Logic

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.

Comments

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 might be simultaneously in SF and NYC?

No. Let's take the extreme case. Suppose I say: Mary is either in San Francisco or she isn't," still using "or" in the inclusive sense. It would certainly be wrong to infer that if I'm right about Mary, then should could simultaneously be both in SF and not in SF. That's a logical impossibility.

It might be tempting to reply this way: my statement wasn't true after all, and couldn't possibly be true. It couldn't be true because an inclusive "or" would imply in this case that Mary really could be in SF and not in SF. But that's a mistake. All that theh inclusive use of "or" amounts to is this: whenver two statements are linked by inclusive "or," the compound will be true whenever at least one of the parts is true. But this says nothing at all about whether the parts actually could be true together. The statement "Mary is in SF or she isn't" is true. Mary is not in San Fransisco, as it happens. And so at least one of the disjuncts of the "or" statement is true. Of course, exactly one of them is true, and in this case, we know that before we know anything about Mary's whereabouts. But that's because "Mary is in SF" and "Mary is not in SF" contradict one another.

So "3 is less than or equal to pi" is true. It's true because 3 is less than pi. And it doesn't follow from the truth of "3 is less than or equal to pi" that it's possible for 3 to be less than and equal to pi. The meanings of "less than" and "equal to" rule that out.

Now of course, there might be cases where someone undersands enough to know that an "or" statement has been made, but lacks the knowlede to know whether the two parts could both be true. For example, consider "Stanley's pet is oviparous or mammalian." Could it be both? Or consider "The particle was a boson or a fermion." Could it have been both? Though the answer happens to be "yes" for Stanley's pet (a platypus would be an example) and "n"o for the particle, we can well imagine that someone might not know. They might accept the authority of the person who makes the statement, and might say to themselves "For all I know, Stanley's pet could be both oviparous and mammalian, and for all I know, the particle may have been both a boson and a fermion." But the logic of "or" doesn't tell us which possibilities are compatible, and it's no fault of "or" that it sticks to its own more limited business: declaring that at least one of two claims is correct.

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