOW COOPER AND I WAS OUT OF TOWN THAT DAY. WILLIAMS SAYS: I SAW BOTH SMITH & JONES WITH COOPER THE DAY OF THE MURDER; ONE OF THEM MUST HAVE KILLED HIM. WHO IS GUILTY?
HELPFUL ASSUMPTIONS INNOCENT PEOPLE DO NOT LIE. INNOCENT PEOPLE ARE NOT MISTAKEN. THERE IS ONLY ONE MURDERER. THE MURDERER IS ONE OF THE SUSPECTS.
REASONING REASONING IS THE PROCESS OF MOVING TOWARD CONCLUSIONS USING CLEAR, RELEVANT AND COMPELLING SUPPORTIVE STATEMENTS.
EXAMPLE REASONING IF INNOCENT PEOPLE DO NOT LIE, AND SINCE SMITH AND JONES CONTRADICT EACH OTHER, THEN EITHER SMITH OR JONES IS GUILTY. IF INNOCENT PEOPLE DO NOT LIE, AND SINCE WILLIAMS AND JONES CONTRADICT EACH OTHER, THEN EITHER WILLIAMS OR JONES IS GUILTY. SINCE THERE IS ONLY ONE MURDERED, THEN SMITH AND WILLIAMS CANNOT BOTH BE GUILTY. JONES IS GUILTY.
WHAT IS LOGIC? LOGIC IS A MECHANISM FOR PROVIDING PRECISE MEANING TO STATEMENTS. IT INCLUDES: A FORMAL LANGUAGE FOR EXPRESSING STATEMENTS. RULES FOR OBJECTIVELY REASONING ABOUT THE TRUTH OF STATEMENTS.
STATEMENTS PROPOSITIONS ARE DECLARATIVE SENTENCES WHICH ARE EITHER TRUE (T) OR FALSE (F). PROPOSITIONS 2+3=5 FORSTER WROTE BOOKS. FYF101 IS GREAT. NOT PROPOSITIONS DID YOU DO THE READING? DO THE READING! X + Y = 4
LOGICAL OPERATORS A LOGICAL OPERATOR IS DEFINED BY A TRUTH TABLE.
NEGATION, NOT, S: TODAY IS FRIDAY. S: TODAY IS NOT FRIDAY. TRUTH TABLE P P NEGATION SWITCHES A STATEMENT S TRUTH VALUE. T F F T
NEGATION, NOT, IT IS NOT TRUE THAT... THAT IS IRRELEVANT. MY EXAMPLES ARE UNSATISFACTORY.
CONJUNCTION, AND, S: TODAY IS FRIDAY T: IT IS RAINING. S T: TODAY IS FRIDAY AND IT IS RAINING. TRUTH TABLE P Q P Q T T T T F F F T F F F F
DISJUNCTION, INCLUSIVE OR, S: TODAY IS FRIDAY T: IT IS RAINING. S T: TODAY IS FRIDAY OR IT IS RAINING. TRUTH TABLE P Q P Q T T T T F T F T T F F F
IN A RESTAURANT... WAITER OFFERS YOU CHOICE OF FRIES OR A BAKED POTATO. THIS IS NOT A DISJUNCTION - BECAUSE YOU CANNOT HAVE BOTH.
EXCLUSIVE OR, XOR, S: TODAY IS FRIDAY T: IT IS RAINING. S T: EITHER TODAY IS FRIDAY OR IT IS RAINING. TRUTH TABLE P Q P Q T T F T F T F T T F F F
(ENGLISH) LANGUAGE IS AMBIGUOUS. YOU OR I WILL WIN THE CONTEST. JOE S MAJOR IS ART OR PHARMACY. LOGIC IS MORE PRECISE. S T = (S T) (S T)
IMPLICATION, CONDITIONAL, IF S THEN T S IMPLIES T T IS NECESSARY FOR S S IS SUFFICIENT FOR T S IS ANTECEDENT, T IS CONSEQUENT. P Q P Q IF YOU STUDY, YOU WILL DO WELL. T T T T F F F T T F F T
BICONDITIONAL, IFF, S IF AND ONLY IF T S IS NECESSARY AND SUFFICIENT FOR T S: YOU CAN TAKE A BUS T: YOU CAN BUY A TICKET S T: YOU CAN TAKE A BUS ONLY IF YOU BUY A TICKET. P Q P Q T T T T F F F T F F F T
BICONDITIONAL, IFF, S AND T ARE EQUIVALENT - BOTH TRUE OR FALSE TOGETHER. S P = (S P) (P S)
OPERATOR PRECEDENCE ALLOWS YOU TO ORDER OPERATIONS HIGH TO LOW:,,,,,, ARE LEFT ASSOCIATIVE: S T U (S T) U
PROPOSITIONS ACCORDING TO UNITY (ABSOLUTE, RELATIVE) ACCORDING TO QUALITY (AFFIRMATIVE, NEGATIVE) ACCORDING TO QUANTITY (ALL, SOME, NONE) ACCORDING TO MODALITY (NECESSARY, POSSIBLE)
UNITY: ABSOLUTE OR RELATIVE ABSOLUTE - WE SAY ONE THING OF ANOTHER WITHOUT QUALIFICATION. (THIS IS A CATEGORICAL PROPOSITION.) ALL HUMANS ARE CAPABLE OF USING LOGIC. RELATIVE - THE COMBINATION OF CATEGORICAL PROPOSITIONS BY CERTAIN CONNECTIVES. (THIS IS A COMPOUND OR HYPOTHETICAL PROPOSITION.) IF MAN IS RATIONAL, THEN HE IS CAPABLE OF USING LOGIC.
