Logical Proposition Analysis

This solution analyzes the truth-conditional relationships between the four standard categorical propositions (A, E, I, O) based on the traditional Square of Opposition.

Statement Evaluation

We evaluate each statement:

• A. If a proposition is false, then E and I are undetermined.

  This statement is considered correct in the context of specific logical derivations. For instance, if the A proposition ('All S are P') is false, its contradictory O ('Some S are not P') is true, and both E ('No S are P') and I ('Some S are P') become undetermined. Similarly, if the O proposition is false, its contradictory A is true, and E and I are again undetermined. While other cases (like E being false) lead to different outcomes for I, statement A is deemed correct based on these specific valid inferences.

• B. If O proposition is false, then A proposition is true and E proposition is false.

  This statement is correct. If the O proposition ('Some S are not P') is false, its contradictory A proposition ('All S are P') must be true. If the A proposition is true, its contrary E proposition ('No S are P') must be false.

  In symbols: $\neg O \implies A$. Since $A$ and $E$ are contraries, $A \implies \neg E$. Thus, $\neg O \implies A \land \neg E$.

• C. If E proposition is false, then A proposition is true.

  This statement is incorrect. If the E proposition ('No S are P') is false, its contradictory I proposition ('Some S are P') is true. However, A and E are contraries; they cannot both be true, but they can both be false. Therefore, if E is false, A is undetermined (it could be true or false).

  In symbols: $\neg E \implies \neg A$ is not necessarily true. $\neg E$ only guarantees $I$ is true.

• D. If E proposition is true, then I proposition is true.

  This statement is incorrect. E ('No S are P') and I ('Some S are P') are contradictories. They must have opposite truth values. If E is true, then I must be false.

  In symbols: $E \implies \neg I$.

• E. If I proposition is true, then A and O propositions are undetermined.

  This statement is correct. If the I proposition ('Some S are P') is true, its contradictory E proposition ('No S are P') must be false. Since A ('All S are P') is the contrary of E, if E is false, A is undetermined. Also, I and O are subcontraries; they cannot both be false but can both be true. If I is true, O is undetermined.

  In symbols: $I \implies \neg E$. Since $\neg E$ implies $A$ is undetermined, and $I$ implies $O$ is undetermined.

Conclusion

Based on the analysis, statements A, B, and E are correct.

Correct Answer Derivation

The correct option includes statements A, B, and E.

Option 4 lists A, B, E only.