Logical Reasoning · Logical Connectives, Syllogisms and Venn Diagrams

Syllogisms - Basic Concepts, Formulas, Key Points and Examples

Explanation

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SYLLOGISMS

🟢 1. BASIC CONCEPT A syllogism is a logical reasoning problem in which conclusions are drawn from two or more given statements. A typical syllogism contains statements and conclusions.

\text{Statements}\Rightarrow\text{Conclusion}

The objective is to determine whether the given conclusion logically follows from the statements. 🟡 KEY POINT A conclusion must be based only on the information given in the statements. Do not use outside knowledge while solving syllogism questions

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🟢 2. CATEGORICAL STATEMENTS The four basic types of categorical statements are:

\text{All A are B}

\text{No A are B}

\text{Some A are B}

\text{Some A are not B}

🟡 KEY POINT These four forms are the foundation of most syllogism problems.

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🟢 3. ALL A ARE B "All A are B" means every member of A belongs to B.

A\subseteq B

This means A is completely contained within B. 🔴 IMPORTANT From:

\text{All A are B}

we cannot conclude:

\text{All B are A}

The reverse statement is not necessarily true.

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🟢 4. NO A ARE B "No A are B" means A and B have no common members.

A\cap B=\varnothing

This means the two groups do not overlap. 🔴 IMPORTANT If:

\text{No A are B}

then:

\text{Some A are B}

cannot be true.

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🟢 5. SOME A ARE B "Some A are B" means at least one member of A is also a member of B.

A\cap B\neq\varnothing

This statement establishes that at least one common member exists. 🟡 KEY POINT "Some" means at least one.

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🟢 6. SOME A ARE NOT B "Some A are not B" means at least one member of A does not belong to B.

A-B\neq\varnothing

This establishes that at least one member of A exists outside B.

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🟢 7. UNIVERSAL STATEMENTS Statements beginning with "All" or "No" are called universal statements. Examples:

\text{All A are B}

\text{No A are B}

Universal statements describe a relationship involving the entire group.

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🟡 KEY POINT "All" and "No" are universal statements.

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🟢 8. PARTICULAR STATEMENTS Statements beginning with "Some" are called particular statements.

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Examples:

\text{Some A are B}

\text{Some A are not B}

Particular statements refer to at least one member of a group.

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🟡 KEY POINT "Some" indicates existence of at least one member.

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🟢 9. POSITIVE STATEMENTS Positive statements do not contain a negative condition.

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Examples:

\text{All A are B}

\text{Some A are B}

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🟢 10. NEGATIVE STATEMENTS Negative statements contain "No" or "Not". Examples:

\text{No A are B}

\text{Some A are not B}

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🟢 11. CONVERSION Conversion means interchanging the subject and predicate of a statement. For:

\text{No A are B}

we can conclude:

\text{No B are A}

For:

\text{Some A are B}

we can conclude:

\text{Some B are A}

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🔴 IMPORTANT From:

\text{All A are B}

we cannot generally conclude:

\text{All B are A}

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🟢 12. TRANSITIVE RELATION If:

\text{All A are B}

and:

\text{All B are C}

then:

\text{All A are C}

In set form:

A\subseteq B

B\subseteq C

Therefore:

A\subseteq C

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🟡 KEY POINT Look for a common term connecting the statements.

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🟢 13. CHAINING WITH ALL If:

\text{All A are B}

and:

\text{All B are C}

then:

\text{All A are C}

Similarly, if:

\text{All A are B}

\text{All B are C}

\text{All C are D}

then:

\text{All A are D}

🟢 14. CHAINING WITH NO If:

\text{All A are B}

and:

\text{No B are C}

then:

\text{No A are C}

Because:

A\subseteq B

and:

B\cap C=\varnothing

Therefore:

A\cap C=\varnothing

🟢 15. CHAINING WITH SOME If:

\text{Some A are B}

and:

\text{All B are C}

then:

\text{Some A are C}

The members common to A and B must also belong to C.

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🟡 KEY POINT "Some + All" can establish a particular conclusion when the common member is carried through the chain.

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🟢 16. SOME WITH NO If:

\text{Some A are B}

and:

\text{No B are C}

then:

\text{Some A are not C}

Because the A members that are B cannot belong to C.

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🟢 17. VENN DIAGRAM METHOD Syllogisms can be solved using Venn diagrams. For:

\text{All A are B}

place A completely inside B. For:

\text{No A are B}

keep A and B separate. For:

\text{Some A are B}

place at least one element in the common region. For:

\text{Some A are not B}

place at least one element inside A but outside B.

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🟡 KEY POINT Use Venn diagrams when the relationships between three or more groups become difficult to visualize.

