DEDUCTIVE REASONING: PROPOSITIONAL LOGIC

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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

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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

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A COMPLEX ARGUMENT

• p v q
• p r
• q s
• Therefore, r v s
• r
• Therefore s
• valid argument

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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

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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

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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

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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.

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TRUTH TABLE OF CONJUNCTIONS

p q p q
T T T
F T F
T F F
F F F
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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.

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DISJUNCTION

• TRUTH TABLE

p q p v q
T T T
T F T
F T T
F F F
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NEGATION

• SIGNIFIES NEGATING OR DENYING THE PROPOSITION,
  WHETHER COMPONENT OR COMPOUND
• VARIETIES OF NEGATING
• A. ITS NOT THE CASE THAT THE PRICE OF EGGS IN
  CHINA IS STEEP.
• B. ITS FALSE THAT THE PRICE OF EGGS IN CHINA IS
  STEEP
• THE PRICE OF EGGS IN CHINA IS NOT STEEP.

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TRUTH TABLE FOR NEGATION

• P.337
• Opposite truth value

p p
T F
F T
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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.

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DISJUNCTION

• TRUTH TABLE

p q p v q
T T T
T F T
F T T
F F F
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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

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TRUTH TABLE FOR CONDITIONALS
p q p q
T T T
T F F
F T T
F F T
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NON-STANDARD FORMS

• Page 220 in text
• Six cases
• Translation
• 6 p q

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TRUTH TABLE
p q p p q
T T F T
T F F T
F T T T
F F T F
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USING TRUTH TABLES

• A method to test for validity
• Important to know how to formulate

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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.

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TRUTH TABLES cont.

• 5. Write in possible truth values for each
  column.
• 6. Rotation principle
• First column TTFF
• Second columnTFTF
• With 8 rows TTTTFFFF, TTFFTTFF,TFTFTFTF, for
  each row

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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.

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COMPLEX ARGUMENTS

• More components
• i.e. P (q r)
• p
• Hence (q r)
• Notice still a modus ponens.

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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

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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)

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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)

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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!

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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