Lesson 11 - Standard Categorical Proposition

A Word Before We Start

• This is the start of our series of topics on logical reasoning which is a part of critical thinking skills
• There will be plenty of terms, and each lesson builds toward another
• Important: We must start mastering the basic concepts as we move all the way to syllogisms in deductive reasoning

Agenda

• What is a Categorical Proposition
• Components of Standard Categorical Proposition
• Natural Language and Standard Categorical Propositions

1. What is a Categorical Proposition?

Definition

Categorical Proposition: A proposition that relates to two classes or categories of things

Examples of Categorical Propositions in Natural Language

Example Statements and Their Meanings

1. "TikTok stars hope for recognition."

   • States that all TikTok stars are included in the set of people who want recognition

2. "Junk foods do not belong in school cafeterias."

   • States that the entire class of junk foods is excluded from the things that belong in school cafeterias

3. "Many dropouts have given up education."

   • States that a part of the students who quit school are included in the class of people who have given up on education

4. "Not all viral videos make you feel good."

   • States that a part of the group of viral videos is not included in the group of things that make you feel good

5. "Selena Gomez loves Hailey Bieber."

   • States that a group with only one member, Selena, is included in the group that loves Hailey Bieber

2. Components of Standard Categorical Proposition

Four Components of a Standard Categorical Proposition (SCP)

As you noticed, the categorical propositions asserted either all or a part of a class was included or excluded in another class. This proposition can be expressed in a standard way using these 4 components:

1. Subject Term
2. Predicate Term
3. Copula (indicating the quality)
4. Quantifiers

The Terms: Subject and Predicate (Part 1)

From Grammar Lessons

• All complete sentences (like a proposition) must have a subject and a predicate
• The subject is what we are talking about
• The predicate is what you want to talk about it

Examples:

1. TikTok stars hope for recognition.
2. Junk foods do not belong in school cafeterias.
3. Many dropouts have given up education.
4. Not all viral videos make you feel good.
5. Selena Gomez loves Hailey Bieber.

The Terms: Subject and Predicate (Part 2)

In Logic, the Most Important Parts Are:

Subject Term

• The category or class of things
• The set or class of things you want to talk about

Predicate Term

• What is said about those things
• The things you want to say about that set

The Copula

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Definition: The word or words that link the subject and the predicate terms

• Expresses the positive or negative quality (is or is not)
• In Standard Categorical Proposition (SCP), the be verbs and modal verbs are the required forms of the copula

Examples:

• All cars are vehicles.
• No car is a natural thing.
• Some cars are expensive vehicles.
  • (Things) are (things)
• Some cars are not heavy pollutants.

The Quantifier

Key Concepts:

• Categorical propositions talk about whether all the members or some of the members of the subject class are included or not included in the predicate class

Two Types of Quantities in SCP:

Universal Quantity

• Indicates that all members are included or not included

Particular Quantity

• Means any quantity that is between 100% to 0%

Standard Terms for Quantity: all, some, and no

Examples:

• All cars are vehicles.
• No car is a natural thing.
• Some cars are expensive vehicles.
• Some cars are not heavy pollutants.

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• “Some good tablet computers are not Apple products “ - is a particular negative proposition
• “No freshman is a versaity player” is a universal negative

The Standard Categorical Proposition (SCP) Structure

Quantifier + Subject + Copula + Predicate

Example: All whales are mammals.

• Quantifier: All
• Subject: whales
• Copula: are
• Predicate: mammals

Important Note: Since we are talking about two groups (usually people or things/events), SCP requires that our predicate is stated in noun form.

Completing the SCP with a Noun Predicate

Practice Examples:

1. All business meetings are events that can be boring.
2. No politicians are perfect humans.
3. Some workers are employees who are far from home.
4. Some unhappy moments are not worth remembering event that you’d like!

Worksheet Practice 1: Making Your Own SCP

(Practice activity)

3. The 4 Forms of Standard Categorical Propositions

Overview

Applying the 4 components:

1. A subject term
2. A predicate term
3. A copula, which is either affirmative or negative in quality
4. One or more words indicating the quantity, universal or particular

Result: This gives us the 4 forms of SCP

The 4 Forms Chart

Logical Symbols for Each Form

What the Forms Mean

Form A: All S are P (Universal Affirmative)

Meaning:

• Every member of the S group is included in the P group
• Not one member of the S group is excluded from the P group

Examples:

• All SIIT students are Thammasat University students.
• All SIIT students are applicants who passed the entrance exams.
• All subjects at SIIT are lessons taught in English.

