# Categorical Reasoning

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Categorical Reasoning
GXEX1406 Thinking and Communication Skills Categorical Reasoning

INTRODUCTION Arguments generally are divided into two types: deductive and inductive. Deductive arguments? Valid or invalid? Inductive arguments? Strong or weak? GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Valid or invalid DR? If Flipper is a dolphin, then Flipper is a mammal. Flipper is a dolphin. So Flipper is a mammal. If Bigfoot is human, then Bigfoot has a heart. Bigfoot is not human. So Bigfoot doesn’t have a heart Bill: I guess some of the seniors were late to practice this morning. Diane: how do u know? Bill: Because the coach said that anyone late to practice this morning would’ve to do sprints and I just saw some of the seniors doing sprints. That’ll teach them. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Categorical Reasoning
Getting deeper into logic

CATEGORICAL REASONING
A form of deductive argument. Also called syllogism. Consists of two or more premises that precede the conclusion. Reasoning conclusions about the properties of individuals from more general premises that concern all the members of the relevant categories. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Example: All whales live in water (Premise) All fish live in water, too (Premise) All fish must be whales. (Conclusion) GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
If the conclusion of an argument is not guaranteed by the truth of the premises then the syllogism is not valid. An invalid argument involving categories is called a categorical fallacy. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Example: All German cars are reliable All BMW are German cars All BMW will be reliable In this case, the conclusion is valid because the premises are true and the conclusion follows necessarily from the premises. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Generally, the paradigms for a valid universal syllogism are as follows: All X are Y. Z is X. Therefore, Z is Y. Or No Z is Y. Therefore, no Z is X. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
From the general form of categorical reasoning in the previous slide, notice that: Each argument has 3 terms which are X, Y and Z. Each term occurs exactly twice in exactly two different claims. Example: All X are Y. Z is X. Z is Y. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
A syllogism is a two premise deductive argument which are: Major premise Minor premise The parts of a categorical syllogisms are labeled as follows: Major term – the term that occurs as a predicate term of the syllogism’s conclusion (cont) GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Minor term – the term that occurs as the subject term of the syllogism’s conclusion Middle term – the term that occurs in both of the premises but not at all in the conclusion. The most frequently used symbols for these three terms are: P –> major term S –> minor term M –> middle term GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
In a categorical syllogism, each of the premises states a relationship between the middle term and one of the others. M premise premise P S conclusion GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Categorical syllogism features: Every proposition is in standard categorical form There are three terms. The major premise is listed first, the minor is listed second. Each term is used in the same sense throughout the argument. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Example: All German cars are reliable (major premise) All BMW are German cars (minor premise) All BMW will be reliable (conclusion – derived from relationship between minor premise and major premise) GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
Explanation: Reliable is the major term since it is the predicate of the conclusion BMW is the minor term since it is the subject of the conclusion German cars is the middle term since it is the term which occurs in both of the premises but not in the conclusion All German cars are reliable (major premise) All BMW are German cars (minor premise) All BMW will be reliable (conclusion – derived from relationship between minor premise and major premise) GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

