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Chapter 3: Problem 91

Explain when conditional statements are true and when they are false.

Short Answer

Expert verified
In a conditional statement structured as 'If P, then Q,' the statement is true when either the hypothesis (P) is false or the conclusion (Q) is true. The statement is false only when a true hypothesis leads to a false conclusion.

Step by step solution

01

Introduction to Conditional Statements

To start, conditional statements are a type of logic used in mathematics, philosophy and computer programming, among others. A typical format is 'If P, then Q', where P is a hypothesis and Q is a conclusion. This statement is true unless a true hypothesis leads to a false conclusion.
02

True Conditions

A conditional statement is true if the if part (the hypothesis) is false or the then part (the conclusion) is true. For example, consider the statement 'If it is raining, then the ground is wet.' If it is not raining (false hypothesis), the statement is true regardless of whether the ground is wet or not. Additionally, if it is raining (true hypothesis) and the ground is wet (true conclusion), the statement is also true.
03

False Conditions

A conditional statement is false only when a true hypothesis leads to a false conclusion. In the previous example, the statement would be false if it is raining (true hypothesis) but the ground is not wet (false conclusion).

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I watch Schindler's List and Milk, I am aware of the destructive nature of intolerance. Today I did not watch Schindler's List or I did not watch Milk. \(\therefore\) Today I am not aware of the destructive nature of intolerance.

Determine whether each argument is valid or invalid. All \(A\) are \(B\), all \(B\) are \(C\), and all \(C\) are \(D\). Thus, all \(A\) are \(D\).

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If I am a full-time student, I cannot work. If I cannot work, I cannot afford a rental apartment costing more than \(\$ 500\) per month. Therefore, ...

Use Euler diagrams to determine whether each argument is valid or invalid. All professors are wise people. Some wise people are actors. Therefore, some professors are actors.

In Exercises 25-36, determine whether each argument is valid or invalid. All natural numbers are whole numbers, all whole numbers are integers, and \(-4006\) is not a whole number. Thus, \(-4006\) is not an integer.
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