Logic and Reasoning
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Logic and Reasoning. Objective. Spot valid and invalid reasoning. Be able to construct a valid reasoning . Make appropriate predictions based on acceptable premises. Logically draw conclusions from experimental result. Statement VS Reasoning Statement – True or False - PowerPoint PPT Presentation
Transcript of Logic and Reasoning
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Logic and Reasoning
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Objective
Spot valid and invalid reasoning.
Be able to construct a valid reasoning.
Make appropriate predictions based on acceptable premises.
Logically draw conclusions from experimental result.
Statement VS Reasoning
Statement – True or False
Reasoning – Valid or Invalid
Logic and Reasoning
Premise
Conclusion
Reasoning
In math term, Premise is called Axiom, Conclusion is called Theorem, Lemma, Reasoning is called Proof.
(something assumed to be true)
(something derived from the premises)
You will get A
If you study hard, you will get A.
You study hard.
Conclusion/Premise: True/False (T/F)
Reasoning: Valid/Invalid (V/I)
Premise
Conclusion
Reasoning
In experimental science, Empirical scientists tell us whether statements are true. Logicians tell us whether reasoning is valid.
“False conclusion may comes from invalid reasoning or false premises”.
Only all true premises and valid reasoning canguarantee true conclusion.
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Truth VS Validity
They are not the same.
For further clarification, see lecture note.
Premises: Dogs have eight legs. [If x is a dog, then x has eight legs.]
Spooky is a dog.
Conclusion: Spooky has eight legs.
Truth for statements.Validity for argument/reasoning.
The argument is valid.
However, the conclusion is false.
p q
p
q
valid
Premises Conclusion Reasoning
Premises Conclusion ReasoningNote
No valid argument can have true premise and false conclusion.
Valid reasoning does not guarantee a true conclusion.
Invalid reasoning does not guarantee a false conclusion.
A false conclusion does not guarantee invalidity.
True premises and a true conclusion together do not guarantee validity.
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pp )(
pqqp
pqqp
)()( rqprqp
)()( rqprqp
)()()( rpqprqp
)()()( rpqprqp
qpqp
pqqp
pqqp
)()( pqqpqp
( )p q p q
( )p q p q
qpqp )(
)()()( qpqpqp
1. Double Negation
2. Commutative Law
3. Associative Law
4. Distributive Law
6. Contra-positive
9. De Morgan’s Law
Some Important Equivalent … from checking the truth table …
5.
7.
8.
10.
11.
( )
( )
p q
p q
p q
q p
q p
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p q
q
p
valid?
p q
p
q
valid?
p q
p
q
valid?
p q
q
p
valid?
Are these arguments/reasoning valid or invalid?
Argument 1 Argument 2
Argument 3 Argument 4
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Argument 1: Are these arguments/reasoning valid or invalid?
Premises: If it rains, then the garden is wet.
The garden is wet.
Activity: Class Discussion
Invalid e.g., x =
Premises: If x = 2, then sin x = 0.
sin x = 0.
Conclusion: Therefore, x = 2.
Conclusion: It rains.
p q
q
p
invalid
Showing one counter-example is enough for confirming invalid reasoning.
Ex)
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Argument 2: Are these arguments/reasoning valid or invalid?
Premises: If it rains, the garden is wet.
It rains.
Activity: Class Discussion
Conclusion: The garden is wet.
Premises: If x = 2, then sin x = 0.
x = 2
Conclusion: Therefore, sin x = 0.
Valid?
p q
p
q
valid (next page)
Showing one true examples is not enough for confirming invalid reasoning.
You need to show that all possible cases are true.
Ex)
p q
p
q
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( )
p q
p q
Valid?How to investigate validity of the reasoning (argument)
Logic Derivation
p q p q p q p p q p q
T
T
F
F
T
F
T
F
T
F
T
T
T
F
F
F
T
T
T
T
is Tautology?p q p q
p q p q Truth Table
Try to find Counter-Example, then show the Contradiction
F
T
T
T
ContradictionT
T
p q p q
( ) ( )
( )
( )
( )
( )
p q p q
p p q p q
F q p q
q p q
q p q
q p q
q q p
T p
T
q
p
x
No valid argument can has true premise and false conclusion.
Proof of Valid Reasoning by Contradiction MethodP Q
No valid argument can have true premise and false conclusion.
QP is invalid FT
QP at least one case that
[Using Contra-positive Equivalence]
FT
QP no one case that QP is valid
FT
QP Assume that there is one case that
Then show that this is not possible – there is no such case -
by (finding) contradiction.
Proof by Contradiction Method
In this case, we write
A reasoning that is not valid is said to be invalid.
