1. Logic and Reasoning

2. 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
Statement (or Proposition) = ประพจน์
Reasoning = การให้เหตุผล
Premise = เหตุ (or หลักฐาน)

3. Logic and Reasoning
Premise
Premise
(something assumed
to be true)
If you study hard, you will get A.
You study hard.
Reasoning
Reasoning
You will get A
Conclusion
Conclusion
(something derived
from the premises)
“False conclusion may comes from invalid reasoning or false premises”.
Conclusion/Premise: True/False (T/F)
Reasoning: Valid/Invalid (V/I)
Axiom : Relativity ( The speed of light is constant and not depending with the speed of observer)
Euclid’s Element ( Things which are equal to the same thing are also equal to one another)
Only all true premises and valid reasoning canguarantee true conclusion.
In math term,
Premise is called Axiom,
Conclusion is called Theorem, Lemma,
Reasoning is called Proof.
In experimental science,
Empirical scientists tell us whether statements are true.
Logicians tell us whether reasoning is valid.
Premise
Statement: True/False
Conclusion
Logically draw conclusions / Construct a valid reasoning.
Spot valid and invalid reasoning.
Statement VS Reasoning
Statement – True or False
Reasoning – Valid or Invalid
Premises: What we assume to be true / given.
Conclusion: What we derive from the premises.

4. Truth VS Validity They are not the same.
Reasoning
Premises
Conclusion
They are not the same.
Truth for statements.
Validity for argument/reasoning.
Premises: Dogs have eight legs. [If x is a dog, then x has eight legs.]
Spooky is a dog.
Conclusion: Spooky has eight legs.
valid
The argument is valid.
However, the conclusion is false.
For further clarification, see lecture note.

5. Completeness and Soundness Theorems
An argument is valid if and only if it is derivable.
Validity: Truth of premise guarantees truth of conclusion
Derivability: kinda like we derive fluid,
according to some rules of inference

6. Note Valid reasoning does not guarantee a true conclusion.
Premises
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.
No valid argument can have true premise and false conclusion.

7. Some Important Equivalent … from checking the truth table …
1. Double Negation
2. Commutative Law
3. Associative Law
4. Distributive Law 5.
6. Contra-positive 7. 8.
9. De Morgan’s Law
10. 11
11. 11

8. Are these arguments/reasoning valid or invalid?

9. Argument 1: Are these arguments/reasoning valid or invalid?
Premises: If it rains, then the garden is wet.
The garden is wet.
invalid
Conclusion: It rains.
Activity: Class Discussion
Ex)
Premises: If x = 2p, then sin x = 0.
sin x = 0.
Conclusion: Therefore, x = 2p.
Invalid
e.g., x = p
Showing one counter-example is enough for confirming invalid reasoning.

10. Argument 2: Are these arguments/reasoning valid or invalid?
Premises: If it rains, the garden is wet.
It rains.
valid (next page)
Conclusion: The garden is wet.
Activity: Class Discussion
Ex)
Premises: If x = 2p, then sin x = 0.
x = 2p.
Conclusion: Therefore, sin x = 0.
Valid?
Showing one true examples is not enough for confirming invalid reasoning.
You need to show that all possible cases are true.

11. How to investigate validity of the reasoning (argument)
Truth Table
No valid argument can has
true premise and false conclusion.
Logic Derivation T T T T T T F F F T F T T F T F F F T T
Try to find Counter-Example,
then show the Contradiction p
Contradiction T T F x q T T T

12. Proof of Valid Reasoning by Contradiction Method
No valid argument can have true premise and false conclusion.
at least one case that
is invalid
[Using Contra-positive Equivalence]
no one case that
is valid
Proof by Contradiction Method
Assume that there is one case that
Then show that this is not possible – there is no such case - by (finding) contradiction.

13. 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.
In this case, we write
A reasoning that is not valid is said to be invalid.

14. Argument 3: Are these arguments/reasoning valid or invalid?
Premises: If it rains, the garden is wet.
It does not rain.
invalid
Conclusion: The garden is not wet.
Activity: Class Discussion
Premises: If x = 2p, then sin x = 0.
x p.
Conclusion: Therefore, sin x
Invalid
e.g., x = p, sin x = 0

15. Argument 4: Are these arguments/reasoning valid or invalid?
Premises: If it rains, the garden is wet.
The garden is not wet.
valid (next page)
Conclusion: It does not rain.
Activity: Class Discussion
Premises: If x = 2p, then sin x = 0.
sin x
Conclusion: Therefore, x p.
Valid?

16. How to investigate validity of the reasoning (argument)
Try to find Counter-Example, then show the Contradiction
Contradiction T F p T q x F T T F
Truth Table
No valid argument can has
true premise and false conclusion.
Logic Derivation T F T T F T T F F T F T F T F T F T F F T T T T

17. How to investigate validity of the reasoning (argument)
Contra-positive Equivalent
Argument2
(already proofed)
valid
valid?
valid

18. Rule of Inference Logical Fallacies Modus Ponendo Ponens
Modus Tollendo Tollens
valid
valid
Logical Fallacies
Fallacy of The Converse
Fallacy of The Converse

19. Some Important Implications
1. Modus Ponens
2. Modus Tollens
3. Simplification
4. Addition
5. Modus Tollendo Ponens
6. Hypothetical Syllogism
7. Biconditional-Conditional
8. Conditional- Biconditional
9. Constructive dilemma

20. 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

21. A = She likes A.
B = She likes B.
U = She likes U.
H = U are happy.
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: ?

