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I have a small question about logic. In my text, "3 is less than or equal to pi" is translated as PvQ, where P is "3 is less than pi" and Q is "3 is equal to pi." Seems simple enough. But why isn't the statement better translated as (PvQ)&~(P&Q)?
Of course, if you know what "less than" and "equal to" really mean, you'll understand that P&Q is precluded; but it bothers me that this is not explicitly stated in the translation. Someone who understands logic but not English might infer from PvQ that 3 may be simultaneously "less than" and "equal to" pi, and this strikes me as problematic.

Just to be sure I'm addressing your worry: it's often said that there are two senses of "or": an inclusive sense, where "P or Q" means "At least one of the statements 'P' and 'Q' is true, and an exclusive sense, where "P or Q" means "exactly one of the statements 'P' and 'Q' is true." Let's suppose I'm the sort of person who makes it a practice of always using "or" in the inclusive sense. Someone who knows this hears me say: "Mary is in San Francisco or in New York City." The logic of my statement doesn't rule out all by itself the possibility that Mary is in both places. What rules that possibility out are the facts of geography and of how people fit into space and time. (It's been claimed that some saints were capable of bilocation, but we'll assume that Mary is, at least in that respect, no saint.) Could someone who knew that I'm an inclusive "or" sort of guy but didn't know much about geography and the relationship between people and space correctly infer that if my statement is true, then Mary...

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