8th Grade LogicDate: Thu, 15 Dec 94 09:52:21 EST Subject: Questions from the honors 8th grade at MDS From: Anonymous Dear Dr. Math, Here are our questions: 1) Logic - Why in a conditional statement if the "p" in the hypothesis is false, then the entire statement is true? Why isn't it undecided? 2) Why do you use the symbols p and q for logic statements? Thank you for your time. Date: Sat, 17 Dec 1994 15:29:46 -0500 (EST) From: Dr. Sydney Subject: Re: Questions from the honors 8th grade at MDS Greetings! Thanks for the question about logic. I am not normally a Dr. of Math, but my friend called me in for this question, so I am making a guest appearance as Dr. In order to answer your question, you must first understand the property of "or statements." In order for an or statement to be true, one of the atomic sentences which composes it must be true. For instance, if you have the statement, "P or Q," it will be true if either P or Q is true. Of course, if both sentences are true, the sentence will be true as well. OK, on to the conditionals.............. Another way of thinking about conditional sentences is as or sentences. If you have a conditional of the form P==>Q, another way of writing it is "not P or Q" where the not applies only to the sentence P. This statement, then, will be true any time the sentence "not P" is true. Something to remember about conditionals is that they cannot be evaluated merely by looking at the truth values of the atomic sentences that compose them. Therefore, if you are thinking of the sentence, in English, as "if P, then Q" it will seem illogical for the statement to be true if P is false. Because of this confusion, it is a good idea to think of the sentence as "not P or Q." In fact, computer programs that are written to evaluate the truth values of logic statements are programmed to approach conditional statements in this way. Just as x, y, and z are the common variables in math, P and Q are the common letters chosen to represent atomic sentences in logic. We do not really know why P and Q were chosen for this, but if one of the other math doctors knows why, he or she will write. I hope that this answer helps you. Thanks for writing the question, because it gave me a chance to make this guest appearance on Dr. Math. Good Luck, Megin "Really a doctor of English and Philosophy" Charner Date: Tue, 3 Jan 1995 12:55:18 -0500 (EST) From: Dr. Ken Subject: Re: Questions from the honors 8th grade at MDS Hello there! I thought I'd add to my esteemed colleague Megin. Sometimes it's not so clear just what we mean by the concept of a conditional statement. It's true that we can translate each conditional statement into an "or" statement: (P) => (Q) means exactly the same thing as the statement (Not P) or (Q). Here's what this means. A conditional statement means "whenever (P) is true, (Q) is true too." So if we can show that (Q) is always true, then we're all set; whenever (P) is true, (Q) will be true, since (Q) is always true. Likewise, if we can show that (P) is never true, which is the same thing as showing that (Not P) is true, then we're all set again; whenever (P) is true, (Q) will be true, since (P) is never true. Here are a couple of examples that I hope will help illustrate what's going on. Let's think about the conditional statement If pigs can fly then Ken eats his hat. Or if you prefer, (Pigs can fly) => (Ken eats his hat) Now, I'm not going to have to eat my hat. You see, pigs can't fly. I'm just making sure we have some of the basics down here. Moreover, not only will I not have to chow cap, I can make this claim with complete moral impunity. No, I'm not going to eat my hat, but if pigs flew, I would do it. Here's another example. If 1=0 then 938493849384938=7 or (1=0) => (938493849384938=7) And here's a proof. Start out with the equality 7=7, and then keep adding the equation 1=0 to it. You'll get 8=7, 9=7, 10=7, and so on, as high as we want. Pretty soon, we'll get the desired result, that 938493849384938=7. In fact, since (1=0) => (Q) no matter what (Q) is, I should be able to prove ANYTHING IN THE WORLD if I get to start with the premise that 1=0. I can prove that 8x5=9, that Elvis is still alive, and that your teeth are green. So there you have it. I hope this helps some. -Ken "Dr." Math |
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