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PROPOSITIONAL LOGIC, CHAPTER 12

PROPOSITIONAL LOGIC, CHAPTER 12. PURPOSE: TO CAPTURE MORE COMPLEX FORMS OF ARGUMENTS. THESE ARGUMENTS MAY MIX DISJUNCTIVE STATEMENTS WITH HYPOTHETICAL ONES. E.G.: IF P OR Q, THEN R P R SYMBOLIC LOGIC: REPLACES ALL TERMS AND FORMS IN ANY ARGUMENT WITH SYMBOLS. 1. SYMBOLIC LOGIC.

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PROPOSITIONAL LOGIC, CHAPTER 12

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  1. PROPOSITIONAL LOGIC, CHAPTER 12 • PURPOSE: TO CAPTURE MORE COMPLEX FORMS OF ARGUMENTS. • THESE ARGUMENTS MAY MIX DISJUNCTIVE STATEMENTS WITH HYPOTHETICAL ONES. • E.G.: IF P OR Q, THEN R • P • R • SYMBOLIC LOGIC: REPLACES ALL TERMS AND FORMS IN ANY ARGUMENT WITH SYMBOLS. 1

  2. SYMBOLIC LOGIC • AS MODERN LOGIC, IT ATTEMPTS TO UNDERSTAND THE FORMS OF ARGUMENTS BY ELIMINATING WORDS AND REPLACES EACH WORD WITH A TERM. • IT REPLACES OTHER WORDS WE ENCOUNTERED, I.E. “IF, THEN” “OR” “AND” WITH CONNECTIVES • IT INTRODUCES METHODS AND RULES TO FURTHER DETERMINE WHETHER ANY ARGUMENT SYMBOLICALLY CAPTURED IS VALID OR NOT 2

  3. PROPOSITIONAL LOGIC AND SYMBOLIC LOGIC • THE USE OF SYMBOLS TO CAPTURE PROPOSITIONS AND THE LINKAGES BETWEEN THEM. • OUR DISTINCTION BETWEEN CONSTITUENT AND COMPOUND PROPOSITIONS WILL HELP HERE • WHY? PROPOSITIONAL LOGIC IS CONCERNED WITH HOW THE TRUTH VALUE OF COMPONENT OR CONSTITUENT PROPOSITIONS EFFECTS THE TRUTH OF THE COMPOUND PROPOSITION • TRUTHTABLES: HELP US SEE THIS 3

  4. SYMBOLIC LOGIC/PROPOSITIONAL LOGIC • CONNECTIVES: 4 TYPES. • 3 LINK TWO PROPOSITIONS AND 1 NEGATES PROPOSITIONS • 1. CONJUNCTION: “AND” (MY SISTER IS HAPPY, AND SHE’S THROWING A PARTY) • 2. DISJUNCTION: “OR” • 3. CONDITIONAL: “IF, THEN” 4

  5. SYMBOLIC LOGIC/PROPOSITIONAL LOGIC • SYMBOLIZING CONNECTIVES • “AND” BECOMES DOT “.” • “OR” BECOMES “V” (WEDGE) • “IF, THEN” BECOMES “HORSESHOE” • WE CAN ALSO USE • NEGATION: (NOT) 5

  6. THE CONJUNCTION OR DOT • ASSERTS TWO COMPONENT OR CONSTITUTIVE PROPOSITIONS • EG. “THE RENT IS DUE, AND I HAVE NO MONEY” • TRUTH VALUE: RECALL, FOR THE WHOLE PROPOSITION TO BE TRUE BOTH PROPOSITIONS MUST BE TRUE • LET US NAME THE COMPONENT PROPOSITIONS p, q 6

  7. THE CONJUNCTION OR DOT • OTHER INDICATIONS OF CONJUNCTIONS • “BUT”: “ALTHOUGH”: “NEVERTHELESS” • EACH CAPTURES HOW FOR THE WHOLE PROPOSITION TO BE TRUE, EACH COMPONENT PART MUST BE TRUE. 7

  8. TRUTH TABLE OF CONJUNCTIONS • PAGE 336. 8

  9. NEGATION • SIGNIFIES NEGATING OR DENYING THE PROPOSITION, WHETHER COMPONENT OR COMPOUND • VARIETIES OF NEGATING: • A. IT’S NOT THE CASE THAT THE TEMPERATURE IS RISING. • B. IT’S FALSE THAT THE TEMPERATURE IS RISING. • THE TEMPERATURE IS NOT RISING. 9

