1. LOGIC and reasoning
MATH 10

2. LOGIC “science” of reasoning non-empirical
the basic task is to distinguish correct from incorrect reasoning

3. Reasoning
mental activity of inferring (drawing conclusions from premises)
Note: premises→conclusion is called an argument

4. STATEMENTS IN AN ARGUMENT
should be a declarative sentence, capable of being TRUE or FALSE (even if we don’t know its truth value)
Note: an interrogative, imperative, or exclamatory sentence cannot be a statement in an argument

5. STATEMENTS IN AN ARGUMENT
We will focus on classical logic where a statement can have only one value: True or False. Non-classical logic: e.g., Fuzzy logic

6. The concern of Logic
The concern of logic is the FORM (premises support/justify the conclusion)
It does NOT focus on the CONTENT

7. The concern of Logic
The psychology and neurophysiology of the mental activity is also NOT a concern of logic.

8. Branches of logic Inductive logic Deductive logic (our focus)
investigates the process of drawing probable though fallible conclusions from premises
Deductive logic (our focus)

9. Deductive logic
if the premises were true, then the conclusion would certainly also be true

10. Deductive logic
Note: If an argument is judged to be deductively correct, then it is also judged to be inductively correct as well. The converse is not true.

11. Deductive logic
Note: Although an argument may be judged to be deductively incorrect, it may still be reasonable, that is, it may still be inductively correct.

12. Levels of deductive logical analysis
Syllogistic/Categorical: “all, some, no, not”
-By Aristotle ( BC)
Sentential (propositional): “and, or, if-then, only if”
Predicate: syllogistic+sentential terms, and universal and existential quantifiers

13. FORM VS Content Focus is FORM (a concern of logic):
An argument is valid if and only if its conclusion follows from its premises.

14. FORM VS Content Focus is CONTENT (not a concern of logic):
An argument is factually correct if and only if all of its premises are true.
<<<The truth or falsity of statements is the subject matter of the sciences>>>

15. FORM VS Content
An argument is sound if and only if it is both factually correct and valid.

16. Example 1 All UPLB students are males. All males have long hair.
Therefore, all UPLB students have long hair.
This is valid but not factually correct (hence, not sound)

17. Example 2 All dogs are animals. All mammals are animals.
Therefore, all dogs are mammals.
This is factually correct but not valid (hence, not sound)

18. Example 3 All pigs are mammals. All mammals are animals.
Therefore, all pigs are animals.
This is factually correct and valid (hence, sound)

19. Example 4 Some circles are big. No big stuffs are small.
Therefore, some circles are not small.
This is valid (is this sound?)

20. MATHEMATICAL REASONING
Well-defined (“precise”) statements are necessary.
e.g.,
collection of cute dogs (not well-defined)
collection of numbers larger than 2 (well-defined)

21. Example 5 Some circles are big. No big stuffs are small.
Therefore, some circles are small.
This is not valid

22. LOGICAL CONSISTENCY
Suppose, you want to write a fiction story involving two groups: the Jologs and the Jejes. In Chapter 1, you declared that “all Jologs are Pachoochies” and “no Jejes are Pachoochies”. These tell the the readers (without explicitly writing) to conclude that Jologs are not the same as Jejes. In Chapter 5, you introduced a character named Hypebeast, and declared that he is both a Jolog and a Jeje. The readers would accuse you of logical inconsistency in the story.

23. Importance of TRAINING
Persons, even “intelligent” ones, without training in logic might commit logical errors.

24. LOGIC and reasoning
MATH 10

25. FUNDAMENTAL PRINCIPLE OF LOGIC
If an argument is valid, then every argument with the same form is also valid.
If an argument is invalid, then every argument with the same form is also invalid.

26. Negation –A A –A T F

27. “and” is commutative and associative
CONJUNCTION
A and B A B
A∧B T F

28. Disjunction A or B (Inclusive) A B A∨B T F
“or” is commutative and associative
Disjunction
A or B (Inclusive) A B
A∨B T F

29. Example
Joe was able to attend his classes on time. Either he slept early last night or he woke up early today.

30. Exclusive or A B
A 𝑿𝑶𝑹 B T F

31. Example
Joe was able to attend his classes on time. Either he rode his bicycle, or he rode a jeep.

32. Material implication (conditional)
A → B
A implies B
If A then B
B because A A B
A→B T F
INVALID Argument

33. Example If it rains then the pavement is wet. A: It rains
B: Pavement is wet

34. Material Equivalence (Biconditional)
A ↔ B
A ≡ B
A if and only if B
A is equivalent to B
A implies B and B implies A A B
A↔B T F

35. TRY THESE
(A ∧ B) → C
A ↔ ((B→C) ∨ –( A→C))

36. TRY THESE
If it rains then the pavement is wet. The pavement isn’t wet because it didn’t rain.
If it rains then the pavement is wet. The pavement isn’t wet; hence, it didn’t rain.
INVALID
VALID

37. TRY THESE
Assume this is true: “If OFW remittances increase, then the economy grows.”
Is the following statement true?
“The economy is slowing down because OFW remittances decline.”