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🟢 18. EXISTENCE OF A GROUP An "All" statement by itself does not necessarily establish that members of the subject group exist. For example:

\text{All A are B}

does not by itself establish:

\text{Some A are B}

However, a statement such as:

\text{Some A are B}

explicitly establishes the existence of A.

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🔴 IMPORTANT Do not automatically convert a universal statement into a particular statement.

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🟢 19. INTERSECTION The symbol:

A\cap B

represents the members common to A and B. If:

A\cap B\neq\varnothing

then A and B have at least one common member. Therefore:

\text{Some A are B}

🟢 20. DISJOINT SETS If:

A\cap B=\varnothing

then A and B have no common members. Therefore:

\text{No A are B}

🟢 21. SUBSET RELATION If:

A\subseteq B

then every member of A is also a member of B. Therefore:

\text{All A are B}

🟢 22. DIFFERENCE OF SETS The expression:

A-B

represents members that belong to A but do not belong to B. Therefore:

A-B\neq\varnothing

means:

\text{Some A are not B}

🟢 23. THREE-LEVEL CHAIN If:

A\subseteq B

B\subseteq C

C\subseteq D

then:

A\subseteq D

Therefore:

\text{All A are D}

🟢 24. CONCLUSION MUST NECESSARILY FOLLOW A conclusion follows only when it must be true based on the given statements. For example:

\text{All A are B}

\text{All B are C}

Therefore:

\text{All A are C}

The conclusion necessarily follows.

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🔴 IMPORTANT A conclusion that is merely possible is not sufficient. The conclusion must be definitely established by the statements.

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🟢 25. INVALID REVERSE RELATION Given:

\text{All A are B}

we know:

A\subseteq B

But we cannot conclude:

B\subseteq A

Therefore:

\text{All B are A}

does not necessarily follow.

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🟢 26. TWO GROUPS INSIDE A COMMON GROUP If:

A\subseteq C

and:

B\subseteq C

we cannot conclude:

A\subseteq B

or:

B\subseteq A

A and B may be separate groups within C.

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🟡 KEY POINT Two groups having the same larger group does not mean that they have a relationship with each other.

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🟢 27. CONTRADICTION Two statements are contradictory when they cannot both be true. For example:

\text{No A are B}

and:

\text{Some A are B}

cannot both be true. Because:

A\cap B=\varnothing

contradicts:

A\cap B\neq\varnothing

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🟢 28. CONCLUSION TYPES A conclusion may be: 1. Universal positive

\text{All A are B}

2. Universal negative

\text{No A are B}

3. Particular positive

\text{Some A are B}

4. Particular negative

\text{Some A are not B}

🟢 29. IMPORTANT SYLLOGISM RELATIONSHIPS All A are B:

A\subseteq B

No A are B:

A\cap B=\varnothing

Some A are B:

A\cap B\neq\varnothing

Some A are not B:

A-B\neq\varnothing

🟢 30. COMMON VALID PATTERNS Pattern 1:

\text{All A are B}

\text{All B are C}

Therefore:

\text{All A are C}

Pattern 2:

\text{Some A are B}

\text{All B are C}

Therefore:

\text{Some A are C}

Pattern 3:

\text{All A are B}

\text{No B are C}

Therefore:

\text{No A are C}

Pattern 4:

\text{Some A are B}

\text{No B are C}

Therefore:

\text{Some A are not C}

🟢 31. COMMON INVALID PATTERNS Invalid Pattern 1:

\text{All A are B}

Therefore:

\text{All B are A}

This does not necessarily follow. Invalid Pattern 2:

\text{All A are B}

\text{All C are B}

Therefore:

\text{All A are C}

This does not necessarily follow. Invalid Pattern 3:

\text{Some A are B}

Therefore:

\text{Some A are C}

without any relationship between B and C. This does not follow.

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🟢 32. QUICK SOLVING METHOD Step 1: Identify the subject and predicate of each statement. Step 2: Identify whether the statement is All, No, Some, or Some ... not. Step 3: Convert the statements into set relationships. Step 4: Connect statements using common terms. Step 5: Check whether the conclusion is necessarily true. Step 6: Reject conclusions based only on possibility or assumption.

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🟡 KEY POINT The safest approach is to translate every statement into a logical relationship before checking the conclusion.

