Conditional and Biconditional Statements

Conditional Statement (Implication): "If p, then q" Form ($p \implies q$)

One of the most prevalent forms of reasoning in mathematics and everyday life involves cause and effect, or condition and consequence. Statements that express this relationship are called conditional statements or implications. They link two statements in such a way that the truth of the first somehow guarantees the truth of the second.

Definition of a Conditional Statement

A compound statement formed by combining two simple statements, say $p$ and $q$, with the phrase "if... then..." is defined as a conditional statement or an implication. It asserts that if the first statement ($p$) is true, then the second statement ($q$) must also be true.

• Symbols: The common symbols used to represent a conditional statement are $\implies$ or $\rightarrow$.
• Notation: The conditional statement "If $p$, then $q$" is written as $p \implies q$ or $p \rightarrow q$.
• Read as: "$p$ implies $q$", "If $p$, then $q$".

In the conditional statement $p \implies q$:

• The statement $p$ is called the hypothesis, the antecedent, or the premise. It is the condition.
• The statement $q$ is called the conclusion or the consequent. It is what is asserted to follow from the hypothesis.

So, $p \implies q$ means "If the hypothesis $p$ is true, then the conclusion $q$ must be true."

Alternative Ways to Express $p \implies q$

The structure "If $p$, then $q$" can be expressed in many grammatically different but logically equivalent ways in English. Understanding these variations is important for interpreting logical statements correctly. Here are some common ways to phrase $p \implies q$:

• If $p$, then $q$. (Standard form)
• $p$ implies $q$.
• $q$ if $p$. (Note the order reversal - $q$ is true if $p$ is true).
• $p$ only if $q$. (This means if $p$ is true, the ONLY way that is possible is if $q$ is also true. If $q$ were false, $p$ couldn't be true. This is equivalent to "If $p$, then $q$").
• $q$ is necessary for $p$. (For $p$ to be true, $q$ is required. If $q$ is false, $p$ cannot be true. This is equivalent to "If $p$, then $q$").
• $p$ is sufficient for $q$. (The truth of $p$ is enough to guarantee the truth of $q$. This is equivalent to "If $p$, then $q$").
• Whenever $p$ happens, $q$ happens.
• $q$ follows from $p$.
• $q$ provided $p$.

Recognising these different phrasings is vital when translating statements from natural language into logical symbols.

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Understanding the Meaning of "If-Then" Statements

The truth value of a conditional statement $p \implies q$ is not always intuitive, especially when the hypothesis $p$ is false. It's best understood as a formal definition of the implication relationship in logic, which models a specific type of guarantee or promise.

The "Promise" Analogy

Consider the statement as a promise: "If you meet condition $p$, then I guarantee $q$ will happen."

Let's analyze when this promise is kept (making the implication True) and when it is broken (making the implication False).

1. Case 1: Hypothesis True, Conclusion True ($p$=T, $q$=T)

   You met the condition ($p$ is True), and the guaranteed result happened ($q$ is True). The promise was kept.

   Example: Promise: "If you score 90%, I will buy you a bike." You scored 90% and I bought you a bike. (Promise kept).

   Result: T $\implies$ T is True.

2. Case 2: Hypothesis True, Conclusion False ($p$=T, $q$=F)

   You met the condition ($p$ is True), but the guaranteed result did NOT happen ($q$ is False). The promise was broken.

   Example: Promise: "If you score 90%, I will buy you a bike." You scored 90% but I did not buy you a bike. (Promise broken).

   Result: T $\implies$ F is False.

3. Case 3: Hypothesis False, Conclusion True ($p$=F, $q$=T)

   You did NOT meet the condition ($p$ is False). The promise was only about what would happen *if* you met the condition. Since you didn't meet the condition, the promise doesn't apply to this situation. Therefore, the promise cannot be considered broken, regardless of what happens with $q$. If $q$ happens anyway, the promise is still not broken.

   Example: Promise: "If you score 90%, I will buy you a bike." You scored 80% (didn't meet condition). I unexpectedly bought you a bike anyway. (Promise not broken).

   Result: F $\implies$ T is True.

4. Case 4: Hypothesis False, Conclusion False ($p$=F, $q$=F)

   You did NOT meet the condition ($p$ is False). The promise doesn't apply. What happens with $q$ is irrelevant to whether the promise was kept or broken based on the condition $p$. If $q$ also doesn't happen, the promise is still not broken.

   Example: Promise: "If you score 90%, I will buy you a bike." You scored 80% (didn't meet condition). I did not buy you a bike. (Promise not broken).

   Result: F $\implies$ F is True.

This truth condition, where an implication is true whenever its hypothesis is false (Cases 3 and 4), is sometimes called "vacuously true" or "true by default". It ensures logical consistency and avoids contradictions when dealing with statements whose conditions are not met. In logic, we define $p \implies q$ based purely on the truth values of $p$ and $q$, not on a causal connection or relevance between $p$ and $q$ in the real world.

Competitive Exam Pointer: Implications

Truth Values of Conditional Statements and their Truth Table

Based on the understanding of the meaning of "if... then..." statements discussed above, we can formally define the truth value of a conditional statement $p \implies q$ using a truth table.

