1. All / Some / No Statements

How It Works

Syllogisms describe relationships between groups of objects.

The easiest way to solve them is by drawing Venn diagrams.

Basic Rules

All A are B

        B
    ┌─────────┐
    │   A     │
    │ ┌─────┐ │
    │ │     │ │
    │ └─────┘ │
    └─────────┘

A is completely inside B.

---

Some A are B

     A       B
   ┌─────┐ ┌─────┐
   │     ├─┤     │
   └─────┘ └─────┘

The circles overlap.

---

No A are B

   ┌─────┐   ┌─────┐
   │  A  │   │  B  │
   └─────┘   └─────┘

The circles never touch.

---

Example Problem

Statements

• All Cats are Mammals.
• Some Mammals are Pets.

Conclusion

Some Cats are Pets.

---

Step 1

Draw the first statement.

      Mammals

   ┌──────────────┐
   │    Cats      │
   │  ┌────────┐  │
   │  └────────┘  │
   └──────────────┘

---

Step 2

Add Pets.

          Mammals
      ┌──────────────┐
      │   Cats       │
      │ ┌──────┐     │
      │ └──────┘──┐  │
      └───────────┼──┘
                  │
               Pets

The Pets circle overlaps Mammals.

But the overlap does not have to touch Cats.

---

Final Answer

The conclusion does NOT follow.

---

2. Possibility Cases (May Be)

How It Works

Possibility questions ask

Can this happen?

Not

Must this happen?

If nothing prevents it,

the possibility follows.

---

Example Problem

Statement

Some Pens are Pencils.

Conclusion

All Pens may be Pencils.

---

Step 1

Given statement

 Pens      Pencils

┌─────┐ ┌─────┐
│     ├─┤     │
└─────┘ └─────┘

Only some overlap is compulsory.

---

Step 2

Can all Pens be inside Pencils?

      Pencils

 ┌─────────────┐
 │   Pens      │
 │ ┌─────────┐ │
 │ └─────────┘ │
 └─────────────┘

Yes.

The original statement is still true.

---

Final Answer

The possibility follows.

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3. Either-Or Cases

How It Works

Sometimes two conclusions are opposites.

Example

• Some A are B
• No A are B

Both cannot be true together.

One of them must be true.

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Common Complementary Pairs

Conclusion I	Conclusion II
Some A are B	No A are B
Some A are not B	All A are B

These are called complementary pairs.

---

Example Problem

Statements

• All Doctors are Engineers.
• No Engineer is a Pilot.

Conclusions

I. Some Doctors are not Pilots.

II. All Doctors are Pilots.

---

Diagram

 Pilots

        Engineers

   ┌──────────────┐
   │  Doctors     │
   │ ┌─────────┐  │
   │ └─────────┘  │
   └──────────────┘


(Pilots never touch Engineers)

---

Doctors cannot become Pilots.

Therefore

Conclusion I is true.

Conclusion II is false.

---

Final Answer

Only Conclusion I follows.

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4. Reverse Syllogism

How It Works

Normally,

Statements → Conclusion

Reverse syllogism works backwards.

Conclusion → Find the correct statements.

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Example Problem

Conclusion

All Dogs are Animals.

Which statements produce this conclusion?

Option 1

• All Dogs are Mammals.
• All Mammals are Animals.

Option 2

• Some Dogs are Mammals.
• All Mammals are Animals.

---

Option 1

Animals

┌───────────────┐
│ Mammals       │
│ ┌───────────┐ │
│ │ Dogs      │ │
│ └───────────┘ │
└───────────────┘

Dogs are inside Animals.

Works perfectly.

---

Option 2

Animals

┌──────────────┐
│ Mammals      │
└──────┬───────┘
       │
      Dogs

Only some Dogs are Mammals.

Other Dogs may lie outside.

So the conclusion is not guaranteed.

---

Final Answer

Option 1

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5. Only / A Few Statements

Important Meaning

Many students confuse these.

Only Pens are Inks

Means

All Inks are Pens

NOT

All Pens are Inks

Diagram

      Pens

 ┌─────────────┐
 │   Inks      │
 │ ┌─────────┐ │
 │ └─────────┘ │
 └─────────────┘

---

Only a Few

Means

Some A are B

AND

Some A are NOT B

Diagram

A

┌─────────────┐
│ ┌───────┐   │
│ │   B   │   │
│ └───────┘   │
└─────────────┘

Part of A is inside B.

Part remains outside.

---

Example Problem

Statements

Only Pens are Inks.

Only a few Pencils are Pens.

---

Diagram

Pens

┌─────────────────────┐
│      Inks           │
│   ┌────────────┐    │
│   └────────────┘    │
│                     │
└──────────┬──────────┘
           │
        Pencils

Pencils overlap Pens but also extend outside.

---

Final Answer

All Inks being Pencils is not possible.

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6. Coded Syllogism

How It Works

Instead of normal English,

symbols are used.

First decode them.

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Example Codes

Symbol	Meaning
@	All
#	Some
$	No
^	Some…Not
&	Only

---

Example Problem

Codes

X @ Y

means

All X are Y

Y # Z

means

Some Y are Z

Conclusion

X # Z

means

Some X are Z

---

Draw the Diagram

            Y

     ┌────────────┐
     │     X      │
     │ ┌──────┐   │
     │ └──────┘───┐
     └────────────┼──
                  │
                  Z

Y overlaps Z.

But that overlap may never touch X.

---

Final Answer

The conclusion does NOT follow.