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Chapter 3: Problem 15

Select the statement that is equivalent to I saw the original King Kong or the 2005 version. a. If I did not see the original King Kong, I saw the 2005 version. b. I saw both the original King Kong and the 2005 version. c. If I saw the original King Kong, I did not see the 2005 version. d. If I saw the 2005 version, I did not see the original King Kong.

Short Answer

Expert verified
The statement that is equivalent to the original statement 'I saw the original King Kong or the 2005 version' is option a: 'If I did not see the original King Kong, I saw the 2005 version.'

Step by step solution

01

Analysis of Option a

Look at option a: 'If I did not see the original King Kong, I saw the 2005 version.' This statement is the logical construct of an implication, where the second event (seeing the 2005 version) occurs if the first event (not seeing the original King Kong) is true. This mirrors the original statement because in both cases, at least one version of the movie is seen.
02

Analysis of Options b, c and d

Now look at options b, c, and d. Option b: 'I saw both the original King Kong and the 2005 version' suggests both events occurred, which is a logical conjunction, not disjunction. It could be untrue even if the original statement is true. Therefore, it is not equivalent. For option c: 'If I saw the original King Kong, I did not see the 2005 version', this implies a logical construct that contradicts the original statement. Option c suggests that both events cannot occur simultaneously, whereas the original statement doesn't exclude this possibility. Option d: 'If I saw the 2005 version, I did not see the original King Kong' presents the same contradiction as option c. Therefore, options b, c, and d are not equivalent to the original statement.
03

Selection of the Equivalent Statement

After analyzing all the options, it is clear that option a: 'If I did not see the original King Kong, I saw the 2005 version' is the only statement equivalent to the original statement: 'I saw the original King Kong or the 2005 version.' This is because it preserves the logical construct of a disjunction, where at least one of the two events must occur.

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Key Concepts

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

Disjunction
A disjunction is a fundamental concept in logic that refers to the use of the word "or" to combine two statements. In the statement, "I saw the original King Kong or the 2005 version," the disjunction implies that at least one of these events must be true. This means you could have seen either one of the versions, or even both, for the statement to hold true.

Disjunctions are represented symbolically using the logical "or," expressed in logic notation as \( p \lor q \), where \( p \) and \( q \) are separate statements. A disjunction is true if at least one of the components \( p \) or \( q \) is true. This makes it a very flexible logical construct, allowing for multiple possible truths without exclusion.
Logical Implication
Logical implication is another key concept in logic. It involves a situation where the truth of one statement guarantees the truth of another. This is denoted with an "if...then..." structure. For example, "If I did not see the original King Kong, then I saw the 2005 version."

This statement can be expressed symbolically as \( eg p \rightarrow q \), which reads "if not \( p \), then \( q \)." Here, \( eg p \) signifies the negation of the first statement, and \( q \) stands for the second statement. A logical implication is true in all scenarios except when the first statement is true and the second is false.
  • This means if the original movie wasn't seen, the implication dictates that the 2005 version must have been.
  • This maintains logical consistency with a disjunction due to the inclusive nature of "or."
Equivalence Analysis
Analyzing logical equivalence involves identifying statements that reveal the same truth regardless of how they are expressed. In our problem, 鈥淚 saw the original King Kong or the 2005 version" needs to be equivalent to the options provided.

In the analysis:
  • Option a, "If I did not see the original King Kong, then I saw the 2005 version," accurately captures the essence of the original disjunction. This logical implication implies at least one version is seen, aligning with the original statement's meaning.
  • Option b suggests both movies were seen, representing a conjunction, which is not equivalent because the original disjunction allows for seeing just one version.
  • Options c and d imply exclusivity, which contradicts the original inclusive disjunction.
Logical equivalence requires identifying the transformation or reinterpretation of a statement that retains the original truth values under all circumstances.
Logical Constructs
Logical constructs are the foundational building blocks of logical reasoning and analysis. In the given exercise, the main constructs include disjunctions and implications.

  • Disjunction allows for multiple possibilities, accommodating flexibility and inclusiveness in logical statements.
  • Implication introduces a cause-effect relationship between statements, where one being true influences or necessitates the truth of another.
Understanding these logical constructs allows us to determine truth values across various permutations of statements. This is crucial when analyzing complex logical expressions or arguments in mathematics or computer science.

By mastering logical constructs, like those demonstrated in the problem, one becomes adept at breaking down and interpreting intricate information, which is a valuable skill across numerous disciplines.

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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 am tired or hungry, I cannot concentrate. I cannot concentrate. \(\therefore\) I am tired or hungry.

Determine whether each argument is valid or invalid. All \(A\) are \(B\), no \(C\) are \(B\), and all \(D\) are \(C\). Thus, no \(A\) are \(D\).

In this section, we used a variety of examples, including arguments from the Menendez trial, the inevitability of Nixon's impeachment, and Spock's (fallacious) logic on Star Trek, 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 Euler diagrams to determine whether each argument is valid or invalid. All dancers are athletes. Savion Glover is an athlete. Therefore, Savion Glover is a dancer.

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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