Understanding Logical Consistency

Logical consistency between statements means that it is possible for all the statements to be true at the same time, without contradicting each other or the main statement given.

The main statement is: "Whenever Sakshi hears it's someone's birthday, she orders a cake." This can be represented as a conditional statement:

• Let H be the statement: "Sakshi hears it's someone's birthday."
• Let C be the statement: "Sakshi orders a cake."

The main statement is logically equivalent to "If H is true, then C is true," which is written in propositional logic as $\text{H} \rightarrow \text{C}$.

Now let's look at the given statements:

• A: Sakshi heard of someone's birthday ($\text{H}$)
• B: Sakshi did not hear of someone's birthday ($\neg\text{H}$)
• C: Sakshi ordered a cake ($\text{C}$)
• D: Sakshi did not order a cake ($\neg\text{C}$)

We need to find an ordered pair of statements (from the options) that can both be true simultaneously, given that the main statement ($\text{H} \rightarrow \text{C}$) is true.

Analyzing the Consistent Pair (D, B)

Let's examine the ordered pair (D, B) from Option 1. This pair consists of Statement D followed by Statement B:

• Statement D: Sakshi did not order a cake ($\neg\text{C}$)
• Statement B: Sakshi did not hear of someone's birthday ($\neg\text{H}$)

We need to check if it's possible for both $\neg\text{C}$ and $\neg\text{H}$ to be true at the same time that the main statement $\text{H} \rightarrow \text{C}$ is true.

Recall the truth table for a conditional statement $\text{P} \rightarrow \text{Q}$:

In our case, P is H and Q is C. The main statement $\text{H} \rightarrow \text{C}$ is true in the first, third, and fourth rows.

We are checking if $\neg\text{C}$ (C is False) and $\neg\text{H}$ (H is False) can both be true. Look at the fourth row of the truth table:

• H is False ($\neg\text{H}$ is True)
• C is False ($\neg\text{C}$ is True)
• $\text{H} \rightarrow \text{C}$ is True

This row shows a scenario where both $\neg\text{H}$ and $\neg\text{C}$ are true, and the main statement $\text{H} \rightarrow \text{C}$ is also true. Therefore, the pair of statements (D, B) is logically consistent with the main statement.

Another way to think about this is using the contrapositive. The statement $\text{H} \rightarrow \text{C}$ ("If she hears, she orders") is logically equivalent to its contrapositive $\neg\text{C} \rightarrow \neg\text{H}$ ("If she did not order a cake, then she did not hear it was a birthday"). If the main statement is true, its contrapositive is also true. The pair (D, B) states that $\neg\text{C}$ is true and $\neg\text{H}$ is true. If $\neg\text{C}$ is true, then according to the contrapositive $\neg\text{C} \rightarrow \neg\text{H}$, $\neg\text{H}$ must follow. So, if $\neg\text{C}$ is true, then $\neg\text{H}$ must also be true in any situation where the main statement holds. The possibility that both $\neg\text{C}$ and $\neg\text{H}$ are true is entirely consistent with the main statement.

Analyzing Other Options for Consistency

• Option 3: AB (H, $\neg$H)

  This pair consists of Statement A (H) and Statement B ($\neg$H). It is impossible for a statement and its negation to both be true at the same time. H and $\neg$H are contradictory. This pair is logically inconsistent.

• Option 4: CA (C, H)

  This pair consists of Statement C (C) and Statement A (H). Is it possible for C and H to both be true when $\text{H} \rightarrow \text{C}$ is true? Yes. The first row of the truth table shows H is True, C is True, and $\text{H} \rightarrow \text{C}$ is True. So, this pair is also logically consistent.

• Option 2: BD ($\neg$H, $\neg$C)

  This pair consists of Statement B ($\neg$H) and Statement D ($\neg$C). This is the same set of statements as in Option 1 (D, B), just in a different order. As shown when analyzing (D, B), the scenario where $\neg\text{H}$ and $\neg\text{C}$ are both true is consistent with $\text{H} \rightarrow \text{C}$ (the fourth row of the truth table). So, this pair is also logically consistent.

Based on standard logical consistency, both (D, B), (B, D), and (C, A) are consistent with the main statement. However, since only one option is given as correct, and Option 1 (DB) is the indicated answer, the primary focus should be on explaining why (D, B) is consistent, as detailed above using the truth table and the contrapositive relationship.

Revision Table: Key Logic Concepts

Additional Information on Conditional Logic

Understanding conditional statements is fundamental in logic. The statement "If P then Q" does not mean P causes Q, only that in any case where P is true, Q must also be true for the statement to hold. If P is false, the statement "If P then Q" is always considered true, regardless of whether Q is true or false. This can sometimes seem counterintuitive in natural language but is a defined rule in propositional logic. This rule is why the scenarios where Sakshi does not hear ($\neg\text{H}$) are consistent with the main statement regardless of whether she orders a cake or not.

The equivalence between a conditional statement and its contrapositive is a very useful rule. $\text{H} \rightarrow \text{C}$ and $\neg\text{C} \rightarrow \neg\text{H}$ always have the same truth value. If it's true that "If she hears, she orders," then it must also be true that "If she didn't order, she didn't hear." This second form directly supports why the pair (D, B) ($\neg\text{C}$, $\neg\text{H}$) is consistent, as $\neg\text{C}$ necessitates $\neg\text{H}$ if the main statement is true.