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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

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

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

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

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

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

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.

TRUTH TABLE OF CONJUNCTIONS pqp & q TTT FTF TFF FFF

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.

DISJUNCTION TRUTH TABLE: pqp v q TTT TFT FTT FFF

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.

TRUTH TABLE FOR NEGATION P.337 Opposite truth value p~p TF FT

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.

DISJUNCTION TRUTH TABLE: pqp v q TTT TFT FTT FFF

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

TRUTH TABLE FOR CONDITIONALS pqp q TTT TFF FTT FFT

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

TRUTH TABLE pq~p~p q TTFT TFFT FTTT FFTF

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

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.

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

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.

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

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

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)

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)

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!

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

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