Slide on identifying counterexamples in logical reasoning, showing three arguments and instructions for evaluating their structure and validity.

Let's carefully analyze the task in your image: "Identifying Counterexamples", which involves evaluating logical arguments for validity, soundness, and comparing their logical forms.

We are given three arguments:

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Argument 1

- All dogs are cats.
- Some dogs are reptiles.
- ∴ Some cats are reptiles.

Argument 2

- All men are mortals.
- Some mortals are Greeks.
- ∴ Some Greeks are men.

Argument 3

- All mammals are vertebrates.
- Some vertebrates are cold-blooded animals.
- ∴ Some cold-blooded animals are mammals.

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We are to:
1. Determine the truth value of premises and conclusions.
2. Check if the arguments are valid and sound.
3. Compare logical forms.
4. Identify whether any argument is a counterexample to another.
5. Reflect on what this shows about validity.

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## 🔍 Step-by-Step Analysis

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✔ Step 1: Truth Value of Premises and Conclusions

We assess the truth of each statement based on real-world knowledge (though in logic, we sometimes abstract away from truth and focus on form).

#### Argument 1
- "All dogs are cats." → False (Dogs and cats are different species).
- "Some dogs are reptiles." → False (Dogs are mammals, not reptiles).
- "Some cats are reptiles." → False (Cats are mammals).

→ All premises false, conclusion false.
But since the premises are false, the argument is not sound, even if it’s valid.

#### Argument 2
- "All men are mortals." → True (All humans die).
- "Some mortals are Greeks." → True (There are living Greeks).
- "Some Greeks are men." → True (Yes, there are male Greeks).

→ All premises true, conclusion true.
But we must check validity — does the conclusion follow logically?

Wait! Let’s examine this more closely.

> Premise 1: All men are mortals.
> Premise 2: Some mortals are Greeks.
> Conclusion: Some Greeks are men.

This is not necessarily valid.
Just because some mortals are Greeks doesn’t mean that any of those Greeks are men.
It could be that all the Greeks who are mortals are women or non-binary people.

So:
- Premises are true
- Conclusion is true
- But does it follow? No — it’s invalid due to fallacy of affirming the consequent in categorical form.

Example:
All A are B.
Some B are C.
∴ Some C are A. → This is invalid.

So Argument 2 is invalid, though its premises and conclusion happen to be true.

Thus, not sound.

#### Argument 3
- "All mammals are vertebrates." → True (Mammals are vertebrates).
- "Some vertebrates are cold-blooded animals." → True (e.g., fish, amphibians).
- "Some cold-blooded animals are mammals." → False (No mammal is cold-blooded; mammals are warm-blooded).

So:
- Premises: True
- Conclusion: False

→ Invalid argument, because from true premises, a false conclusion follows.

Hence, not valid, and thus not sound.

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✔ Step 2: Logical Form Comparison

Let’s extract the logical structure of each argument.

We use standard categorical logic notation:

- All A are B → ∀x(A(x) → B(x))
- Some A are B → ∃x(A(x) ∧ B(x))

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#### Argument 1
- All dogs are cats. → All D are C.
- Some dogs are reptiles. → Some D are R.
- ∴ Some cats are reptiles. → Some C are R.

Logical form:
> All D are C.
> Some D are R.
> ∴ Some C are R.

This is a valid form? Let’s test.

Suppose:
- All dogs are cats → Every dog is a cat.
- Some dogs are reptiles → There exists a dog that is a reptile.
- Then that dog is both a cat and a reptile → So some cats are reptiles.

✔ Valid.

But as we saw, the premises are false, so not sound.

But the form is valid.

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#### Argument 2
- All men are mortals. → All M are O.
- Some mortals are Greeks. → Some O are G.
- ∴ Some Greeks are men. → Some G are M.

Form:
> All M are O.
> Some O are G.
> ∴ Some G are M.

This is invalid.

Why? Because “some O are G” doesn’t tell us anything about whether those Greeks are men. The Greeks could be women, children, etc.

