Identifying Denying Propositions

The question asks us to find two propositions from the given list that are denials of each other. In logic, two statements are denials (contradictories) if they cannot both be true and cannot both be false simultaneously. If one is true, the other must be false.

Analyzing Propositions

Let's analyze the logical form of each proposition:

• (a) All elephants are equal to cats: This is a universal affirmative statement. Standard form: All S are P.
• (b) Some elephants are not equal to cats: This is a particular negative statement. Standard form: Some S are not P.
• (c) Some elephants are equal to cats: This is a particular affirmative statement. Standard form: Some S are P.
• (d) Few elephants are also equal to cats: "Few" implies "some". So, this is also a particular affirmative statement, equivalent to (c). Standard form: Some S are P.

Finding the Denial Pair

We need to find a pair that represents contradiction:

• The contradictory of a universal affirmative statement ("All S are P") is a particular negative statement ("Some S are not P").
• Applying this, the contradictory of proposition (a) "All elephants are equal to cats" is "Some elephants are not equal to cats", which is proposition (b).

Let's check the relationships:

• Pair (a) and (b): "All elephants are equal to cats" (A form) and "Some elephants are not equal to cats" (O form). These are contradictories.
• Pair (a) and (c): "All elephants are equal to cats" (A form) and "Some elephants are equal to cats" (I form). These are not contradictories.
• Pair (c) and (d): Both are "Some elephants are equal to cats" (I form). They are essentially the same statement, not denials.

Therefore, propositions (a) and (b) are denials of each other.

Conclusion

The propositions that are denials of each other are (a) and (b).