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Chapter 3: Problem 84

Write each compound statement in symbolic form. Let letters assigned to the simple statements represent English sentences that are not negated. If commas do not appear in compound English statements, use the dominance of connectives to show grouping symbols (parentheses) in symbolic statements. The lines go down, or if the transformer blows then we do not have power.

Short Answer

Expert verified
The symbolic form of the statement 'The lines go down, or if the transformer blows then we do not have power' is \(p \vee (q \rightarrow r)\).

Step by step solution

01

Identifying The Simple Statements

The simple statements here can be identified as: 'The lines go down' and 'The transformer blows' and 'We do not have power'. We can symbolize these as p, q, and r respectively
02

Determining The Connectives And Grouping

In the compound statement given - 'The lines go down, or if the transformer blows then we do not have power' - we notice the use of the 'or' connective and 'if...then'. According to dominance of connectives, 'if...then' takes precedence over 'or' hence it must be within parentheses.
03

Writing the Symbolic Form

After identifying the simple statements and logical connectives, the symbolic form will be: p or (if q then r). In symbolic representation, 'or' is often symbolized as 鈭�, 'if...then' as 鈫�. So the symbolic form becomes: \(p \vee (q \rightarrow r)\).

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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 used Euler diagrams to determine that an argument is valid, but when I reverse one of the premises and the conclusion, this new argument is invalid.

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 all people obey the law, then no jails are needed. Some people do not obey the law. \(\therefore\) Some jails are needed.

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders are insects. Therefore, no spiders have six legs.

Translate the argument below into symbolic form. Then use a truth table to determine if the argument is valid or invalid. It's wrong to smoke in public if secondary cigarette smoke is a health threat. If secondary cigarette smoke were not a health threat, the American Lung Association would not say that it is. The American Lung Association says that secondary cigarette smoke is a health threat. Therefore, it's wrong to smoke in public.

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 a metrorail system is not in operation, there are traffic delays. Over the past year there have been no traffic delays. \(\therefore\) Over the past year a metrorail system has been in operation.
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