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Chapter 3: Problem 95

Based on the meaning of the inclusive or, explain why if \(p \vee q\) is true, then \(p \rightarrow \sim q\) is not necessarily true.

Short Answer

Expert verified
The truth of \(p \vee q\) doesn't necessarily make \(p \rightarrow \sim q\) true because the former can be true if either or both of the elements are true, without establishing a direct causal connection. On the other hand, the latter directly links the truth of \(p\) to the truth of \(\sim q\). Thus, a situation when \(p\) and \(q\) are both true, makes \(p \vee q\) true but \(p \rightarrow \sim q\) to be false, subjecting to the negative of \(q\), \(\sim q\), being false.

Step by step solution

01

Understanding the Meaning of 'Inclusive Or'

The 'inclusive or' operation, \(p \vee q\), is true whenever either \(p\) is true, or \(q\) is true, or both are true. This operation doesn't specify a strict causal connection between the elements, meaning either can be true or both can be, for the operation to be true.
02

Understanding the Meaning of 'Implication'

The implication operation, \(p \rightarrow \sim q\), means that if \(p\) is true, then \(\sim q\) must also be true. It establishes a more direct connection between the two elements; the truth of \(p\) directly leads to the truth of \(\sim q\). If \(p\) is true but \(\sim q\) isn't, then the implication is false.
03

Analyzing the Contrast Between Both Operations

Having clarified the meanings of \(p \vee q\) and \(p \rightarrow \sim q\), it's clear that they express different relationships. The truth of \(p \vee q\) doesn't guarantee that the truth of \(p\) implies the truth of \(\sim q\). This is because, while all elements can be true in the 'inclusive or' operation, the implication operation requires that if \(p\) is true, then \(\sim q\) must also be true. For instance, if we consider \(p\) and \(q\) to be true, this makes \(p \vee q\) to be true but \(p \rightarrow \sim q\) to be false since \(\sim q\) would be false while \(p\) is true.

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