Question

\(p \Rightarrow(q \Rightarrow p) \Rightarrow r\) is (a) Contradiction (b) tautology (c) Neither contradiction Nor tautology (d) Both contradiction \& tautology

Short answer

(c) Neither contradiction Nor tautology

Step-by-step solution

Setting up the truth table

To evaluate the truth of the given statement, we have to consider all possible values of \(p\), \(q\), and \(r\). We will create the following columns: 1. \(p\) 2. \(q\) 3. \(r\) 4. \(q \Rightarrow p\) 5. \((q \Rightarrow p) \Rightarrow r\) 6. \(p \Rightarrow ((q \Rightarrow p) \Rightarrow r)\) Now, we will fill up the table with all possible values of \(p\), \(q\), and \(r\).

Filling in all possible values

Fill in the table with all possible combinations of \(p\), \(q\), and \(r\). There are 2 possible values (True or False) for each variable, so there will be \(2^3 = 8\) rows in the table. $$ \begin{array}{ccc|c|c|c} p & q & r & q \Rightarrow p & (q \Rightarrow p) \Rightarrow r & p \Rightarrow ((q \Rightarrow p) \Rightarrow r)\\ \hline T & T & T & T & T & T \\ T & T & F & T & F & F \\ T & F & T & T & T & T \\ T & F & F & T & F & F \\ F & T & T & F & T & T \\ F & T & F & F & F & T \\ F & F & T & T & T & T \\ F & F & F & T & F & T \\ \end{array} $$

Analyzing the truth table

Now that we have the truth table completed, let's analyze the final column (the statement \(p \Rightarrow ((q \Rightarrow p) \Rightarrow r)\)). If every entry in the column is true, then the statement would be a tautology. If every entry is false, it's a contradiction. If there's a mix of true and false values, it's neither. In our case, we have a mix of true and false values in the final column. Therefore, the statement is neither a contradiction nor a tautology. #Answer#: (c) Neither contradiction Nor tautology

Key concepts

Truth Table

A truth table is a powerful tool used in propositional logic to analyze the truth value of complex logical expressions based on their simplest components. Each row of a truth table represents a unique combination of truth values for all the components involved, allowing us to see how these combinations affect the truth value of the overall expression.

To construct a truth table, follow these steps:

• Identify all the propositional variables in the expression.
• List all possible truth value combinations for these variables. If there are three variables, such as \( p, q \), and \( r \), you should have \( 2^3 = 8 \) rows because each variable can be either true (T) or false (F).
• Calculate the truth values of any intermediate expressions, using logical rules, for each combination of variable truth values.
• Finally, determine the truth value of the entire expression by considering the truth values of its components for each row.

The completed truth table allows you to determine whether a complicated expression is a tautology, contradiction, or neither.

Tautology

A tautology in propositional logic is a statement that is always true, regardless of the truth values of its individual components. This means if you make a truth table for the statement, the final column in the table will be all true (T) entries.

Tautologies are important in logic because they represent universally true propositions. They can help simplify logical expressions and prove logical equivalencies. Common examples of tautologies include statements like "\( p \lor eg p \)" (a statement is either true or its negation is true) and "\( p \Rightarrow p \)" (a statement implies itself).

Identifying a tautology involves:

• Creating a truth table for the statement.
• Checking whether all of the resulting truth values in the final column are true.

If they are, you have a tautology!

Contradiction

A contradiction is the opposite of a tautology; it is a statement that is always false, regardless of the values of its component propositions. When you look at the truth table for a contradictory statement, the final column will consist entirely of false (F) entries.

Contradictions are not useful in practice, but they help in theoretical logic to define inconsistent states or conditions that cannot logically occur. For example, the statement "\( p \land eg p \)" (a statement and its negation are both true) is a classic contradiction, as it can never be true.

• To check for contradiction, you construct a truth table for the proposition.
• If all entries in the result column are false, the proposition is a contradiction.

Such propositions help define the boundaries of logical systems.

Logical Implication

Logical implication, often symbolized as \( \Rightarrow \), is a fundamental concept in propositional logic. It describes a relationship between two statements, where the truth of one (the consequent) depends on the truth of another (the antecedent).

In the expression \( p \Rightarrow q \), \( p \) is the antecedent and \( q \) is the consequent. This expression is considered false only when \( p \) is true and \( q \) is false. In all other scenarios, \( p \Rightarrow q \) is true.

Logical implications are essential when examining conditional statements, and they form the basis for logical reasoning.

• If you encounter an implication like \( (q \Rightarrow p) \Rightarrow r \), it illustrates a more complex chain of conditions, requiring careful analysis.
• Resolving implications usually involves considering all possible truth combinations in a truth table, ensuring accurate logical deductions.

Understanding logical implication allows you to see how different statements can influence each other's truth.