22. She doesn’t like A. She likes B. She likes U. U are happy. ……… (1)
Premises are assumes
to be true.
……… (2)
……… (3)
She doesn’t like A.
From (1) with Simplification
……… (4)
……… (5)
She likes B.
From (2) and (5)
with Modus Tollendo Ponens
She likes U.
……… (6)
From (3) and (6)
with Modus Ponens
……… (7)
U are happy.

23. 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

24. Activity: Class Discussion
R = It rains.
H = It is humid.
B = I wear blue shirt.
C = It is cold.
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: ?
Activity: Class Discussion

25. It rains I wear blue shirt. It is not cold. ……… (1) ……… (2) ……… (3)
Premises are assumed
to be true.
It rains
……… (3)
From (3) with addition
……… (4)
I wear blue shirt.
From (1),(4) with Modus Ponens
……… (5)
From (2),(5) with Modus Tollens
……… (6)
It is not cold.
However, we can’t determine the truth value of H.
(we don’t know whether it is humid or not.

26. 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

27. Activity: Class Discussion
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: ?
Activity: Class Discussion

28. I goes to a movie. ……… (1) ……… (2) ……… (3) ……… (4) ……… (5) ……… (6)
Premises are assumed
to be true.
……… (3)
……… (4)
From (3) with Contrapositive
From (2),(4) with
Hypothetical Syllogism
……… (5)
By Tertium non datur
(Principle of Excluded Middle)
……… (6)
From (1),(5),(6) with
Constructive dilemma
……… (7)
I goes to a movie.

29. If-then Example

30. Necessary and Sufficient Condition
Example) The one who graduates, must pass this course.
P: Graduator, Q: the one who passes this course. x
What is the relation
between P and Q? p x q
P is necessary or sufficient condition of Q ?
P is sufficient condition of Q.
Q is necessary or sufficient condition of P ?
Q is necessary condition of P. 35 35

31. Example of statement usually used in conversation
Example) P only if Q
“จะเป็น p ได้ ต้องเป็น q (เท่านั้น)”
แต่การที่เป็น q ไม่ได้แปลว่าจะเป็น p โดยอัตโนมัติ p q
การที่ไม่ใช่ q นั้น แสดงว่าไม่ใช่ p
p only if q
- จะเป็น p ได้ ต้องเป็น q (เท่านั้น) [แต่การที่เป็น q ไม่ได้แปลว่าจะเป็น p automatic]
ประโยคนี้ จริงแล้ว ความหมายน่าจะตรงและสอดคล้องกับ sense ของ contrapositive มากกว่า คือ
If not q, then not p.
ถ้าไม่ q แล้ว จะไม่ p.
Alernative คือ
Only if q, that p (is possible). ต้องเป็น q เท่านั้น จึงมีโอกาสจะเป็น p
Only under the condition of q, that p <---- Here is where "only" is better understood.
[Only if q] ภายใต้เงื่อนไขของ q เท่านั้น ที่จะ p
What is necessary / sufficient condition of what ?
P is sufficient condition of Q.
Q is necessary condition of P. 36

32. Example of statement usually used in conversation
Example) P if and only if Q q
P if Q p
P only if Q q
p only if q
- จะเป็น p ได้ ต้องเป็น q (เท่านั้น) [แต่การที่เป็น q ไม่ได้แปลว่าจะเป็น p automatic]
ประโยคนี้ จริงแล้ว ความหมายน่าจะตรงและสอดคล้องกับ sense ของ contrapositive มากกว่า คือ
If not q, then not p.
ถ้าไม่ q แล้ว จะไม่ p.
Alernative คือ
Only if q, that p (is possible). ต้องเป็น q เท่านั้น จึงมีโอกาสจะเป็น p
Only under the condition of q, that p <---- Here is where "only" is better understood.
[Only if q] ภายใต้เงื่อนไขของ q เท่านั้น ที่จะ p p 37

33. Some Logic: Necessary and Sufficient Conditions (Deductive Reasoning)
Implication (Conditional Statement): p  q
Note: There is also “inductive reasoning.” p q p q p q
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.
p is a sufficient condition for 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.
q whenever p 38 38

34. Conditional Statements: If p, then q: PV = mRT
(If/Under-the-condition-of/) For a fixed gas and mass of the gas,
(if/under-the-condition-of/) and for a fixed temperature:
If volume increases, then pressure decreases.
If pressure does not decrease, then volume does not increase.
Pressure decreases or volume does not increase.
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.

35. Real Life Example Objective: Design Exp: ….. Doing Exp: ….. Result:
by experiment
varying P
Design Exp: …..
Doing Exp: …..
Result:
By the way,
the experimental result
should be ….?
Basic Knowledge
Of Mech Material
Premises
lab
conclusion
Predicted Result

36. Discussion: Cause of Error? inaccurate E inaccurate I inaccurate L
theory
theory
Discussion:
lab
Cause of Error?
lab
inaccurate E
Not likely possible
maybe possible
inaccurate I
Not likely possible
maybe possible
maybe possible
inaccurate L
Not likely possible

37. Rule of Inference Modus Tollendo Tollens Modus Ponendo Ponens
valid
valid
- Investigate the validity of argument (reasoning).
- Make a theoretical predictions. // Logically draw conclusions.
- Hypothesis Testing 42 42