  10. TRUTH TABLE FOR NEGATION • P.337 • Opposite truth value 10

  11. PRACTICE QUIZ 12.1 • 1. M . A • 2. TRUTH TABLE 11

  12. DISJUNCTION • COMPONENTS p, q, EACH A DISJUNCT • STATEMENTS WITH DISJUNCTS DO NOT ASSERT THESE BUT EXPRESS THEM • TRUTH VALUE: IF EITHER ONE OR BOTH EXPRESSED PROPOSITIONS IS TRUE, THEN THE WHOLE PROPOSITION IS TRUE. • IF NEITHER IS TRUE, THEN THE WHOLE PROPOSITION IS FALSE. 12

  13. DISJUNCTION • TRUTH TABLE: 13

  14. CONDITIONAL • “IF p, THEN q” • MEANING: IF ANTECEDENT IS TRUE, THEN CONSEQUENT IS ALSO TRUE. • E.G. “IF I STUDY HARD, I WILL PASS THE EXAM.” • WE ASK ABOUT THE TRUTH OF EACH COMPONENT PROPOSITION. • IF P IS TRUE AND Q IS TRUE, IS THE CONDITIONAL TRUE? IS THE WHOLE STATEMENT TRUE? • RULE OF THUMB: THE ONLY CONDITION UNDER WHICH A CONDITIONAL PROPOSITION IS FALSE IS WHEN THE ANTECEDENT IS TRUE BUT THE CONSEQUENT IS FALSE 14

  15. TRUTH TABLE FOR CONDITIONALS 15

  16. NON-STANDARD FORMS REPEATED • STRATEGY BOX 12.1 P. 341. MEMORIZE!! • PRACTICE QUIZ 12.2, #9 • IF ~Q, THEN P • TRUTH TABLE: NEXT PAGE 16

  17. TRUTH TABLE 17

  18. TRUTH FUNCTION • APPRECIATING WHAT IS MEANT BY THE TRUTH VALUE OF CONNECTIVES • NOT RELATED TO THE WORLD OF EXPERIENCE • THE CONJUNCTION: RELATES ANY TWO PROPOSITIONS. EASY TO SEE THAT BOTH MUST BE TRUE FOR WHOLE TO BE TRUE. 18

  19. TRUTH FUNCTION • DISJUNCTION: • E.G. THE EARTH IS ROUND V 2 + 2= 5 • A BIT TOUGH TO SEE THAT THE WHOLE IS TRUE EVEN THOUGH ONE DISJUNCT IS FALSE. • REMEMBER THE MEANING OF DISJUNCTION • CONDITIONAL: EVEN TRICKIER • EG. IF GRASS IS RED, THEN HUMANS CAN BREATHE UNDER WATER 19

  20. TRUTH VALUE AND CONDITIONALS • IF BOTH ARE FALSE, THEN THE STATEMENT AS A WHOLE IS TRUE. • SAME AS DISJUNCT! • P ⊃ Q IS EQUIVALENT TO ~P V Q • DON’T WORRY TOO MUCH ABOUT THIS PRINCIPLE. MEMORIZE RULE OF THUMB • PRACTICE QUIZ 12.3 20

  21. STATEMENT FORMS OR RULES OF PUNCTUATION • PURPOSE: TO CAPTURE THE COMPONENT PROPOSITION OF MORE COMPLEX PROPOSITIONS. • USE OF PARENTHESIS OR BRACKETS TO DO SO. • 2 STATEMENTS: • 1. EITHER I’LL GO HOME AND WATCH T.V. OR I’LL THINK ABOUT THE ELECTION • 2. I’LL GO HOME, AND I’LL EITHER WATCH TV OR THINK ABOUT THE ELECTION • BOTH HAVE THE SAME COMPONENT PROPOSITIONS BUT SAY DIFFERENT THINGS. 21

  22. PUNCTUATION CONT. • STRATEGY: USE PARENTHESES SO CONNECTIVES JOIN TWO COMPONENTS AND LEAVE MAIN CONNECTIVE OUTSIDE PARENTHESES • TECHNIQUE: • 1. IDENTIFY MAIN CONNECTIVE • 2. IDENTIFY COMPONENT CONNECTIVES AND TERMS. • GO BACK TO OUR EXAMPLE: • LET H= GO HOME • LET T= WATCH TV • LET E= THINK ABOUT THE ELECTION 22

  23. PUNCTUATION CONT. • 1. (H . T) V E • 2. H . (T V E) • CONJUNCTIONS: EASY. NO NEED TO WORRY ABOUT PLACEMENT OF PARENTHESES OR HOW WE GROUP THEM. • EG. I LIKE LOGIC AND I DRINK TEA AND I READ BOOKS BY HEIDEGGER. • L . T. H (L . T) . H OR L . (T . H) 23

