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Chapter 3: Problem 94

Based on the meaning of the inclusive or, explain why it is reasonable that if \(p \vee q\) is true, then \(\sim p \rightarrow q\) must also be true.

Short Answer

Expert verified
In the context of logic, if the 'inclusive or' statement \(p \vee q\) is true, this means that either 'p' is true, 'q' is true, or both are true. If 'p' is false (\(\sim p\) is true), then 'q' must be true to make \(p \vee q\) true. Therefore, \(\sim p \rightarrow q\) is true because if we assume that 'p' is false, 'q' must be true, following the rules of logical implication, where the implication is false only if 'p' is true and 'q' is false.

Step by step solution

01

Defining 'Inclusive Or'

In logic, 'inclusive or' is represented by the symbol \(\vee\). The statement 'p \(\vee\) q' is true if 'p' is true, or 'q' is true, or both are true.
02

Understanding the Implication

A logical implication 'p \(\rightarrow\) q' is false only when 'p' is true and 'q' is false. In all other cases, it's true.
03

Connecting the 'Inclusive or' to the Implication

If \(p \vee q\) is true, then there are three possibilities: 'p' is true, 'q' is true, or both 'p' and 'q' are true. If 'p' is false, i.e., \(\sim p\) is true, then 'q' must be true for 'p \(\vee\) q' to hold, since one or the other must be true. Thus, if \(p \vee q\) is true, then \(\sim p \rightarrow q\) must also be true.

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