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Chapter 3: Problem 28

Construct a truth table for the given statement. \((p \rightarrow q) \leftrightarrow \sim r\)

Short Answer

Expert verified
The truth table is as follows: For p=T, q=T, r=T: \((p \rightarrow q) \leftrightarrow \sim r = F\). For p=T, q=T, r=F: \((p \rightarrow q) \leftrightarrow \sim r = T\). For p=T, q=F, r=T: \((p \rightarrow q) \leftrightarrow \sim r = F\). For p=T, q=F, r=F: \((p \rightarrow q) \leftrightarrow \sim r = F\). For p=F, q=T, r=T: \((p \rightarrow q) \leftrightarrow \sim r = F\). For p=F, q=T, r=F: \((p \rightarrow q) \leftrightarrow \sim r = T\). For p=F, q=F, r=T: \((p \rightarrow q) \leftrightarrow \sim r = F\). For p=F, q=F, r=F: \((p \rightarrow q) \leftrightarrow \sim r = T\).

Step by step solution

01

Define the Variables

Firstly, the propositional variables need to be defined to construct the truth table. Here, the variables are p, q, and r. Define the possible values that these variables can take, i.e., either True (T) or False (F). Generate all possible combinations of these variables.
02

Calculate \(p \rightarrow q\)

Next, calculate the values for the implication logical operator \(p \rightarrow q\). This operator is defined as follows: if p is true and q is true, then \(p \rightarrow q\) is true. If p is true and q is false, then \(p \rightarrow q\) is false. If p is false, then \(p \rightarrow q\) is always true, regardless of the truth value of q.
03

Calculate \(\sim r\)

Now calculate the values for the negation \(\sim r\). This operator simply inverts the truth value of r. If r is true \(\sim r\) is false, and if r is false, then \(\sim r\) is true.
04

Calculate \((p \rightarrow q) \leftrightarrow \sim r\)

Finally, calculate the values for the compound logical statement \((p \rightarrow q) \leftrightarrow \sim r\). This operator is defined as follows: If both sides share the same truth value, then the biconditional is true. If one side is true and the other is false, then the biconditional is false.

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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 tell you I cheated, I'm miserable. If I don't tell you I cheated, I'm miserable. \(\therefore\) I'm miserable.

Write a valid argument on one of the following questions. If you can, write valid arguments on both sides. a. Should the death penalty be abolished? b. Should Roe v. Wade be overturned? c. Are online classes a good idea? d. Should marijuana be legalized? e. Should grades be abolished? f. Should same-sex marriage be legalized?

In the Sixth Meditation, Descartes writes I first take notice here that there is a great difference between the mind and the body, in that the body, from its nature, is always divisible and the mind is completely indivisible. Descartes's argument can be expressed as follows: All bodies are divisible. No minds are divisible. Therefore, no minds are bodies. Use an Euler diagram to determine whether the argument is valid or invalid.

In this section, we used a variety of examples, including arguments from the Menendez trial, the inevitability of Nixon's impeachment, Spock's (fallacious) logic on Star Trek, and even two cartoons, to illustrate symbolic arguments. a. From any source that is of particular interest to you (these can be the words of someone you truly admire or a person who really gets under your skin), select a paragraph or two in which the writer argues a particular point. (An intriguing source is What Is Your Dangerous Idea?, edited by John Brockman, published by Harper Perennial, 2007.) Rewrite the reasoning in the form of an argument using words. Then translate the argument into symbolic form and use a truth table to determine if it is valid or invalid. b. Each group member should share the selected passage with other people in the group. Explain how it was expressed in argument form. Then tell why the argument is valid or invalid.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &(p \rightarrow q) \wedge(q \rightarrow p) \\ &\frac{p}{\therefore p \vee q} \end{aligned} $$
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