Problem 44 - Thinking Mathematically

Question

Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \(\sim(p \vee q) \leftrightarrow(\sim p \wedge \sim q)\)

Step-by-Step Solution

Answer

The logical statement \(\sim(p \vee q) \leftrightarrow(\sim p \wedge \sim q)\) is a tautology. As the truth table reveals, regardless of the truth values of \(p\) and \(q\), the outcome of the entire statement is always true.

Step 1: Setting up the truth table

Start by setting up a truth table with a row for every combination of truth values for variables \(p\) and \(q\). The variables \(p\) and \(q\) can each be true or false, so there will be \(2^2 = 4\) rows in the table. List all combinations in the rows.

Step 2: Calculate negations

Next, find the truth values of \(\sim p\) and \(\sim q\) for each row: the negation of a statement is true if the original statement is false and vice versa.

Step 3: Calculate disjunction and conjunction

Now figure out the truth values of \(p \vee q\) and \(\sim p \wedge \sim q\) for each row.

Step 4: Calculate biconditional statement

Next, calculate the truth values of the complete statement \(\sim(p \vee q) \leftrightarrow(\sim p \wedge \sim q)\) for each row. This is true only if both \(p \vee q \rightarrow false\) and \(\sim p \wedge \sim q \rightarrow false\), or when \(p \vee q \rightarrow true\) and \(\sim p \wedge \sim q \rightarrow true\) i.e. when both sides of \(\leftrightarrow\) are equivalent.

Step 5: Determine nature of statement

Lastly, determine whether the statement is a tautology, contradiction, or neither. If every row of its truth table results in 'true', it's a tautology; if no row results in 'true', it's a contradiction, and if some rows result in 'true', and some in 'false', it is neither

Key Concepts

Tautology

Understanding the concept of a tautology is fundamental in logic and critical thinking. A tautology is a logical statement that is always true, regardless of the truth values of its components. This means that when you assess the statement under all possible scenarios, it produces a truth value of 'true'. It's akin to saying 'it will either rain or not rain tomorrow,' which is an undisputable fact because there are no other possibilities.

To recognize a tautology, you can use a truth table to evaluate the statement with every possible combination of truth values for its constituent variables. If every outcome in the truth table is 'true', then you have a tautology. This concept is not just a theoretical construct; it has practical implications, particularly in mathematics and computer science, where tautologies are used to ensure that logical arguments are foolproof and error-free.

Logical Contradiction

In stark contrast to a tautology, a logical contradiction is a statement that is never true, no matter what the circumstances are. It's an assertion that inherently contains a conflict, like saying 'I am lying right now,' which creates a paradox. If you told the truth about lying, then you're not lying, but if you're lying about lying, then you're telling the truth - and around we go.

To illustrate a contradiction with a truth table, none of the possible combinations of truth values for its variables will result in a 'true' outcome. Each instance will yield 'false', demonstrating that the statement cannot ever be true under any circumstances. Recognizing such contradictions is vital, as it allows us to discard invalid arguments and focus on constructing logical, consistent theories.

Biconditional Statement

A biconditional statement is a bit like a two-way street in logic. It represents a relationship between two statements where either both are true, or both are false. This can be phrased as 'if and only if', which signifies that the two statements are linked in such a way that one guarantees the other, and vice versa.

For example, a biconditional statement could be 'You can drive this car if and only if you have a valid driver's license.' In this case, having a license ensures the right to drive, and driving the car implies possession of that license. In a truth table, a biconditional statement is true when both parts have matching truth values. If one part is true and the other part is false, or vice versa, the biconditional statement is false. Symbolically, this is represented as 'P ↔ Q'. It is vital for articulating precise conditions and defining equivalences in logical operations and reasoning.