Lesson 12 - Categorical Propositions & Immediate Inference

Agenda

• Venn Diagram & SCP
• Distribution
• Immediate Inference

1. Illustrating SCPs - The Venn Diagram & SCPs

SCPs and Their Meaning

Practice Drill

Help Yourselves:

• Pair up
• Drill each other
• Student A gives the letter symbol
• Student B gives the structure
• Or you can do it in reverse

Example: Proposition (O) - Some S are not P

Illustrating SCP

It's important that we all master the symbols and their structures, including where the symbols are if we are to put them on a square.

The Four Forms:

• A: All S are P
• E: No S are P
• I: Some S are P
• O: Some S are not P

Since these propositions assert that one group is included in or excluded from another group, or if a member of one group is included or not included in another group, these propositions can be illustrated using the Venn diagram.

The Venn Diagram

Four Areas Explained

Area 1: Anything in this area is an SIIT student but not an online shopper

Area 2: Anything in this area is both an SIIT student and an online shopper

Area 3: Anything in this area is an online shopper but not an SIIT student

Area 4: This is outside of both circles. Anything that is neither an SIIT student nor an online shopper

A Word on Our Venn Diagram

• Shade means none
• X means at least one

Venn Diagram: Proposition (A)

All S are P

Explanation:

• (A) asserts that every member of the S class is in the P class
• This means that (A) also asserts that there is no member of the S class that is outside of the P class
• Therefore, Area 1 is empty
• We shade Area 1 to show that there is nothing there
• Does (A) say that there are no P members outside of S (in Area 4)? No, it doesn't

Example:

"All Valentines-themed items are discounted products"

• S: Valentines-themed items
• P: discounted products

Key Points:

• All members of the S class are included in the P class
• This means that all items that are Valentine-themed are included in the group of discounted products
• Since every member of the S class is inside the P class, we are also asserting that there is no S class member that is outside of the P class
• Area 1 is empty (denoted by shading that area)
• Thus, our proposition asserts that the attribute of being a discounted product is distributed to all Valentine-themed items
• The subject is distributed

Distribution Pattern:

• All members inside the S circle are SP
• Those inside the P circle are SP and P
• Therefore, the SUBJECT IS DISTRIBUTED

Venn Diagram: Proposition (E)

No S are P

Explanation:

• (E) asserts that no member of the S class is in the P class
• At the same time, it asserts that no member of the P class is in the S class either
• Therefore, Area 2 is empty
• Again, we shade Area 2 to show that there is nothing inside it
• In effect, the attribution "S member only" applies to everyone found in the S area
• At the same time, the attribution "P member only" applies to everyone found in the P area

Example:

"No discounted items were returned products"

• S: discounted items
• P: returned products

Key Points:

• No members of the S class are included in the P class
• None of the discounted items are in the class of returned products
• This also means that there are no returned products that are in the discounted class
• Therefore Area 2 is empty
• This proposition asserts two things:
  1. Discounted items are not in the returned products class
  2. Returned products are not in the discounted items class
• In short, the attribute "Not P" is distributed to all members of the S class
• At the same time, the attribute "Not S" is distributed to all members of the P class
• Therefore in an (E) proposition, BOTH S and P are distributed

Distribution Pattern:

• All members inside the S circle are S only
• All members inside the P circle are P only as well
• BOTH SUBJECT TERM AND PREDICATE TERM ARE DISTRIBUTED

Venn Diagram: Proposition (I)

Some S are P

Explanation:

• (I) asserts that parts of the S class are in the P class
• However, we must remember that "some" in SCP means at least one
• We use X to denote that one member
• In essence, (I) asserts that at least one member of the S class is included in the P class
• Therefore, we put X in Area 2
• As a result, this also tells us that not everyone in the S class is just an S member
• At least one is both an S and P
• The same is also true for the P class

Example:

"Some expensive imported bags are fake items"

• S: expensive imported bags
• P: fake items

Key Points:

• Some members of the S class are included in the P class
• There's a class/group of expensive imported bags. Not all of those bags are fake. Some are fake. How many?
• In categorical propositions, "some" means at least one
• There is at least one expensive imported bag that is in the fake items class/group
• We place X, the symbol of at least one member, in AREA 2
• This X represents a member that is both S and P at the same time
• As a result, there is no attribution that we can distribute to all members of S or to all members of P
• In (I) propositions, neither S nor P is distributed

Distribution Pattern:

• The members in the S circle are S only and SP
• The members in the P circle are P only and SP
• NEITHER SUBJECT TERM NOR PREDICATE TERM IS DISTRIBUTED

Venn Diagram: Proposition (O)

Some S are not P

Explanation:

• (O) asserts that at least one member of the S class is not in the P class
• Consequently, the attribution "not P" applies to anyone in area outside P

Example:

"Some imported items are not valuable things"

• S: imported items
• P: valuable things

Key Points:

• Some members of the S class are not included in the P class
• There is at least one imported item that is not a valuable thing
• Again, in SCP, "some" means at least one, and the symbol X represents this
• Since (O) asserts that there is at least one S member that is not in the P class, we place X in Area 1
• Consequently, the attribute "not P" is distributed to all members of the P class
• Therefore, in (O) propositions, P is distributed

Distribution Pattern:

• The X outside of the P circle distributes a new characteristic "not X" to all members inside the P circle
• THE PREDICATE TERM IS DISTRIBUTED

SCP in Other Words

Alternative Formulations:

• All S are P = No members of S are outside P
• No S are P = No members of S are inside P
• Some S are P = At least one S exists that is a P
• Some S are not P = At least one S exists that is not a P

2. Distribution in SCP

What Does Distribution Mean?

Distribution₁:

• Distribution is an attribute of the terms (subject and predicate) of propositions

• A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term

• A term is distributed if and only if the statement assigns (or distributes) an attribute to every class member denoted by the term

• Thus, if a statement asserts something about every member of the S class, then S is distributed

• If it asserts something about every member of the P class, then P is distributed

• Otherwise S and P are undistributed

Remember OR ELSE

Two Mnemonic Devices for Distribution

"Unprepared Students Never Pass"

• Universals distribute Subjects
• Negatives distribute Predicates

"Any Student Earning B's Is Not On Probation"

• A distributes Subject
• E distributes Both
• I distributes Neither
• O distributes Predicate

Practice B

Read each SCP. Identify the form (A, E, I, or O). Then, double-underline the distributed term.

Example: Some items at Kingpower are not things that cost 30% less - O
(The predicate "things that cost 30% less" would be double-underlined)

3. Immediate Inference

What is Immediate Inference?

An argument with a single premise that allows us to draw conclusions based on logical relationships between categorical propositions.

Let's Consider This

Proposition 1: "Every one of your classmates gave your project a 5-star rating"

Proposition 2: "A few of your classmates did not give your project a 5-star rating"

Clearly, these two propositions cannot be both true at the same time.

Modern/Boolean Square of Opposition

Background:

• There are actually 2 Squares of Opposition
• The first one is by Aristotle, but his method follows the Aristotelian Standpoint that (A) and (E) assert the existence of their subjects even when these subjects are fantastical or imaginary
• The Modern or Boolean Square of Opposition applies the Boolean Standpoint that premises are "closed" or unreceptive to existence
• Both of these squares are used in making Immediate Inferences
• For our TU104 class, we will focus only on the Modern Square of Opposition

Structure of Modern Square:

          Logically
        undetermined
    A ______________ E
     \             /
      \  Contra   /
       \  dictory/
Logically\      /Logically
undetermined\  /undetermined
            \/
            /\
    Contra /  \dictory
          /    \
         /      \
    I ______________ O
          Logically
        undetermined

Key Points:

• Contradictory relation is the only relationship considered in the Modern Square of Opposition
• The propositions with contradictory relationships have opposite truth values
• If one is true, the other is false
• No other inferences are possible

Testing Immediate Inference₁

Steps for Testing:

1. Since the source text did not indicate which is true and which is false, begin by assuming that the premise is true
2. Enter the pertinent truth value in the square
3. Use the square to compute the truth value of the conclusion
4. If the square indicates that the conclusion is true (correct), the argument is valid
5. Arguments that are valid from the Boolean standpoint are unconditionally valid since they ignore the importance of the existence of things

Example:

Premise: All of your classmates are raters who gave your project a 5-star rating
Conclusion: Some of your classmates are not raters who gave your project a 5-star rating

Analysis:

• (If) A is true
• O is false
• These are contradictories, so the inference is VALID

Example 1 in Other Words

Formal Statement:
"(Indeed/It is true that) all of your classmates are raters who gave your project a 5-star rating. Therefore, it is (false/not true/incorrect/wrong) that some of your classmates are not raters who gave your project a 5-star rating."