UNITY: ABSOLUTE OR RELATIVE COPULA - CONNECTS SUBJECT & PREDICATE ABSOLUTE - WE SAY ONE THING OF ANOTHER WITHOUT QUALIFICATION. (THIS IS A CATEGORICAL PROPOSITION.) ALL HUMANS ARE CAPABLE OF USING LOGIC. RELATIVE - THE COMBINATION OF CATEGORICAL PROPOSITIONS BY CERTAIN CONNECTIVES. (THIS IS A COMPOUND OR HYPOTHETICAL PROPOSITION.) IF MAN IS RATIONAL, THEN HE IS CAPABLE OF USING LOGIC.
QUALITY: AFFIRMATIVE OR NEGATIVE ASK IF PREDICATE BEING AFFIRMED OR DENIED. AFFIRMATIVE - EXPRESSES UNION OF PREDICATE WITH SUBJECT. LOGIC IS FUN. NEGATIVE - EXPRESSES SEPARATION OF PREDICATE FROM SUBJECT. LOGIC IS NOT FUN. NO LOGIC IS FUN.
EXAMPLE PEOPLE WHO ARE NOT FRIENDLY ARE UNSOCIABLE.
EXAMPLE PEOPLE WHO ARE NOT FRIENDLY ARE UNSOCIABLE. EVEN THOUGH THIS SOUNDS NEGATIVE, THE NEGATIVE PREDICATE IS BEING AFFIRMED OF THE NEGATIVE SUBJECT.
QUANTITY: ALL, SOME, NONE ALL - UNIVERSAL - PREDICATE BELONGS TO THE SUBJECT AND ALL ITS SINGULAR MEMBERS. ALL TIGERS ARE MAMMALS. NONE - UNIVERSAL - PREDICATE DENIED OF THE SUBJECT AND ALL ITS SINGULAR MEMBERS. NO TIGERS ARE LIONS. SOME - PARTICULAR - PREDICATE SAID OF THE SUBJECT BUT IS MEANT TO BE LIMITED TO SOME OF ITS SINGULAR MEMBERS. SOME TIGERS ARE FIERCE.
FOUR TYPES OF CATEGORICAL PROPOSITIONS - FOR ALL - THERE EXISTS AT LEAST ONE... - EQUAL BY DEFINITION - SUBSET - SET INTERSECTION - THE EMPTY SET.
MODALITY: NECESSARY, POSSIBLE, IMPOSSIBLE NECESSARY - S - S MUST BE TRUE POSSIBLE - S - S COULD BE TRUE IMPOSSIBLE - S - S COULD NEVER BE TRUE
WHY CARE? (AGAIN) WE BUILD ARGUMENTS OUT OF PROPOSITIONS. SOME OF THESE PROPOSITIONS (THE PREMISES) SHOULD EPISTEMICALLY SUPPORT OTHER PROPOSITIONS (THE CONCLUSIONS) OF THE ARGUMENT. AN ARGUMENT IS VALID IFF THE PREMISES THE CONCLUSIONS. AN ARGUMENT IS SOUND IFF IT IS VALID AND THE PREMISES ARE TRUE. LOGIC WINS ARGUMENTS.
EXERCISE LET P Q & R BE THE FOLLOWING PROPOSITIONS: P: YOU GET AN A ON THE FINAL EXAM. Q: YOU DO EVERY EXERCISE IN THE BOOK. R: YOU GET AN A IN THIS CLASS. WRITE AS A LOGICAL STATEMENT: YOU GET AN A IN THE CLASS, BUT YOU DO NOT DO EVERY EXERCISE IN THE BOOK.
EXERCISE LET P Q & R BE THE FOLLOWING PROPOSITIONS: P: YOU GET AN A ON THE FINAL EXAM. Q: YOU DO EVERY EXERCISE IN THE BOOK. R: YOU GET AN A IN THIS CLASS. WRITE AS A LOGICAL STATEMENT: YOU GET AN A IN THE CLASS, BUT YOU DO NOT DO EVERY EXERCISE IN THE BOOK. R Q
EXERCISE LET P Q & R BE THE FOLLOWING PROPOSITIONS: P: YOU GET AN A ON THE FINAL EXAM. Q: YOU DO EVERY EXERCISE IN THE BOOK. R: YOU GET AN A IN THIS CLASS. WRITE AS A LOGICAL STATEMENT: TO GET AN A IN THIS CLASS, IT IS NECESSARY FOR YOU TO GET AN A ON THE FINAL.
EXERCISE LET P Q & R BE THE FOLLOWING PROPOSITIONS: P: YOU GET AN A ON THE FINAL EXAM. Q: YOU DO EVERY EXERCISE IN THE BOOK. R: YOU GET AN A IN THIS CLASS. WRITE AS A LOGICAL STATEMENT: TO GET AN A IN THIS CLASS, IT IS NECESSARY FOR YOU TO GET AN A ON THE FINAL. R P
EXERCISE LET P Q & R BE THE FOLLOWING PROPOSITIONS: P: YOU GET AN A ON THE FINAL EXAM. Q: YOU DO EVERY EXERCISE IN THE BOOK. R: YOU GET AN A IN THIS CLASS. WRITE AS A LOGICAL STATEMENT: GETTING AN A ON THE FINAL & DOING EVERY EXERCISE IN THE BOOK IS ENOUGH TO GET AN A IN THIS CLASS.
EXERCISE LET P Q & R BE THE FOLLOWING PROPOSITIONS: P: YOU GET AN A ON THE FINAL EXAM. Q: YOU DO EVERY EXERCISE IN THE BOOK. R: YOU GET AN A IN THIS CLASS. WRITE AS A LOGICAL STATEMENT: GETTING AN A ON THE FINAL & DOING EVERY EXERCISE IN THE BOOK IS ENOUGH TO GET AN A IN THIS CLASS. P Q R