  24. MORE COMPLEX CASES • P. 347. • FIRST STEP: IDENTIFY MAIN CONNECTIVE • SECOND STEP: IDENTIFY IF COMPONENT PART CONTAINS OTHER COMPONENT PARTS AND… • THIRD STEP: CONTINUE ON UNTIL ALL COMPONENT CONNECTIVES ARE SYMBOLIZED. • NEGATION: RULE OF THUMB: A NEGATION SIGN IN FRONT OF A COMPONENT STATEMENT IS A NEGATION OF THE COMPONENT ONLY. BUT NEGATION SIGN IN FRONT OF A COMPOUND STATEMENT IS NEGATION OF WHOLE STATEMENT. • I.E. EITHER LESLIE IS NOT SAD OR SHE’S A GOOD ACTOR. ~S V G • ~(S V G) WOULD MEAN “LESLIE IS NEITHER SAID NOR A GOOD ACTOR” 24

  25. MORE COMPLEX CASES: CONDITIONAL CONNECTIVES • P. 247-348 • RULE OF THUMB REPEATED: START WITH MAIN CONNECTOR AND IDENTIFY EACH BASIC COMPONENT AND THEN DETERMINE WHETHER THERE ARE COMPONENTS WITHIN THESE. 25

  26. LOGICAL STRUCTURE • MEANING: REPLACING STATEMENTS WITH TERMS. • PURPOSE: TO RECOGNIZE BASIC FORMS AND TEST VALIDITY. • I.E. NO LONGER THE G, P, A, D AND O, BUT p, q, r, s, t TO CAPTURE ANY STATEMENTS. 26

  27. THE BICONDITIONAL • ALREADY TOUCHED ON THIS WITH THE CO-CONDITIONAL STATEMENTS. • I.E. IF AND ONLY IF YOU ATTEND ALL OF MY CLASS WILL YOU HAVE A CHANCE OF PASSING THE CLASS. • MEANS: IF p, THEN q and IF q, THEN p. • SYMBOLIZED: (p ⊃ q) . q ⊃ p • Easier way: p q (use of 3 parallel lines) • equivalence 27

  28. TRUTH TABLE FOR BICONDITIONAL • PRACTICE QUIZ. 12.4 28

  29. COMPUTING TRUTH VALUES FOR MORE COMPLEX STATEMENTS • PURPOSE: DETERMINING THE TRUTH OF MORE COMPLEX OR COMPOUND STATEMENTS ON THE BASIS OF FIRST CALCULATING THE TRUTH VALUES OF THE COMPONENT OR INTERNAL PARTS. • METHOD: NOTE: WE WILL BE USING A SLIGHTLY DIFFERENT VERSION OF KELLEY’S METHOD 1 AND 2. THIS COMBINES THE STRENGTH OF EACH. 29

  30. TRUTH TABLES/METHOD • 1. GIVE EACH TERM A SEPARATE COLUMN • 2. IF ANY TERM IS NEGATED, GIVE THIS NEGATION A SEPARATE COLUMN • 3. GIVE THE COMPONENT CONNECTIVES A SEPARATE COLUMN • 4. GIVE THE MAIN CONNECTIVE THE LAST COLUMN • 5. FILL IN THE COLUMNS WITH THE TERMS AND TERMS ALONE BY ROTATING TRUTH VALUES. 30

  31. TRUTH TABLES/METHOD • PRINCIPLE OF ROTATION: IF THERE ARE ONLY 2 TERMS, ROTATE THIS WAY: FIRST TERM ROTATE WITH TTFF. SECOND TERM, TFTF. • IF THERE ARE THREE TERMS, ROTATE AS SUCH: FIRST TERM: TTTTFFFF. SECOND TERM TTFFTTFF AND THIRD TERM TFTFTFTF (THREE TERMS REQUIRE 8 ROWS) • 6. FILL IN THE TRUTH VALUES FOR THE OTHER COLUMNS BASED ON THE ORIGINAL TERMS. • 7.FILL IN THE MAIN CONNECTIVE COLUMN BY RELATING THE CORRECT COMPONENTS. 31

  32. TRUTH TABLES/METHOD • STATEMENT: • A (A . B) 32

  33. TRUTH TABLE TECHNIQUE APPLIED • PRACTICE QUIZ 12.5 • 1. A V (B V A) 33

  34. TRUTH TABLE TECHNIQUE APPLIED • PRACTICE QUIZ 12.5, 7. • (P ~Q) V (R . P) NEXT PAGE 34

  35. TRUTH TABLE QUIZ 12.5, 7 35

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