Valid Reasoning (Argument)
A reasoning (an argument)
is said to be valid if and only if, by virtue of logic,
the truth of the premise P guarantees the truth of the conclusion Q,
if P is true, Q is necessarily/always true,
is a tautology.
QP
QP
QP
QP
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Argument 3: Are these arguments/reasoning valid or invalid?
Premises: If it rains, the garden is wet.
It does not rain.
Activity: Class Discussion
Invalid e.g., x = sin
x
Premises: If x = 2, then sin x = 0.
x 2
Conclusion: Therefore, sin x 0.
p q
p
q
invalidConclusion: The garden is not wet.
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Argument 4: Are these arguments/reasoning valid or invalid?
Premises: If it rains, the garden is wet.
The garden is not wet.
Activity: Class Discussion
Valid?
p q
q
p
valid (next page)
Premises: If x = 2, then sin x = 0.
sin x 0.
Conclusion: Therefore, x
2.
Conclusion: It does not rain.
Truth Table
( )
p q
p q
Logic Derivation
is Tautology?p q q p
p q q p
Try to find Counter-Example, then show the Contradiction
FT
T
T
ContradictionF
F
( ) ( )
( )
( )
( )
( )
( )
p q q p
p q q q p
p q F p
p q p
p q p
p q p
p p q
T q
T
T
p q q p q
p
x
How to investigate validity of the reasoning (argument)
p q
q
p
Valid?
p q p q p q q p q q
p
T
T
F
F
T
F
T
F
T
F
T
T
F
F
F
T
T
T
T
T
q
F
T
F
T
No valid argument can has true premise and false conclusion.
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Argument2
(already proofed)
valid
q
p
qp
valid?
p q
q
p
p q q p
How to investigate validity of the reasoning (argument)
Contra-positive Equivalent
q p
valid
is
is
q
p
Rule of Inference
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Modus Ponendo Ponens
valid
q
p
qp Modus Tollendo Tollens
valid
p
q
qp
Logical Fallacies
Fallacy of The Converse
p
q
qp
q
p
qp
Fallacy of The Converse
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qpqp )(
pqqp )(
pqp
qqp
qpp
qpq
qpqp )(
pqqp )(
qpqp
pqqp
qppqqp )()(
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sqrpsrqp )()()(
1. Modus Ponens
2. Modus Tollens
3. Simplification
4. Addition
5. Modus Tollendo Ponens
7. Biconditional-Conditional
8. Conditional- Biconditional
6. Hypothetical Syllogism
9. Constructive dilemma
Some Important Implications
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Logically Draw Conclusions
Premises:She does not like A and she likes B.
She does not like B or she likes U.
If she likes U, then U are happy.
Conclusions: She likes who?
and Who are happy?
Activity: Class Discussion
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Premises:
She does not like A and she likes B.
She does not like B or she likes U.
If she likes U, then U are happy.
Conclusions: ?
BA
UB
HU
A = She likes A.
B = She likes B.
U = She likes U.
H = U are happy.
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A B
B U
H
U H
U
B
She likes U.
U are happy.
……… (1)
……… (2)
……… (3)
Premises are assumes to be true.
From (1) with Simplification A ……… (4)
……… (5)
……… (6)From (2) and (5) with Modus Tollendo Ponens
She doesn’t like A.
She likes B.
From (3) and (6) with Modus Ponens
……… (7)
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Logically Draw Conclusions
Premises:If it rains or it is humid, then I wear blue shirt.
If it is cold, then I do not wear blue shirt.
It rains.
Conclusions: What is the weather condition?
What color of the shirt I wear?
Activity: Class Discussion
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R = It rains.
H = It is humid.
B = I wear blue shirt.
C = It is cold.
BHR )(
BC R
Activity: Class Discussion
Premises:
If it rains or it is humid, then I wear blue shirt.
If it is cold, then I do not wear blue shirt.
It rains.
Conclusions: ?
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C
B
It is not cold.
……… (1)
……… (2)
……… (3)
Premises are assumedto be true.
From (3) with addition R H ……… (4)
……… (5)
……… (6)From (2),(5) with Modus Tollens
It rains
I wear blue shirt.
However, we can’t determine the truth value of H. (we don’t know whether it is humid or not.
BHR )(
BC R
From (1),(4) with Modus Ponens
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Logically Draw Conclusions
Premises: If I am bored, then I go to a movie.
If I am not bored, then I go to a library.
If I do not go to a movie, then I do not go to a
library.
Conclusions: Where do I go?
Activity: Class Discussion
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B = I am bored.
M = I go to a movie.
L = I go to a library.
Premises:
If I am bored, then I go to a movie.
If I am not bored, then I go to a library.
