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: premisesconclusion is called an argument STATEMENTS IN AN

ARGUMENT should be a declarative sentence, capable of being TRUE or FALSE (even if we dont 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 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, ifthen, 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. <<>> 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. http://clipart-library.com/cartoon-dogs-pics.html This is factually correct but not

valid (hence, not sound) EXAMPLE 3 All pigs are mammals. All mammals are animals. Therefore, all pigs are animals. https://kids.nationalgeographic.com/animals/pig/#pig-fenc

This is factually correct and valid (hence, sound) 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 (welldefined)

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 T F

A F T and is commutative and associative CONJUNCTION A and B

A T T F F B T F

T F AB T F F F

or is commutative and associative DISJUNCTION A or B A T (Inclusive)

T F F B T F T F

AB T T 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 T T F

F B T F T F AB

F T 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 T T F F B AB

T T INVALID Argument F F T T F T

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 T T F F B T

F T F AB T F F T

TRY THESE (A B) C A ((BC) ( AC)) TRY THESE If it rains then the pavement is wet. The

pavement isnt wet because it didnt rain. INVALID If it rains then the pavement is wet. The pavement isnt wet; hence, it didnt rain. 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.

(AB) B) A 1.B. CONTRAPOSITION (AB) (B A) In fact, (B A) (AB) (AB) (B A)

1.B. CONTRAPOSITION (AB) (B A) Remember, (AB) (A B) x 1.B. CONTRAPOSITION

(AB) (B A) Also, (AB) (BA) 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 tautology. Statement 2 is a 2. DISJUNCTIVE SYLLOGISM A B A

((AB) A) B Therefore, B. (AB) (AB) 3. HYPOTHETICAL SYLLOGISM

A B B C Therefore, A C. 4. DE MORGANS 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 its 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 /