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I believe that the confusion that some students experience with conditional statements in math and formal logic courses, over the idea that one cannot assume the inverse/converse of a conditional, stems from their intuitive understanding of Grice's maxim of Quantity. I think it would be most helpful, upon the introduction of this subject, for the teacher (or the course material) to explain roughly as follows:
Conditional statements take the form of "If something, then something else." Like, "If it's raining, then I wear a hat." Let's say I told you, "If anyone's around, I don't pick my nose." So what do you think I do when I'm alone? [Pause.] I pick my nose, right! But do you really know that? If I never picked my nose, my statement would still be true, right? [Longer pause, possibly repeating the original statement for consideration.] You're right: If I never picked my nose, I wouldn't have said it like that. I would have just said, "I don't pick my nose." What we just demonstrated is a linguistic rule called Grice's maxim of Quantity, which states that I don't tack on phrases like "If anyone's around," unless I actually have a reason to do so. I said it, so you figured I had a reason for saying it, and that's how you knew that I pick my nose when nobody's around. Incidentally, the statement "If nobody's around, I pick my nose" is the inverse of my original statement, "If anyone's around, I don't pick my nose." You get the inverse just by negating both parts of the statement.
The maxim of Quantity, the rule that you used to figure out the inverse, is not a rule in formal logic! It's just a rule of natural language. In formal logic, I could say, "If anyone's around, I don't pick my nose," and you would have no way of knowing what I do if nobody's around. Unlike with natural language, when you're given a conditional statement in formal logic, that's not enough information to know whether the inverse of that statement is true. That should be obvious with the statement, "If it's raining, then I wear a hat." Can you conclude the inverse, that if it's not raining, then I don't wear a hat? Of course not! Maybe I'll also wear a hat to keep the sun off my face, or just because it's stylish. So we can't assume the inverse.
Now let's look at the contrapositive, which is just a fancy name for a concept that you already understand, as illustrated by the famous quote by Dan Quayle, "If Al Gore invented the Internet, then I invented spell check..."
Conditional statements take the form of "If something, then something else." Like, "If it's raining, then I wear a hat." Let's say I told you, "If anyone's around, I don't pick my nose." So what do you think I do when I'm alone? [Pause.] I pick my nose, right! But do you really know that? If I never picked my nose, my statement would still be true, right? [Longer pause, possibly repeating the original statement for consideration.] You're right: If I never picked my nose, I wouldn't have said it like that. I would have just said, "I don't pick my nose." What we just demonstrated is a linguistic rule called Grice's maxim of Quantity, which states that I don't tack on phrases like "If anyone's around," unless I actually have a reason to do so. I said it, so you figured I had a reason for saying it, and that's how you knew that I pick my nose when nobody's around. Incidentally, the statement "If nobody's around, I pick my nose" is the inverse of my original statement, "If anyone's around, I don't pick my nose." You get the inverse just by negating both parts of the statement.
The maxim of Quantity, the rule that you used to figure out the inverse, is not a rule in formal logic! It's just a rule of natural language. In formal logic, I could say, "If anyone's around, I don't pick my nose," and you would have no way of knowing what I do if nobody's around. Unlike with natural language, when you're given a conditional statement in formal logic, that's not enough information to know whether the inverse of that statement is true. That should be obvious with the statement, "If it's raining, then I wear a hat." Can you conclude the inverse, that if it's not raining, then I don't wear a hat? Of course not! Maybe I'll also wear a hat to keep the sun off my face, or just because it's stylish. So we can't assume the inverse.
Now let's look at the contrapositive, which is just a fancy name for a concept that you already understand, as illustrated by the famous quote by Dan Quayle, "If Al Gore invented the Internet, then I invented spell check..."
