LOGIC and reasoning MATH 10

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

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

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

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

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

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

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

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

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.

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.

Levels of deductive logical analysis Syllogistic/Categorical: “all, some, no, not” -By Aristotle (384-322 BC) Sentential (propositional): “and, or, if-then, only if” Predicate: syllogistic+sentential terms, and universal and existential quantifiers

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

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>>>

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

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)

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) http://clipart-library.com/cartoon-dogs-pics.html

Example 3 All pigs are mammals. All mammals are animals. Therefore, all pigs are animals. This is factually correct and valid (hence, sound) https://kids.nationalgeographic.com/animals/pig/#pig-fence.jpg

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

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)

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

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.

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

LOGIC and reasoning MATH 10

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.

Negation –A A –A T F

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

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

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

Exclusive or A B A 𝑿𝑶𝑹 B T F

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

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

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

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

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

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

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.”

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.”

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

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

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

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

LOGIC and reasoning MATH 10

TAUTOLOGY true for all possible truth-value assignments

Self-contradiction false for all possible truth-value assignments

contingent neither self-contradictory nor tautological

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

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

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

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

http://www.naturalhealthpractice.com/Health_Detective_P650C340.cfm 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?

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.

LOGIC and reasoning MATH 10

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

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

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”)

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)

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)

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

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

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)

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)

References https://courses.umass.edu/phil110-gmh/text/c01.pdf https://www.iep.utm.edu/prop-log/ https://thebestschools.org/magazine/15-logical-fallacies-know/