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INTRO LOGICINTRO LOGICDAY 04 DAY 04
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Schedule for Unit 1Schedule for Unit 1
Day 1 Intro
Day 2 Chapter 1
Day 3 Chapter 2
Day 4 Chapter 3
Day 5 Chapter 4
Day 6 Chapter 4
Day 7 Chapter 4
Day 8 EXAM #1
40% of Exam 1
60% of Exam 1
warm-up
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CHAPTER 3CHAPTER 3VALIDITY IN VALIDITY IN
SENTENTIAL LOGIC SENTENTIAL LOGIC
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Validity in GeneralValidity in General
an argument is valid
if and only if
it is impossible for
the conclusion to be false
while
the premises are true
an argument is invalid
if and only if
it is possible for
the conclusion to be false
while
the premises are true
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Validity in Sentential LogicValidity in Sentential Logic
an argument is valid
if and only if
there is no casecase in which
the premises are trueand
the conclusion is false
an argument is invalid
if and only if
there is at least one casecase in which
the premises are trueand
the conclusion is false
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What is a Case?What is a Case?
A casecase is a
possible combination of truth-values
assigned to the atomic formulas.
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case 4
case 3
case 2
case 1
SR
Example 1Example 1
If an argument form has 2 atomic sentences, then there are 4 casescases [4 = 22].
F
T
F
T T
T
F
F
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Example 2Example 2
If an argument form has 3 atomic sentences, then there are 8 casescases [8 = 23].
Q R S
case 1 T T T
case 2 T T F
case 3 T F T
case 4 T F F
case 5 F T T
case 6 F T F
case 7 F F T
case 8 F F F
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Example 1 Example 1 Modus TollensModus Tollens
/ not R; not Sif R then S
/ R; SR S
/ conclusion; premisepremise
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Truth-TableTruth-Table
R S ; S
F
T
F
T
S
F4
F3
T2
T1
/ RRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
T
T
F
T
T
F
T
F
T
T
F
F
VALID
NO
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Example 2 Example 2 Evil TwinEvil Twin of Modus Tollens of Modus Tollens
/ not S; not Rif R then S
/ S; RR S
/ conclusion; premisepremise
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CounterexampleCounterexample
/ I don't live in Mass
; I don't live in Boston
if I live in Boston then I live in Mass
FTT
/ not S; not Rif R then S
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Truth-TableTruth-Table
F
T
F
T
S
F4
F3
T2
T1
R S ; R / SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
T
YES
INVALID
T
F
T
T
T
F
F
T
F
T
F
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Example 3 Example 3 Modus PonensModus Ponens
/ S; Rif R then S
/ S; RR S
/ conclusion; premisepremise
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Truth-TableTruth-Table
R S ; R
F
T
F
T
S
F4
F3
T2
T1
/ SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
T
T
F
T
F
F
T
T
F
T
F
T
VALID
NO
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/ R; Sif R then S
/ R; SR S
/ conclusionpremise
Example 4 Example 4 Evil TwinEvil Twin of Modus Ponens of Modus Ponens
; premise
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CounterexampleCounterexample
/ I live in Boston
; I live in Mass
if I live in Boston then I live in Mass
FTT
/ R; Sif R then S
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Truth-TableTruth-Table
R S ; S
F
T
F
T
S
F4
F3
T2
T1
/ RRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
T
YES
INVALID
T
F
T
F
T
F
T
F
F
T
T
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Example 5 Example 5 Modus Tollendo PonensModus Tollendo Ponens(disjunctive syllogism)(disjunctive syllogism)
/ S; not RR or S
/ S; RR S
/ conclusion; premisepremise
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Truth-TableTruth-Table
R S ; R
F
T
F
T
S
F4
F3
T2
T1
/ SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
F
T
T
T
T
T
F
F
F
T
F
T
VALID
NO
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/ not S; RR or S
/ S; RR S
/ conclusion; premisepremise
Example 6 Example 6 Evil TwinEvil Twin of MTP of MTP
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Truth-TableTruth-Table
R S ; R
F
T
F
T
S
F4
F3
T2
T1
/ SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
F
YES
INVALID
T
T
T
F
F
T
T
T
F
T
F
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Example 7Example 7
/
/ not ( R and S)
( R & S )R
not R
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Truth-TableTruth-Table
/
F
T
F
T
S )
F
F
T
T
( R &
F
F
T
T
R
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid?
T
T
F
F
T
T
T
F
F
F
F
T
VALID
NO
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Example 8Example 8
/
/not ( R and S) not R
( R & S ) R
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Truth-TableTruth-Table
/
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or invalid? INVALID
YES
F
F
T
T
R
T
T
F
F
F
T
F
T
S )
F
F
T
T
( R &
T
T
T
F
F
F
F
T
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Logical EquivalenceLogical Equivalence
Two formulas are logically equivalentlogically equivalent
if and only if
they have the same truth-value
no matter what (in every case).
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Examples 7 and 8Examples 7 and 8
not R and not S=not ( R and S)
IT IS JUST LIKE MATH!
ZOMBIE ZOMBIE REASONINGREASONING
not R or not S=not ( R or S)
x2 + y2=( x + y)2
x + y=( x + y)
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Truth-Table for 7Truth-Table for 7
F
F
T
T
R & &
F
T
F
T
S
F
T
F
T
S ) //
F
F
T
T
( R
Are the two formulas logically equivalent?
F
F
F
T
T
T
T
F
T
T
F
F
T
F
T
F
T
F
F
F
Do the formulas match in truth value? NONO
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Truth-Table for 8Truth-Table for 8
F
F
T
T
R
F
T
F
T
S
F
T
F
T
S ) //
F
F
T
T
( R
Are the two formulas logically equivalent?
F
T
T
T
T
F
F
F
T
T
F
F
T
F
T
F
T
T
T
F
Do the formulas match in truth value? NONO
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Valid Equivalence – 1Valid Equivalence – 1
( R & S ) // R S
F T T T T T F F T
T T F F F T T T F
T F F T T F T F T
T F F F T F T T F
Are the two formulas logically equivalent? YES
Do the formulas match in truth value? YES
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Valid Equivalence – 2Valid Equivalence – 2
( R S ) // R & S
F T T T F T F F T
F T T F F T F T F
F F T T T F F F T
T F F F T F T T F
Do the formulas match in truth value?
Are the two formulas logically equivalent? YES
YES
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THE ENDTHE END