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Chapter 3: Problem 14

Let \(p\) and \(q\) represent the following simple statements: \(p\) : This is an alligator. \(q\) : This is a reptile. Write each compound statement in symbolic form. If this is not a reptile, then this is not an alligator.

Short Answer

Expert verified
The compound statement 'If this is not a reptile, then this is not an alligator.' in symbolic form is \(\sim q \rightarrow \sim p\).

Step by step solution

01

Identify the simple statements

The simple statements are \(p\) : This is an alligator and \(q\) : This is a reptile.
02

Understand the compound statement

The compound statement is 'If this is not a reptile, then this is not an alligator.' We need to represent this using negations and implications.
03

Translate the compound statement into symbolic form

According the the command of logic, the compound command 'If this is not a reptile, then this is not an alligator' can be translated to symbolic form as 'If not \(q\), then not \(p\)', which is denoted as \(\sim q \rightarrow \sim p\)

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Key Concepts

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

Compound Statements in Symbolic Logic
In symbolic logic, compound statements are formed by combining two or more simple statements using logical connectives. Simple statements are basic assertions that do not contain any other statements; for example, "This is an alligator" and "This is a reptile." The compound statement combines these simple statements into a more complex proposition.

Common logical connectives used to form compound statements include:
  • Conjunctions (\( p \land q \)), which express "and"
  • Disjunctions (\( p \lor q \)), which stand for "or"
  • Conditionals or implications (\( p \rightarrow q \)), which relate to "if... then..."
Compound statements allow us to express more complex logical relationships and conditions by using these connectors.
Understanding Negations in Logic
A negation in logic simply reverses the truth value of a statement. If a statement is true, its negation is false, and vice versa. In symbolic form, negation is denoted by the symbol "\sim ".

For instance, if the statement is "This is a reptile" (represented by \( q \)), its negation "This is not a reptile" would be symbolized as \( \sim q \). Understanding negation is crucial as it helps to express statements like "not," "no," or "never" in logic.

Negations are significant when working with implications and other compound statements because they affect the truth conditions of the entire expression.
Implications in Logical Statements
Implications, also known as conditional statements, are pivotal in logic as they describe a cause-and-effect relationship between two statements. In abstract terms, an implication can be read as "If the first statement is true, then the second statement is also true." This logical relationship is symbolized as \( p \rightarrow q \).

Using our example: "If this is not a reptile, then this is not an alligator," can be transformed into an implication: "If \( \sim q \), then \( \sim p \)." This suggests that the truth of "This is not an alligator" depends on the truth of "This is not a reptile."
  • "Antecedent" - the first part of the implication (\( \sim q \))
  • "Consequent" - the result that follows (\( \sim p \))
Understanding implications is essential for interpreting and constructing logical arguments.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagrams did not illustrate the conclusion, so the argument is invalid.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of 3 . Therefore, 8 is not a multiple of 6 .

Use Euler diagrams to determine whether each argument is valid or invalid. All physicists arc scientists. All scientists attended college. Therefore, all physicists attended college.

Based on the following argument by conservative radio talk show host Rush Limbaugh and directed at former vice president Al Gore. You would think that if \(\mathrm{Al}\) Gore and company believe so passionately in their environmental crusading that [sic] they would first put these ideas to work in their own lives, right? ... Al Gore thinks the automobile is one of the greatest threats to the planet, but he sure as heck still travels in one of them - a gas guzzler too. (See, I Told You So, p. 168) Limbaugh's passage can be expressed in the form of an argument: If Gore really believed that the automobile were a threat to the planet, he would not travel in a gas guzzler. Gore does travel in a gas guzzler. Therefore, Gore does not really believe that the automobile is a threat to the planet. Use Limbaugh's argument to determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to avoid a long truth table and instead use a standard form of an argument, I tested the validity of Limbaugh's argument using the following representations: \(p\) : Gore really believes that the automobile is a threat to the planet. \(q:\) He does not travel in a gas guzzler.

Conservative commentator Rush Limbaugh directed this passage at liberals and the way they think about crime. Of course, liberals will argue that these actions [contemporary youth crime] can be laid at the foot of socioeconomic inequities, or poverty. However, the Great Depression caused a level of poverty unknown to exist in America today, and yet I have been unable to find any accounts of crime waves sweeping our large cities. Let the liberals chew on that. (See, I Told You So, p. 83) Limbaugh's passage can be expressed in the form of an argument: If poverty causes crime, then crime waves would have swept American cities during the Great Depression. Crime waves did not sweep American cities during the Great Depression. \(\therefore\) Poverty does not cause crime. (Liberals are wrong.) Translate this argument into symbolic form and determine whether it is valid or invalid.
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