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🟢 33. QUICK REVISION FORMULAS All A are B:

A\subseteq B

No A are B:

A\cap B=\varnothing

Some A are B:

A\cap B\neq\varnothing

Some A are not B:

A-B\neq\varnothing

All A are B and All B are C:

A\subseteq B\subseteq C

Therefore:

A\subseteq C

Some A are B and All B are C:

A\cap B\neq\varnothing

and:

B\subseteq C

Therefore:

A\cap C\neq\varnothing

Some A are B and No B are C:

A\cap B\neq\varnothing

and:

B\cap C=\varnothing

Therefore:

A-C\neq\varnothing

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🔴 IMPORTANT KEY POINTS Do not reverse an "All" statement. Do not assume two groups overlap without evidence. Do not assume that "Some A are not B" means "No A are B". Do not assume that "All A are B" means that A definitely exists. Do not use outside knowledge. A conclusion must necessarily follow from the given statements. Use Venn diagrams when the relationships are complex.

Example

🔵 SYLLOGISMS — SOLVED EXAMPLES 🟢 EXAMPLE 1: ALL A ARE B QUESTION: Statements: All teachers are educated. All educated people are knowledgeable. Conclusion: All teachers are knowledgeable. SOLUTION:

\text{All Teachers are Educated}

\text{All Educated are Knowledgeable}

Therefore:

\text{All Teachers are Knowledgeable}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 2: SIMPLE CHAINING QUESTION: Statements: All cats are animals. All animals are living beings. Conclusion: All cats are living beings. SOLUTION:

\text{Cats}\subseteq\text{Animals}

\text{Animals}\subseteq\text{Living Beings}

Therefore:

\text{Cats}\subseteq\text{Living Beings}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 3: NO A ARE B QUESTION: Statements: No doctors are engineers. Some doctors are teachers. Conclusion: Some teachers are not engineers. SOLUTION:

\text{No Doctors are Engineers}

\text{Some Doctors are Teachers}

The doctors who are teachers cannot be engineers. Therefore:

\text{Some Teachers are not Engineers}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 4: SOME A ARE B QUESTION: Statements: Some students are athletes. All athletes are fit. Conclusion: Some students are fit. SOLUTION:

\text{Some Students are Athletes}

\text{All Athletes are Fit}

Therefore, the students who are athletes are also fit.

\text{Some Students are Fit}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 5: SOME A ARE NOT B QUESTION: Statements: Some students are not athletes. All athletes are fit. Conclusion: Some students are not fit. SOLUTION: We know:

\text{Some Students are not Athletes}

and:

\text{All Athletes are Fit}

Being not an athlete does not necessarily mean being unfit. Therefore, the conclusion cannot be established. 🔵 ANSWER:

\text{Conclusion does not follow}

🟢 EXAMPLE 6: REVERSE RELATION QUESTION: Statements: All doctors are professionals. Conclusion: All professionals are doctors. SOLUTION: Given:

\text{All Doctors are Professionals}

This means:

\text{Doctors}\subseteq\text{Professionals}

It does not mean:

\text{Professionals}\subseteq\text{Doctors}

🔴 IMPORTANT The reverse of an "All" statement cannot be assumed. 🔵 ANSWER:

\text{Conclusion does not follow}

🟢 EXAMPLE 7: NO OVERLAP QUESTION: Statements: No birds are mammals. Some parrots are birds. Conclusion: Some parrots are not mammals. SOLUTION:

\text{No Birds are Mammals}

Therefore:

\text{Birds}\cap\text{Mammals}=\varnothing

Also:

\text{Some Parrots are Birds}

Therefore, those parrots cannot be mammals.

\text{Some Parrots are not Mammals}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 8: THREE-LEVEL CHAIN QUESTION: Statements: All A are B. All B are C. All C are D. Conclusion: All A are D. SOLUTION:

A\subseteq B

B\subseteq C

C\subseteq D

Therefore:

A\subseteq B\subseteq C\subseteq D

Hence:

A\subseteq D

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 9: SOME AND ALL QUESTION: Statements: Some engineers are managers. All managers are graduates. Conclusion: Some engineers are graduates. SOLUTION:

\text{Some Engineers are Managers}

\text{All Managers are Graduates}

Therefore:

\text{Some Engineers are Graduates}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 10: NO AND ALL QUESTION: Statements: No artists are scientists. All painters are artists. Conclusion: No painters are scientists. SOLUTION:

\text{No Artists are Scientists}

\text{All Painters are Artists}

Therefore:

\text{Painters}\subseteq\text{Artists}

Since artists and scientists do not overlap:

\text{Painters}\cap\text{Scientists}=\varnothing

Therefore:

\text{No Painters are Scientists}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 11: INVALID CONCLUSION QUESTION: Statements: All roses are flowers. Some flowers are red. Conclusion: Some roses are red. SOLUTION: We know:

\text{All Roses are Flowers}

and:

\text{Some Flowers are Red}

The red flowers need not be roses. Therefore, we cannot conclude:

