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Chapter 5: Problem 36

"If \(\mathrm{x}\) is a good actor, then \(\mathrm{y}\) is bad actress" is (1) a tautology (2) a contradiction (3) a contingency (4) None of these

Short Answer

Expert verified
Answer: The statement is a contingency.

Step by step solution

01

Rewriting the statement using logical symbols

The given statement can be rewritten as a logical implication "x 鈫� y", where x represents the statement "x is a good actor" and y represents the statement "y is a bad actress".
02

Creating a truth table for the implication

To determine the property of this statement, we need to create a truth table for the logical implication "x 鈫� y". The truth table looks like this: x | y | x 鈫� y --- | --- | ----- T | T | T T | F | F F | T | T F | F | T
03

Analyzing the truth table

Looking at the truth table, we can see that the statement "x 鈫� y" (If x is a good actor, then y is a bad actress) can be true or false, depending on the truth values of x and y.
04

Identifying the type of statement

Since the statement can be either true or false based on the values of x and y, it is not a tautology or a contradiction. A tautology is a statement that is always true, and a contradiction is a statement that is always false. Therefore, the given statement is a contingency, which is a statement that can be true or false depending on the truth values of its components. So the correct option is: (3) a contingency

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Implication
Logical implication is a fundamental concept in logic. It describes a relationship between two statements or propositions. In our example, "If \( x \) is a good actor, then \( y \) is a bad actress" can be written as \( x \to y \). This means if \( x \) is true, \( y \) must also be true for the implication to hold.

However, if \( x \) is false, the implication \( x \to y \) is considered true, regardless of whether \( y \) is true or false. This might seem counterintuitive, but it follows the rules of logical implication. This relationship is best understood through a truth table, which visually represents the possible outcomes.
Truth Table
A truth table helps us determine the truth value of a logical proposition under different conditions. It is a systematic way to explore all possible combinations of truth values for statements.

To create a truth table for \( x \to y \):
  • List all possible truth values for \( x \) and \( y \).
  • Apply the rules of implication to find \( x \to y \).
Here, when \( x \) is true and \( y \) is false, \( x \to y \) is false. In all other cases, \( x \to y \) is true. Thus, the truth table shows the logical behavior of the implication.
Tautology
A tautology is a statement that is always true, regardless of the truth values of its components. For a logical proposition to be a tautology, every row in its truth table must evaluate to true.

In the exercise, the statement \( x \to y \) is not a tautology. The truth table contains a scenario (\( x = T \), \( y = F \)) where \( x \to y \) is false. This makes it clear that the statement does not meet the criteria for a tautology.
Contradiction
A contradiction is a statement that is always false, no matter the truth values of its individual parts. Essentially, every possible outcome in its truth table results in a false value.

For example, if we have a statement that is false in all scenarios, it is a contradiction. For the given \( x \to y \), there are scenarios where the statement is true, showing it is not a contradiction.
Contingency
A contingency is a logical statement that can be true or false depending on the truth values of its components. Unlike tautologies and contradictions, contingencies have a mix of true and false outcomes in their truth tables.

In the exercise, the statement "If \( x \) is a good actor, then \( y \) is a bad actress" is a contingency. This is because it can be true or false, depending on \( x \) and \( y \). Contingencies show the variability and practical applicability of logical statements in real-world situations.

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Most popular questions from this chapter

What are the truth values of \((\sim \mathrm{p} \Rightarrow \sim \mathrm{q})\) and \(\sim(\sim \mathrm{p} \Rightarrow \mathrm{q})\) respectively, when \(\mathrm{p}\) and \(\mathrm{q}\) always speak true in any argument? (1) \(\mathrm{T}, \mathrm{T}\) (2) \(\mathrm{F}, \mathrm{F}\) (3) \(\mathrm{T}, \mathrm{F}\) (4) \(\mathrm{F}, \mathrm{T}\)

If \(\mathrm{p}:\) In a triangle, the centroid divides each median in the ratio \(1: 2\) from the vertex and \(\mathrm{q}: \mathrm{In}\) an equilateral triangle, each median is perpendicular bisector of one of its sides. The truth values of inverse and converse of \(\mathrm{p} \Rightarrow \mathrm{q}\) are respectively (1) \(\mathrm{T}, \mathrm{T}\) (2) \(\mathrm{F}, \mathrm{F}\) (3) \(\mathrm{T}, \mathrm{F}\) (4) \(\mathrm{F}, \mathrm{T}\)

Which of the following is the negation of the statement, "All animals are carnivores"? (1) Some animals are not carnivores. (2) Not all animals are carnivores. (3) There is atleast one animal which is not carnivores. (4) All the above.

If \(\mathrm{p}\) is negation of \(\mathrm{q}\), then \((\mathrm{p} \Rightarrow \mathrm{q}) \vee(\mathrm{q} \Rightarrow \mathrm{p})\) is \(\mathrm{a}\) (1) tautology (2) contradiction (3) contingency (4) none of these

When does the value of the statement \((\mathrm{p} \wedge \mathrm{r}) \Leftrightarrow(\mathrm{r} \wedge \mathrm{q})\) become false? (1) \(\mathrm{p}\) is \(\mathrm{T}, \mathrm{q}\) is \(\mathrm{F}\) (2) \(\mathrm{p}\) is \(\mathrm{F}, \mathrm{r}\) is \(\mathrm{F}\) (3) \(\mathrm{p}\) is \(\mathrm{F}, \mathrm{q}\) is \(\mathrm{F}\) and \(\mathrm{r}\) is \(\mathrm{F}\) (4) None of these
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