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Chapter 1: Problem 5

Use rules of inference to show that the hypotheses 鈥淩andy works hard,鈥� 鈥淚f Randy works hard, then he is a dull boy,鈥� and 鈥淚f Randy is a dull boy, then he will not get the job鈥� imply the conclusion 鈥淩andy will not get the job.鈥�

Short Answer

Expert verified
By applying Modus Ponens twice, we conclude that Randy will not get the job.

Step by step solution

01

Identify the Hypotheses

Express the given hypotheses in symbolic form: Let鈥檚 denote the statements as follows: 1. 'Randy works hard' as \(P\)2. 'If Randy works hard, then he is a dull boy' as \(P \rightarrow Q\)3. 'If Randy is a dull boy, then he will not get the job' as \(Q \rightarrow R\)
02

Use Modus Ponens on Hypothesis 1 and 2

Modus Ponens states that if \(P\) and \(P \rightarrow Q\) are true, then \(Q\) must be true. So from \(P\) and \(P \rightarrow Q\), we can infer \(Q\):\[P\] \[P \rightarrow Q\] \[\therefore Q\]
03

Apply Modus Ponens again using Step 2 result

Using the result \(Q\) from the previous step and the third hypothesis \(Q \rightarrow R\), apply Modus Ponens again to infer \(R\). \[Q\] \[Q \rightarrow R\] \[\therefore R\]Since \(R\) corresponds to 'Randy will not get the job', the conclusion is 'Randy will not get the job'.

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Key Concepts

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

Modus Ponens
Modus Ponens is an important rule of inference in logic. It is often used in mathematical proofs and logical reasoning. The rule states:

If we have two statements:
  • Statement 1: A premise, let's call it P.
  • Statement 2: A conditional statement, P implies Q (symbolically, this is written as \(P \rightarrow Q\)).

If both of these statements are true, then we can conclude that Q must also be true. This is written symbolically as:

\[ P, \ P \rightarrow Q \ \therefore Q \]

To better understand this, consider the example provided in the exercise. We know that Randy works hard (\
Logical Implication
Logical implication is a fundamental concept in symbolic logic. It connects two statements with an 鈥渋f-then鈥� relationship.

In logical terms, if statement P implies statement Q, it means that if P is true, then Q must also be true. This is symbolically written as \(P \rightarrow Q\).

In our exercise example, the hypotheses include logical implications such as:
  • \
Symbolic Logic
Symbolic logic uses symbols to represent logical statements and their relationships. This makes complex logical expressions easier to read and manipulate.

In the exercise, the statements are represented using symbols like P, Q, and R. This notation helps to clearly see the logical structure and apply rules of inference like modus ponens.

For instance, the exercise translates the statements into symbolic form:
  • Randy works hard is represented as P.
  • If Randy works hard, then he is a dull boy is represented as \(P \rightarrow Q\).

  • If Randy is a dull boy, then he will not get the job is represented as \(Q \rightarrow R\).


With these symbols, the steps in the solution become straightforward.

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Most popular questions from this chapter

Show that the equivalence \(p \wedge \neg p \equiv \mathbf{F}\) can be derived using resolution together with the fact that a conditional statement with a false hypothesis is true. [Hint: Let \(q=\) \(r=\mathbf{F}\) in resolution.

Determine the truth value of the statement \(\exists x \forall y\left(x \leq y^{2}\right)\) if the domain for the variables consists of a) the positive real numbers. b) the integers. c) the nonzero real numbers.

Show that \(\forall x P(x) \vee \forall x Q(x)\) and \(\forall x(P(x) \vee Q(x))\) are not logically equivalent.

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) No one has lost more than one thousand dollars playing the lottery. b) There is a student in this class who has chatted with exactly one other student. c) No student in this class has sent e-mail to exactly two other students in this class. d) Some student has solved every exercise in this book. e) No student has solved at least one exercise in every section of this book.

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.
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