Form E: No S is P (Universal Negative)

Meaning:

• No member of the S group is included in the P group

Examples:

• No legit streaming service is free content.
• No student in this class is a confused learner.
• No unused ticket was a thing that given a refund.

Form I: Some S are P (Particular Affirmative)

Meaning:

• At least one member of the S group is included in the P group

Examples:

• Some BTS members are artists who performing again in 2024.
• Some parts of the lesson are challenging stuff.
• Some SIIT students are people from other countries.

Form O: Some S are not P (Particular Negative)

Meaning:

• At least one member of the S class is excluded in the P class

Examples:

• Some Thais are not spicy food lovers.
• Some SIIT students are not tutorial class members.
• Some LAS101 students are not people from the freshman class.

4. Natural Language & SCP

Challenge Question

"Not one of the gifts I got was exciting."

What SCP form is this?

The Reality of Natural Language

"Obviously, we do not talk in SCP. But if we are going to analyze categorical statements, it will be easier if we are dealing with statements that are written in SCP."

Translating Categorical Statements in Natural Language into SCP

Requirements for Standard Form

To work with a categorical statement, we need to formulate it as one of the four standard forms: A, E, I, or O

Three Key Requirements:

1. Quantifier

   • The quantity must be explicitly indicated by a quantifier
   • Words such as "all," "no," or "some"

2. Terms

   • The subject and predicate terms must be words or phrases designating a class of objects

3. Copula

   • Must always be some form of the verb "to be"
   • Examples: "is," "are," "was," "were," "will be" (and their negative forms)

Translation Structure Table

Applying All Elements in Translating into Standard Form

Converting Predicates to Noun Form

If the predicate of a sentence is not a term designating objects, we must turn it into nouns.

Examples:

Translating into Standard Form: Subject-Predicate Order

Issue

In English, the subject normally comes first in a sentence, with the predicate following it. But there are exceptions.

Examples:

Example 1: "Tender is the night."

• (P) (S)
• Translation: All nights are a time when it is tender.

Example 2: "In the middle of the table were some pears."

• (Practice translation needed)
• ก็คือ Some pears were fruits on middle of the table

Translating into Standard Form: Non-Standard Quantifiers

Universal Propositions

Affirmative:

11. All whales are mammals.
12. A cat is a predator.
13. Every president faces unexpected challenges.
14. Each item on the menu must has a price.
15. Any student using the gym must show an ID.

Negative:

12. No dinosaur ate sushi.
13. Not a creature was stirring.
14. None of the telephone is working.

Particular Propositions

Affirmative:

19. Some apples are good for baking.
20. A car is parked outside.

Negative:

20. Some apples are not good for baking.
21. A wire is not plugged in.

Important Note on "A"

The indefinite article "a" can mean either universal or particular quantity, depending on the context.

We use context in supplying the quantifiers.

Examples:

• A. Beer is an alcohol. (Universal - All beer)
• B. Beer is in the fridge. (Particular - Some beer)

Worksheet Practice 3: Natural Language to SCP

(Practice activity)

A2: Standard Categorical Propositions

Graded In-Class Group Activity (40 minutes)

Instructions:

1. Form a group. You can have 3-4 members only.
2. You can only use the class computer. You can use your practice sheet and other notes on paper.
3. Work as a group, discuss, and agree on the best answer.
4. Each member must complete his/her paper.
5. Collect and staple together your sheets.
6. One paper will be randomly selected as the representative paper.

Summary

Key Takeaways:

• Categorical propositions relate two classes or categories of things
• Four components of SCP: Quantifier, Subject Term, Copula, Predicate Term
• Four forms: A (All S are P), E (No S is P), I (Some S are P), O (Some S are not P)
• Natural language must be translated into standard form for logical analysis
• Predicates must be expressed as nouns or noun phrases
• Context determines whether ambiguous quantifiers like "a" are universal or particular