This is a classic fallacy of the converse or illicit conversion.

✘ Invalid

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#### Argument 3
- All mammals are vertebrates. → All M are V.
- Some vertebrates are cold-blooded animals. → Some V are C.
- ∴ Some cold-blooded animals are mammals. → Some C are M.

Form:
> All M are V.
> Some V are C.
> ∴ Some C are M.

Again, this is invalid.

Why?
- All mammals are vertebrates → M ⊆ V
- Some vertebrates are cold-blooded → Some V ∈ C
- But those cold-blooded vertebrates might not be mammals (e.g., frogs)
- So no guarantee that any cold-blooded animal is a mammal.

In fact, no mammal is cold-blooded, so conclusion is false.

So again, invalid.

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✔ Step 3: Are Arguments of the Same Logical Form?

Now compare the logical structures.

Let’s abstract:

#### Argument 1:
> All A are B.
> Some A are C.
> ∴ Some B are C.

#### Argument 2:
> All A are B.
> Some B are C.
> ∴ Some C are A.

#### Argument 3:
> All A are B.
> Some B are C.
> ∴ Some C are A.

Wait! Arguments 2 and 3 have the same logical form:

> All A are B.
> Some B are C.
> ∴ Some C are A.

But Argument 1 has a different form:
> All A are B.
> Some A are C.
> ∴ Some B are C.

So:
- Argument 1: Different form
- Arguments 2 and 3: Same form

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✔ Step 4: Is One Argument a Counterexample to Another?

A counterexample to an argument form is a case where:
- The premises are true,
- But the conclusion is false,
- Showing that the form is invalid.

Let’s look at Argument 3:
- Premises: True
- Conclusion: False
- And it has the same form as Argument 2

So Argument 3 is a counterexample to the logical form of Argument 2.

Even though Argument 2 has true premises and true conclusion, its form is invalid, and Argument 3 demonstrates that by showing:
- Same form
- True premises
- False conclusion

Therefore, Argument 3 is a counterexample to the argument form used in Argument 2.

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✔ Step 5: What Does This Show About Validity?

This shows that:

> Validity depends only on form, not content.

An argument can have true premises and a true conclusion but still be invalid if the reasoning is flawed.

Moreover, a single counterexample with true premises and a false conclusion proves that a form is invalid.

So:
- Argument 2 has true premises and true conclusion, but is invalid.
- Argument 3 has the same form, but with true premises and false conclusion → proves the form is invalid.

Thus, Argument 3 serves as a counterexample to the validity of the form used in Argument 2.

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## 🟩 Final Summary

| Argument | Valid? | Sound? | Truth of Premises | Truth of Conclusion | Notes |
|--------|--------|--------|-------------------|---------------------|-------|
| 1 | ✔ Valid | ✘ Not sound | False | False | Form: All A are B, Some A are C → Some B are C (valid) |
| 2 | ✘ Invalid | ✘ Not sound | True | True | Form: All A are B, Some B are C → Some C are A (invalid) |
| 3 | ✘ Invalid | ✘ Not sound | True | False | Same form as 2, but conclusion false → counterexample |

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## ✔ Key Takeaways

1. Soundness requires validity + all true premises.
2. Validity is about logical form — does the conclusion follow necessarily?
3. Counterexample = same form, true premises, false conclusion → proves invalidity.
4. Argument 3 is a counterexample to the form of Argument 2, proving that form is invalid.
5. Even if an argument seems plausible (like Argument 2), it may be invalid — truth of premises and conclusion doesn't guarantee validity.

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🧠 Final Answer:

> Argument 3 is a counterexample to Argument 2 because they share the same logical form:
> - All A are B
> - Some B are C
> - ∴ Some C are A
>
> But in Argument 3, the premises are true and the conclusion is false, proving the form is invalid.
>
> This shows that validity depends on logical form, not truth values alone — a single counterexample with true premises and false conclusion suffices to show an argument form is invalid.

✔ Therefore, Argument 3 exposes the flaw in Argument 2's reasoning.