38. TRY THESE
Assume this is true: “If OFW remittances increase, then the economy grows.”
Is the following statement true?
“The economy is slowing down; hence, OFW remittances decline.”

39. 1.A. ModUS TOLLENS (“denying mode”)
A → B –B
Therefore, –A.
(A→B) ∧ –B) → –A

40. 1.b. CONTRAPOSITION (A→B) → (–B → –A) In fact, (–B → –A) → (A→B)

41. 1.B. CONTRAPOSITION
(A→B) ≡ (–B → –A)
Remember,
(A→B) ≡ (–A → –B) x

42. 1.B. CONTRAPOSITION
(A→B) ≡ (–B → –A)
Also,
(A→B) ≡ (B→A) x

43. LOGIC and reasoning
MATH 10

44. TAUTOLOGY
true for all possible truth-value assignments

45. Self-contradiction
false for all possible truth-value assignments

46. contingent
neither self-contradictory nor tautological

47. Logically equivalent statements
Statement 1 and Statement 2 have the same truth values
Statement 1 ≡ Statement 2 is a tautology.

48. 2. Disjunctive syllogism
A ∨ B –A
Therefore, B.
((A∨B) ∧ –A) → B
(A∨B) ≡ (–A→B)

49. 3. Hypothetical syllogism
A → B
B → C
Therefore, A → C.

50. 4. DE Morgan’s law
–(A ∧ B) ≡ (–A ∨ – B)
–(A ∨ B) ≡ (–A ∧ – B)

51.
TRY THIS
Either cat fur or dog fur was found at the scene of the crime. If dog fur was found at the scene of the crime, officer Rock had an allergy attack. If cat fur was found at the scene of the crime, then Pissy the Cat must have entered the scene of the crime. If Pissy the Cat entered the scene then Thirdy the owner of Pissy is responsible for the crime. But officer Rock didn't have an allergy attack. What is the result of the investigation?

52. Principle of counterexamples
If someone claimed “all swans are white”, you could refute that person by finding a swan that isn't white.
However, if you could not find a non-white swan, you could not thereby say that the claim was proved, only that it was not disproven yet.

53. LOGIC and reasoning
MATH 10

54. Fallacies
non sequitur: conclusion that doesn't follow logically from the previous statement

55. Some Informal fallacies
Ad hominem (personal attacks)
Argumentum ad verecundiam (appeal to authority)
Argumentum ad misericordiam (appeal to pity)

56. Some Informal fallacies
Argumentum ad ignorantiam (appeal to ignorance; e.g., “no one has ever been able to prove that extra-terrestrials do not exist, so they must be real”)

57. Some Informal fallacies
Straw man (intentionally misrepresented proposition that is set up because it is easier to defeat than an opponent's real argument)
Ignoratio elenchi (Red herring; distraction that sounds relevant but off-topic)

58. Some Informal fallacies
False Dilemma/False Dichotomy (offering limited options even though there are more)
Slippery Slope (moving from a seemingly benign starting point and working through a number of small steps to an improbable extreme)

59. Some Informal fallacies
Petitio principii (circular argument/begging the question; e.g., “the judge is just because judges cannot be unjust”)
Hasty generalization (general statements without sufficient evidence to support them) - stereotyping, exaggeration

60. Some Informal fallacies
Tu quoque (appeal to hypocrisy; e.g., “Jane committed adultery. Jill committed adultery. Lots of us did”)
Ad populum (bandwagon)

61. Some Informal fallacies
Non causa pro causa (e.g., “since your parents named you ‘Harvest,’ they must be farmers”)
Post hoc ergo propter hoc (because this came first then this caused that; e.g., superstitions)

62. Some Informal fallacies
Cum hoc ergo propter hoc (correlation/coincidence)
Equivocation/ambiguity (word, phrase, or sentence is used deliberately to confuse, deceive, or mislead by sounding like it’s saying one thing but actually saying something else)

63. References https://courses.umass.edu/phil110-gmh/text/c01.pdf