Conclusion: Since this argument follows the Modern Square of Opposition, then the argument is VALID.

Proving the Immediate Inference Using the Venn Diagram

Visual Proof:

Premise (A): All of your classmates are raters who gave your project a 5-star rating

• The (A) diagram shows that Area 1 is shaded (nothing should be found in Area 1)

Conclusion (O): Some of your classmates are not raters who gave your project a 5-star rating

• The (O) diagram would place an X in Area 1

Analysis:

• The premise diagram asserts that there is NONE in area 1
• It does not allow us to assume that something could be in area 1
• Therefore, it is wrong to allow the possibility that an S member might exist in Area 1 according to (O)
• The argument is VALID

Can We Do It the Other Way Around?

Yes! Remember that we are working with pairs of propositions that have opposite truth values. If one is false, the other is true.

Example 1 Reversed

Formal Statement:
"(Actually/In fact) some of your classmates are not raters who gave your project a 5-star rating. Therefore, it (is not true/is false/is wrong) that all of your classmates are raters who gave your project a 5-star rating."

Analysis:

• (If) O is true
• A is false
• These are contradictories, so the inference is VALID

Proving with Venn Diagram:

Premise (O): Some of your classmates are not raters who gave your project a 5-star rating

• The (O) diagram places an X in Area 1

Conclusion (A): All of your classmates are raters who gave your project a 5-star rating

• The (A) diagram would shade Area 1

Analysis:

• The premise (O) asserts that there is at least one S member in Area 1
• Therefore, it is wrong to assume that Area 1 can be empty at the same time, as what (A) proposes
• The argument is VALID

Rules on Contradictory Propositions

1. Both propositions cannot be true
2. Both propositions cannot be false
3. The premise can either be true or false

Practice C

Identify the form of the propositions. Determine their truth value. Use the Modern Square of Opposition to prove that the immediate inference is valid. Then prove your conclusion using Venn Diagrams for the premise and conclusion.

Example Problem:

Statement: "Indeed, all discounted T-shirts are items sold today. Therefore, it is true that some T-shirts are not items sold today."

Step 1: Identify Forms

• Premise: "All discounted T-shirts are items sold today" = A (Truth Value: True)
• Conclusion: "Some T-shirts are not items sold today" = O (Truth Value: True)

Step 2: Check Modern Square

• Simplified argument: "A is true; therefore, O is true"
• According to the Boolean Square, A and O are contradictories
• If A is true, O must be false
• Therefore, this argument is INVALID

Step 3: Prove with Venn Diagrams

Premise (A): All discounted T-shirts are items sold today

• The premise diagram asserts that there is NONE in area 1
• It does not allow us to assume that something could be in area 1

Conclusion (O): Some T-shirts are not items sold today

• This would require an X in Area 1

Final Analysis:

• Concluding that there is at least one in area 1 is impossible when the premise says area 1 is empty
• The Venn diagrams prove the INVALIDITY of this argument

A3: Venn Diagrams & SCP, & Immediate Inferences

This is a 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

Quick Reference Summary

Distribution Summary Table

Venn Diagram Quick Guide

• A (All S are P): Shade area of S outside P (Area 1)
• E (No S are P): Shade overlap area (Area 2)
• I (Some S are P): Place X in overlap area (Area 2)
• O (Some S are not P): Place X in S outside P (Area 1)

Boolean Square Relationships

• A and O are contradictories (opposite truth values)
• E and I are contradictories (opposite truth values)
• All other relationships are logically undetermined in the Modern Square