I’m working through Daniel Velleman’s textbook How to Prove It: A Structured Approach (second edition). In Example 1.1.2 (1). The book analyses the statement:
Either John went to the store, or we’re out of eggs.
It models this as:
$$P \lor Q$$
where:
- P = “John went to the store”
- Q = “We’re out of eggs.”
However, it seems to me that the word “either” often implies an exclusive OR (XOR) — that is, only one of P or Q should be true, but not both.
In this specific real-world context, if John had gone to the store, presumably he would have bought eggs, meaning we would not be out of eggs. Therefore, it feels incorrect to model the statement with inclusive OR, allowing both P and Q to be true simultaneously.
Shouldn’t the more accurate symbolic translation be:
$$(P \land \neg Q) \lor (\neg P \land Q)$$
instead of just $P \lor Q$? Or is Velleman intentionally simplifying here for pedagogical reasons, focusing on introducing basic connectives first? This is ambiguous for me throughout the chapter section, whether translating logical statements into English or vice versa.
