Categorical Proposition Relationships

The question asks to identify propositions that cannot both be true but may both be false. This specific logical relationship is known as contraries.

Understanding Contrary Propositions

Contraries are pairs of universal propositions with the same subject and predicate. They share the characteristic that they cannot both be true simultaneously. However, unlike contradictory propositions, they are capable of both being false at the same time.

Analyzing the Given Propositions

Let's classify the given propositions concerning 'cars' and 'vehicles' based on their logical form:

• A. All cars are vehicles. (This is a Type $A$ proposition - Universal Affirmative)
• B. Some cars are vehicles. (This is a Type $I$ proposition - Particular Affirmative)
• C. No cars are vehicles. (This is a Type $E$ proposition - Universal Negative)
• D. Some cars are not vehicles. (This is a Type $O$ proposition - Particular Negative)

Determining the Contrasting Pair

In the traditional square of opposition in logic, the relationship between Type $A$ and Type $E$ universal propositions (when they have the same subject and predicate) is that of contraries.

• Proposition A: "All cars are vehicles."
• Proposition C: "No cars are vehicles."

These two propositions, $A$ and $C$, are contraries. They cannot both be true: if it's true that all cars are vehicles, it must be false that no cars are vehicles, and vice versa.

Crucially, they can both be false. For example, if some cars were vehicles and others were not, then both the statement "All cars are vehicles" and the statement "No cars are vehicles" would be false.

Other relationships relevant to the options include:

• $A$ and $D$ are contradictories (must have opposite truth values).
• $I$ and $E$ are contradictories.
• $I$ and $O$ are subcontraries (can both be true, cannot both be false).

Therefore, the propositions that fit the description "cannot both be true although they may both be false" are A and C.