If I do not go to a movie, then I do not go to a library.
Conclusions: ?
MB LB
LM
Activity: Class Discussion
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B B
……… (1)
……… (2)
……… (3)
Premises are assumed to be true.
From (3) with Contrapositive L M ……… (4)
……… (5)
……… (6)
I goes to a movie.
From (2),(4) with Hypothetical Syllogism
MB LB
LM
B M
From (1),(5),(6) with Constructive dilemma M ……… (7)
By Tertium non datur (Principle of Excluded Middle)
Necessary and Sufficient Condition
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?
?
?
?
?
p q
p q
p q
q p
p q
q
p
x
Example) The one who graduates, must pass this course.
P: Graduate, Q: the one who passes this course.
What is the relation between P and Q?
p q
Q is necessary condition of P.
P is sufficient condition of Q.
P is necessary or sufficient condition of Q ?
q p x
Q is necessary or sufficient condition of P ?
Example of statement usually used in conversation
q
p
Example) P only if Q
p q
q p แต่�การที่�เป็�น q ไม่�ได้�แป็ลว่�าจะเป็�น p โด้ยอั�ต่โนม่�ต่�
การที่�ไม่�ใช่� q น��น แสด้งว่�าไม่�ใช่� pq p
Q is necessary condition of P.
P is sufficient condition of Q.
What is necessary / sufficient condition of what ?
“จะเป็�น p ได้� ต่�อังเป็�น q เท่�านั้��นั้”
Example of statement usually used in conversation
p
q
Example) P if and only if Q
p q
q pP if Q
P only if Q
q
p p q
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Some Logic: Necessary and Sufficient Conditions (Deductive Reasoning) Implication (Conditional Statement): p q
Note: There is also “inductive reasoning.”
p q Equivalence (~q) (~p)
If p, then q. If not q, then not p.
q if p.
p only if q.
q is a necessary condition for p. If not q, then not p.
p is a sufficient condition for q. If p, then q.
Converse of p q: q p
Contra-positive of p q: ~ q ~ p
p q and its contra-positive ~ q ~ p are equivalent. That is:
If p q is true, ~q ~ p is also true.
If p q is false, ~ q ~ p is also false.
On the other hand, p q does not imply q p.
The truth of p q does not automatically guarantee the truth of q p.
p
q
p
qp
q
q whenever p
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Conditional Statements: If p, then q: PV = mRT
(If/Under-the-condition-of/) For a fixed gas and mass of the gas,
i dV P
d iP V
If volume increases, then pressure decreases.
If pressure does not decrease, then volume does not increase.
Pressure decreases or volume does not increase.
PV mRT
i dV P
For a fixed gas and mass of the gas, and for a fixed pressure:
If temperature decreases, then volume decreases.
If volume does not decreases, then temperature doest not decreases.
Temperature does not decreases, or volume decreases.
(if/under-the-condition-of/) and for a fixed temperature:
Real Life Example
Objective:
( ; ... ; ...)By f P by experiment
y
By
P
Design Exp: …..
Doing Exp: …..
Result:
lab
By the way, the experimental result should be ….?
Basic KnowledgeOf Mech Material
Predicted Result
Premises
conclusion
varying P
Prediction of Expected Result
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( ) ?By f P
y
1xM
2
P
V
1 1 1( ) where 02 2
P LM x x x
Boundary Conditions
( 0, 0), ( , 0),
( , 0)2
x y x L y
L dyx
dx
2 2
12 16
Px x Ly
EI
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1 2
1
2 6
P xy C x C
EI
3
4
11
2912LB x
PLy y
EI
Basic KnowledgeOf Mech Material
Predicted Result
If … some assumptions
…
1In the case of 02
Lx
2
2
1
d y M
dx EI
If … some assumptions …
should have a linear relationship.
and By P
P
By
theory
lab
P
By
theory
lab
inaccurate E
inaccurate I
inaccurate L
inaccurate Position C
Not likely possible
inaccurate load P
Cause of Error?
Not likely possible
Not likely possible
maybe possible
maybe possible
maybe possible
Not likely possible maybe possible
? used realE E
?used realI I
? used realL L
?AC > used realAC
maybe possible
constantreal usedP P
maybe possible
constantreal usedP P j
Err in support dish Err in mass of each dish
Discussion:
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2912B
PLy
EI
2
2
1
d y M
dx EI
2 2
12 16
Px x Ly
EI
Rule of Inference
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Modus Ponendo Ponens
valid
q
p
qp Modus Tollendo Tollens
valid
p
q
qp
p q q p
- Investigate the validity of argument (reasoning).
- Make a theoretical predictions. // Logically draw conclusions.
- Hypothesis Testing
