Thinking Critically
Copyright © 2011 Pearson Education, Inc.
1-B
Unit 1B
Definitions
„
Propositions and
Truth Values
„
„
Slide 1-3
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A proposition makes a claim (either an assertion
or a denial) that may be either true or false. It
must have the structure of a complete sentence.
A proposition
Any
iti h
has ttwo possible
ibl truth
t th values:
l
T = true or F = false.
A truth table is a table with a row for each
possible set of truth values for the propositions
being considered.
Slide 1-4
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1-B
Negation (Opposites)
Double Negation
The negation of a proposition p is another proposition
that makes the opposite claim of p.
p
T
F
not p
F
T
1-B
← If p is true (T), not p is false (F).
← If p is false (F), not p is true (T).
The double negation of a proposition p, not not p,
has the same truth value as p.
p
T
F
not p
F
T
not not p
T
F
Symbol: ~
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Slide 1-5
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Slide 1-6
1
1-B
Logical Connectors
1-B
And Statements (Conjunctions)
Propositions are often joined with logical
connectors—words such as and, or, and if…then.
Given two propositions p and q, the statement p and q
is called their conjunction. It is true only if p and q are
both true.
Example:
p = I won the game.
q = It was fun.
f
p
T
T
F
F
Logical Connector New Proposition
and
I won the game and it was fun.
or
I won the game or it was fun.
if…then
If I won the game, then it was fun.
Slide 1-7
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q
T
F
T
F
p and q
T
F
F
F
Symbol:
Slide 1-8
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1-B
Or Statements (Disjunctions)
Or Statements (Disjunctions)
The word or can be interpreted in two distinct ways:
„
„
1-B
An inclusive or means “either or both.”
Given two propositions p and q, the statement p or q
is called their disjunction. It is true unless p and q
are both false.
p
T
T
F
F
An exclusive or means “one or the other
other, but not
both.”
In logic, assume or is inclusive unless told otherwise.
q
T
F
T
F
p or q
T
T
T
F
Symbol:
Slide 1-9
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Slide 1-10
Copyright © 2011 Pearson Education, Inc.
1-B
If… Then Statements (Conditionals)
Truth Table Practice
A statement of the form if p, then q is called a
conditional proposition (or implication). It is true
unless p is true and q is false.
„
„
p
q
if p, then q
T
T
F
F
T
F
T
F
T
F
T
T
p
q
T
T
F
F
T
F
T
F
~p
~q
Note: ~ signifies NEGATION
∧ signifies AND
∨ signifies OR
Proposition p is called the hypothesis.
Proposition q is called the conclusion.
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1-B
Slide 1-11
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p∧ q
~ p ∨ ~ q ~ ( p ∧ q)
Practice by writing the
truth values of each row
in the table above.
Slide 1-12
2
1-B
Truth Table Practice
p
q
~p
~q
T
T
F
F
T
F
T
F
F
F
1-B
Truth Table Practice
p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T
F
F
Practice by writing the
truth values of each row
in the table above.
Note: ~ signifies NEGATION
∧ signifies AND
∨ signifies OR
q
~p
~q
T
T
F
F
T
F
T
F
F
F
F
T
Note: ~ signifies NEGATION
∧ signifies AND
∨ signifies OR
Slide 1-13
Copyright © 2011 Pearson Education, Inc.
p
p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T
F
F
T
F
T
Practice by writing the
truth values of each row
in the table above.
Slide 1-14
Copyright © 2011 Pearson Education, Inc.
1-B
Truth Table Practice
p
q
~p
~q
T
T
F
F
T
F
T
F
F
F
T
F
T
F
1-B
Truth Table Practice
p ∧ q ~ p ∨ ~ q ~ ( p ∧ q)
T
F
F
F
T
T
F
T
T
Practice by writing the
truth values of each row
in the table above.
Note: ~ signifies NEGATION
∧ signifies AND
∨ signifies OR
q
~p
~q
p∧ q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
F
F
Note: ~ signifies NEGATION
∧ signifies AND
∨ signifies OR
Slide 1-15
Copyright © 2011 Pearson Education, Inc.
p
~ p ∨ ~ q ~ ( p ∧ q)
F
T
T
T
F
T
T
T
Practice by writing the
truth values of each row
in the table above.
Slide 1-16
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1-B
Alternative Phrasings of Conditionals
The following are common alternative ways of stating
if p, then q:
1-B
Variations on the Conditional
Conditional:
If p, then q
If it is raining, then I will bring an
umbrella to work.
„
p is sufficient for q
„
q is necessary for p
Converse:
If q, then p
If I bring an umbrella to work, then it
g
must be raining.
„
p will lead to q
„
q if p
Inverse:
If ~p, then ~q
If it is not raining, then I will not bring an
umbrella to work.
„
p implies q
„
q whenever p
Contrapositive:
If ~q, then ~p
If I do not bring an umbrella to work,
then it must not be raining.
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Slide 1-17
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Slide 1-18
3
1-B
Logical Equivalence
Two statements are logically equivalent if they
share the same truth values.
logically equivalent
p q ~p ~ q if p, then
th q if q, then
th p if ~p, then
th ~q if ~q, then
th
~p
(converse) (inverse) (contrapositive)
TT F F
T
T
T
T
TF F T
F
T
T
F
FT T F
T
F
F
T
FF T T
T
T
T
T
logically equivalent
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Slide 1-19
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