DEDUCTIVE REASONING:
PROPOSITIONAL LOGIC
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Purposes:
To analyze complex claims and deductive argument
forms
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To determine what arguments are valid or not
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To learn further how deduction works
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Logical relationships among statements
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Symbolic Logic: Using symbols instead of words
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SYMBOLIC LOGIC
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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 vq
p
r
q
s
Therefore, r v s
~r
Therefore s
valid argument
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SYMBOLIC LOGIC/PROPOSITIONAL
LOGIC
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CONNECTIVES: 4 TYPES.
3 LINK TWO PROPOSITIONS AND 1
NEGATES PROPOSITIONS
•Conjunction
•Disjunction
•Negation
•Conditional
p&q
pvq
~p
p
q
SYMBOLIC LOGIC/PROPOSITIONAL
LOGIC
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Variable or terms
P and Q
Any term would do
Term represents a claim or statement
Simple and complex statements
Truth value
THE CONJUNCTION
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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.
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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
p
q
p & q
T
T
T
F
T
F
T
F
F
F
F
F
DISJUNCTION
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pvq
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
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TRUTH TABLE:
p
q
pvq
T
T
T
T
F
T
F
T
T
F
F
F
NEGATION
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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
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P.337
Opposite truth
value
p
~p
T
F
F
T
DISJUNCTION
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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
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TRUTH TABLE:
p
q
pvq
T
T
T
T
F
T
F
T
T
F
F
F
CONDITIONAL
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“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
p
q
p
T
T
T
T
F
F
F
T
T
F
F
T
q
NON-STANDARD FORMS
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Page 220 in text
Six cases
Translation
# 6: ~p
q
TRUTH TABLE
p
q
~p
~p
T
T
F
T
T
F
F
T
F
T
T
T
F
F
T
F
q
USING TRUTH TABLES
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A method to test for validity
Important to know how to formulate
TRUTH TABLE METHOD-ARGUMENTS
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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.
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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.
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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
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More components
i.e.
P ~(q & r)
p
Hence: ~(q & r)
Notice: still a modus ponens.
METHOD FOR TRICKY ARGUMENTS
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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
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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.
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Eg. Leo does not sing the blues, and Fats does
not sing the blues.
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~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
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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
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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