CATEGORICAL REASONING
When is a categorical arguments said to be sound and valid? When the conclusion follows necessarily from the premises When the form of the argument guarantees that the conclusion will follow, the argument is considered valid. When the premises can be assumed to be telling the truth, the argument is said to be true. If an argument is both true and valid, that argument is said to be sound. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Let’s think… What can you conclude from the following premises:
“All Communication and Thinking Skill classes are full. Marina found a class that wasn’t full.” A. Not all Communication and Thinking Skill classes are full. B. Marina didn’t find all the classes. C. Marina found a class that wasn’t Communication & Thinking Skill.. D. All Communication and Thinking Skill classes were not found. (cont) GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Exercises 2. Which of the following major premises forms a valid argument with the minor premise, “Malaysians invented the first written language," and the conclusion, “Malaysians created the modern world.“ A. The inventors of written language created the modern world. B. The Malaysian created the modern writing. C. Written language is the foundation of the modern world. D. The argument is invalid. (cont) GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Exercises 3. Consider the following chain argument: "All libraries are quiet. Only quiet places are restful. The places where I like to study are restful." What can you conclude from the minor premise, "My bedroom is a place where I like to study"? A. My bedroom is restful, quiet and a library. B. My bedroom is restful and quiet. C. Libraries are restful and my bedroom is quiet. D. Libraries and my bedroom are places I like to study. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Categorical reasoning
Argument that makes a claim about the relationship between two or more classes or categories of things. The argument has one of the following standard forms: All S are P No S are P Some S are P Some S are not P STANDARD FORM CATEGORICAL STATEMENTS GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Examples All fruits are apples No fruits are apples
Some fruits are apples Some fruits are not apples You may test the validity of these simple categorical arguments by the use of Venn diagrams. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Our everyday statements lack explicit categorical reasoning
Koalas are marsupials. All koalas are marsupials Men are IT savvy Some men are IT savvy Politicians are liars Some politicians are liars GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Variant of “All S are P” Every S is a P Whoever is an S is a P
Whatever is an S is a P If anything is an S, then it is a P If something is not a P, then it is not an S Any S is a P Each S is a P S are all P S are always P GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Variant of “No S are P” No P are S S are not P
Nothing that is an S is a P Not a single S is P All S are non-P If anything is an S, then it is not a P None of the S are P GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Variant of “Some S are P”
Some P are S A few S are P There are S that are P Several S are P Many S are P Most S are P Nearly all S are P GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Variant of “Some S are not P”
Not all S are P S are not always P Some S are non-P Most S are not P Nearly all S are not P Several S are not P A few S are not P There are S that are not P GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Is this argument valid? No sharks are pets, since no barracuda are pets, and no sharks are barracuda. No one who is a Nobel Prize winner is a rock star. A number of astrophysicists are Nobel Prize winner. Therefore a number of astrophysicists are not rock stars GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Is this argument valid? Anyone who eats pizza every night is at risk for heart disease. Some people who are at risk for heart disease are cab drivers. So, some cab drivers are people who eat pizza. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

General advice: Always try to restate the speaker’s or writer’s intended meaning as accurately as possible. E.g. There are messages that are not spell-checked. There are inter-office memos that are messages. Therefore there are interoffice memos that are not spell-checked. Some messages are not spell-checked. Some messages are inter-office memos. Therefore some interoffice memos are not spell-checked. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Exercise 1: Translate the following into standard categorical form
Maples are trees Roses are red Each insect is an animal All that glitters is not gold If anything is an igloo, then it is made of ice Success has ruined many a man Only the educated are free The questions that can be answered is not worth asking There are birds that cannot fly Cheaters never prosper GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Answers All maples are trees Some roses are red
All insects are animals Some that glitters are gold All igloo is made of ice Some men have been ruined by success All free persons are educated persons No questions that have answers are questions that worth asking Some birds cannot fly No cheaters are persons who prosper GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Categorical Syllogisms
Syllogism – a three-line deductive argument (A deductive argument that has 2 premises & a conclusion) Categorical syllogism – a syllogism in which all statements in the argument are categorical reasoning E.g. No doctors are professional wrestlers All cardiologists are doctors So, no cardiologists are professional wrestlers GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

More example All painters are artists Some magicians are artists
So, some magicians are painters Is this argument valid? Invalid GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

The Venn diagram technique to check validity
Translate arguments into standard-form categorical statements Draw & label three overlapping circles, one for each term Use shading to represent the information in “all” or “no” statement. Use Xs to represent the information in “some statements” Diagram the two premises. Place X appropriately Look to see if the completed diagram contains all information in the conclusion. If it does, argument is valid. If it does not, argument is not valid. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

So….are these arguments valid?
No sharks are pets, since no barracuda are pets, and no sharks are barracuda. Invalid No one who is a Nobel Prize winner is a rock star. A number of astrophysicists are Nobel Prize winner. Therefore a number of astrophysicists are not rock stars Valid GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Is this argument valid? Anyone who eats pizza every night is at risk for heart disease. Some people who are at risk for heart disease are cab drivers. So, some cab drivers are people who eat pizza. Invalid GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning

Discuss with friends: At least one lawyer is not a golfer. Only persons who have attended law school are lawyers. So, at least one person who has attended law school is not a golfer. Every person who drinks and drives is an irresponsible person. Not every person who talks on a car phone is an irresponsible person. Hence not every person who talks on a car phone is a person who drinks and drive. GXEX1406 Thinking and Communication Skills – Week 9 Categorical Reasoning