Formal Truth Value Definition

The conditional statement $p \implies q$ is defined to be False if and only if the hypothesis ($p$) is True and the conclusion ($q$) is False. In all other scenarios, the conditional statement $p \implies q$ is defined to be True.

Truth Table for Conditional Statement ($\implies$)

For two simple statements $p$ and $q$, there are $2^2 = 4$ possible combinations of their truth values. The truth table for $p \implies q$ lists each combination and the resulting truth value of the conditional statement:

This table is the definitive rule for determining the truth value of any conditional statement $p \implies q$ given the truth values of its components $p$ and $q$. The only row where $p \implies q$ is False is the second row (T $\implies$ F).

Logical Equivalence: $p \implies q \equiv \sim p \lor q$

A significant logical equivalence states that a conditional statement $p \implies q$ is logically equivalent to the disjunction of "not $p$" and $q$, i.e., $\sim p \lor q$. Logical equivalence means that the two statements always have the same truth value for all possible truth values of their components. We can prove this by constructing the truth table for $\sim p \lor q$ and comparing it to the truth table for $p \implies q$.

Let's construct the truth table for $\sim p \lor q$:

Now, let's compare the final column of this table ($\sim p \lor q$) with the final column of the truth table for $p \implies q$.

Since the truth values in the columns for $p \implies q$ and $\sim p \lor q$ are identical for every combination of truth values of $p$ and $q$, we conclude that $p \implies q$ is logically equivalent to $\sim p \lor q$. This equivalence is very useful in logical proofs and when rewriting statements.

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Competitive Exam Pointer: Truth Table for Implication

Biconditional Statement: "p if and only if q" Form ($p \iff q$)

Beyond simple conditional statements, we often encounter situations in mathematics where two statements are so intrinsically linked that they are true or false together. This mutual dependence is expressed using a biconditional statement.

Definition of a Biconditional Statement

A compound statement formed by connecting two simple statements, $p$ and $q$, using the phrase "if and only if" is called a biconditional statement. It asserts that the truth of $p$ is tied directly to the truth of $q$ – they must have the same truth value for the biconditional to hold true.

• Symbols: The symbols used for a biconditional statement are $\iff$ or $\leftrightarrow$.
• Notation: The biconditional statement "$p$ if and only if $q$" is written as $p \iff q$ or $p \leftrightarrow q$.
• Read as: "$p$ if and only if $q$", "$p$ iff $q$", or "$p$ is equivalent to $q$".

The phrase "if and only if" signifies that the conditional relationship works in both directions. That is, "$p$ if and only if $q$" is logically equivalent to saying "(If $p$, then $q$) AND (If $q$, then $p$)" or $(p \implies q) \land (q \implies p)$. This conjunction of two implications is what defines the biconditional.

Alternative Phrasings for $p \iff q$

Like conditional statements, biconditionals can be expressed in several ways in English. Recognising these is important for translation into logical form:

• $p$ if and only if $q$. (Standard form)
• $p$ iff $q$. (Common abbreviation)
• $p$ is necessary and sufficient for $q$. (This captures the two implications: $p$ is sufficient for $q$ means $p \implies q$; $p$ is necessary for $q$ means $q \implies p$).
• $q$ is necessary and sufficient for $p$. (Equivalent to the above).
• $p$ is equivalent to $q$.
• $p$ holds precisely when $q$ holds.

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Truth Values of Biconditional Statements and their Truth Table

The truth value of a biconditional statement $p \iff q$ is determined by the truth values of its components $p$ and $q$. As the name "equivalence" suggests, the biconditional is true when $p$ and $q$ are equivalent in terms of their truth – they are either both true or both false.

Truth Value Definition for Biconditional

The biconditional statement $p \iff q$ is defined to be True if and only if $p$ and $q$ have the same truth value.

• If $p$ is True and $q$ is True, then $p \iff q$ is True.
• If $p$ is False and $q$ is False, then $p \iff q$ is True.

If $p$ and $q$ have different truth values (one is True and the other is False), then $p \iff q$ is defined to be False.

Truth Table for Biconditional Statement ($\iff$)

The truth table for the biconditional operator shows the resulting truth value of $p \iff q$ for all 4 possible combinations of truth values for $p$ and $q$.

This table clearly shows that $p \iff q$ is True only in the first row (T $\iff$ T) and the last row (F $\iff$ F), where the truth values of $p$ and $q$ match.

Equivalence to Conjunction of Implications

As mentioned in the definition, $p \iff q$ is logically equivalent to $(p \implies q) \land (q \implies p)$. We can demonstrate this equivalence using a truth table. If the truth values in the final columns for both compound statements are identical for all combinations of $p$ and $q$, they are logically equivalent.

Let's construct the truth table for $(p \implies q) \land (q \implies p)$: We need columns for $p$, $q$, $p \implies q$, $q \implies p$, and finally $(p \implies q) \land (q \implies p)$.

Comparing the final column of this table with the final column of the truth table for $p \iff q$ (shown above), we see that the truth values match exactly for all rows. This formally proves the logical equivalence: $p \iff q \equiv (p \implies q) \land (q \implies p)$.

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Competitive Exam Pointer: Conditional and Biconditional