\text{Some Roses are Red}

🔵 ANSWER:

\text{Conclusion does not follow}

🟢 EXAMPLE 12: TWO CONCLUSIONS QUESTION: Statements: All books are useful. Some books are expensive. Conclusions: I. Some useful things are expensive. II. All expensive things are useful. SOLUTION: From:

\text{All Books are Useful}

and:

\text{Some Books are Expensive}

the books that are expensive are also useful. Therefore:

\text{Some Useful Things are Expensive}

Conclusion I follows. However, there is no statement that all expensive things are books. Therefore:

\text{All Expensive Things are Useful}

does not follow. 🔵 ANSWER:

\text{Only Conclusion I follows}

🟢 EXAMPLE 13: SOME A ARE NOT B QUESTION: Statements: Some employees are not graduates. All graduates are qualified. Conclusion: Some employees are not qualified. SOLUTION: We know:

\text{Some Employees are not Graduates}

and:

\text{All Graduates are Qualified}

Being a non-graduate does not necessarily mean being unqualified. Therefore:

\text{Some Employees are not Qualified}

cannot be established. 🔵 ANSWER:

\text{Conclusion does not follow}

🟢 EXAMPLE 14: CONVERSION QUESTION: Statement: No doctors are pilots. Conclusion: No pilots are doctors. SOLUTION: Given:

\text{No Doctors are Pilots}

This means:

\text{Doctors}\cap\text{Pilots}=\varnothing

Therefore:

\text{Pilots}\cap\text{Doctors}=\varnothing

Hence:

\text{No Pilots are Doctors}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 15: SOME A ARE B CONVERSION QUESTION: Statement: Some teachers are writers. Conclusion: Some writers are teachers. SOLUTION: Given:

\text{Some Teachers are Writers}

This means:

\text{Teachers}\cap\text{Writers}\neq\varnothing

Therefore:

\text{Writers}\cap\text{Teachers}\neq\varnothing

Hence:

\text{Some Writers are Teachers}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 16: CHAIN WITH NEGATIVE STATEMENT QUESTION: Statements: All A are B. No B are C. Conclusion: No A are C. SOLUTION:

A\subseteq B

and:

B\cap C=\varnothing

Therefore:

A\cap C=\varnothing

Hence:

\text{No A are C}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 17: SOME A ARE B AND NO B ARE C QUESTION: Statements: Some A are B. No B are C. Conclusion: Some A are not C. SOLUTION:

\text{Some A are B}

and:

\text{No B are C}

Therefore, the A members that are B cannot be C. Hence:

\text{Some A are not C}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 18: INVALID REVERSE CHAIN QUESTION: Statements: All A are B. All C are B. Conclusion: All A are C. SOLUTION: We know:

A\subseteq B

and:

C\subseteq B

Both A and C are subsets of B, but there is no information showing that A is contained in C. Therefore:

A\subseteq C

cannot be established. 🔵 ANSWER:

\text{Conclusion does not follow}

🟢 EXAMPLE 19: THREE STATEMENTS QUESTION: Statements: All students are learners. All learners are readers. Some students are athletes. Conclusion: Some athletes are readers. SOLUTION:

\text{All Students are Learners}

\text{All Learners are Readers}

Therefore:

\text{All Students are Readers}

Also:

\text{Some Students are Athletes}

Therefore, those students who are athletes are readers.

\text{Some Athletes are Readers}

🔵 ANSWER:

\text{Conclusion follows}

🟢 EXAMPLE 20: EXAMINATION TYPE QUESTION: Statements: All managers are graduates. Some graduates are engineers. No engineers are artists. Conclusions: I. Some managers are engineers. II. Some graduates are not artists. SOLUTION: From:

\text{All Managers are Graduates}

and:

\text{Some Graduates are Engineers}

we cannot conclude:

\text{Some Managers are Engineers}

Therefore, Conclusion I does not follow. Now:

\text{Some Graduates are Engineers}

and:

\text{No Engineers are Artists}

Therefore, those graduates who are engineers are not artists. Hence:

\text{Some Graduates are not Artists}

Conclusion II follows. 🔵 ANSWER:

\text{Only Conclusion II follows}

🟡 QUICK EXAM TIP Always convert the statements into simple relationships before checking the conclusion. For example:

\text{All A are B}

means:

A\subseteq B

and:

\text{No B are C}

means:

B\cap C=\varnothing

Therefore:

A\cap C=\varnothing

Hence:

\text{No A are C}

🔴 IMPORTANT Do not assume that two groups overlap unless the statements establish that relationship. Do not reverse "All" statements. Do not use real-world knowledge. Check whether the conclusion is necessarily true from the given statements.