DEDUCTIVE REASONING: PROPOSITIONAL LOGIC

1. DEDUCTIVE REASONING: PROPOSITIONAL LOGIC • Purposes: • To analyze complex claims and deductive argument forms • To determine what arguments are valid or not • To learn further how deduction works • Logical relationships among statements • Symbolic Logic: Using symbols instead of words

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

3. A COMPLEX ARGUMENT • p v q p r q s Therefore, r v s ~r Therefore s valid argument

4. SYMBOLIC LOGIC/PROPOSITIONAL LOGIC • CONNECTIVES: 4 TYPES. • 3 LINK TWO PROPOSITIONS AND 1 NEGATES PROPOSITIONS • Conjunction p & q • Disjunction p v q • Negation ~p • Conditional p q

5. SYMBOLIC LOGIC/PROPOSITIONAL LOGIC • Variable or terms • P and Q • Any term would do • Term represents a claim or statement • Simple and complex statements • Truth value

6. THE CONJUNCTION • 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

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

8. TRUTH TABLE OF CONJUNCTIONS

9. DISJUNCTION • p v q • God is dead or cellphones cause cancer • Truth value: if either disjunct is true, then whole claim is true. • If neither is true, statement is false.

10. DISJUNCTION • TRUTH TABLE:

11. NEGATION • SIGNIFIES NEGATING OR DENYING THE PROPOSITION, WHETHER COMPONENT OR COMPOUND • VARIETIES OF NEGATING: • A. IT’S NOT THE CASE THAT THE PRICE OF EGGS IN CHINA IS STEEP. • B. IT’S FALSE THAT THE PRICE OF EGGS IN CHINA IS STEEP • THE PRICE OF EGGS IN CHINA IS NOT STEEP.

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

13. 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.

14. DISJUNCTION • TRUTH TABLE:

15. 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

16. TRUTH TABLE FOR CONDITIONALS

17. NON-STANDARD FORMS • Page 220 in text • Six cases • Translation • # 6: ~p q

18. TRUTH TABLE

19. USING TRUTH TABLES • A method to test for validity • Important to know how to formulate

20. TRUTH TABLE METHOD-ARGUMENTS • 1. Allocate a column for each component statement. • 2. Allocate a column for each premise and one for the conclusion. • 3. If there are only 2 terms/components, you require 4 rows. • 4. If there are 3 components, you require 8 rows.

21. TRUTH TABLES cont. • 5. Write in possible truth values for each column. • 6. Rotation principle: • First column: TTFF • Second column:TFTF • With 8 rows: TTTTFFFF, TTFFTTFF,TFTFTFTF, for each row

22. TRUTH TABLES, cont. • 7. Negated terms require a separate column. • 8. Fill in the rest of the rows based on your knowledge of the connectives. • 9. Identify any rows with F in the conclusion column. • 10. If on these rows the premises together have a T, then argument is invalid. If not, then argument is valid.

23. COMPLEX ARGUMENTS • More components • i.e. P ~(q & r) • p • Hence: ~(q & r) • Notice: still a modus ponens.

24. METHOD FOR TRICKY ARGUMENTS • Identify the main connective. • It is not in the parentheses. • Work from inside and then outside of parenthesis. • Use columns for each component

25. SYMBOLIZING COMPLEX STATEMENTS • Pp.228-229 text. • Much hinges on where to place parenthesis. • Identify main connective! Crucial • i.e. It is not the case that Leo sings the blues and Fats sings the blues. • Negation is main connective • Hence: ~(L & F)

26. COMPLEX STATEMENTS, cont. • Eg. Leo does not sing the blues, and Fats does not sing the blues. • ~L & ~F • Eg. If the next Prime Minister is from Ontario, then neither the West nor Atlantic Canada will be happy. • Main connective? Conditional • Symbolized: p ~(W v A)

27. TRUTH TABLES AND COMPLEX ARGUMENTS • Please turn to pg. 230 in text. • Notice, need to have column 4 before we cannot negate column 4: column 6 • Options: negate r in row 4: Where you place columns is not important!

28. SHORT METHOD OF TRUTH TABLES • Please note typos, pg. 232 • Main purpose: to discover if there is a way to make the premises true when we assign a value of false to the conclusion. • We need to fill out each column but we eliminate those rows where the conclusion is true. • Turn to pg. 231 bottom