A False Statement Implies Any StatementDate: 09/25/97 at 20:16:57 From: Kemp Subject: Logic Dear Dr.Math, My name is Megan and I'm 15 yrs old. I'm doing Logic in math right now, and were learning about Quantifiers and Laws of Inference. I was wondering if you knew any ways to make learning these and remembering them easier? If you have any suggestions, I'd be very grateful. Thank you, Megan Date: 09/26/97 at 13:47:34 From: Doctor Mike Subject: Re: Logic Dear Megan, I'll send you the e-mail I sent to my own daughter who is doing similar stuff in a college course, and asked me to explain why a False statement implies Any statement. This will not cover all you ask about, but typically this idea often puzzles people for a while. If you still want to know more after this, please write back with a question that is more specific, so we will know better what we are trying to answer. Or, you could check out other books in the library, and compare explanations. So, here goes : "... In case you still are wondering about why a False implies anything, try this explanation on for size. It may help. It's just a little thing .... just HALF of a truth table. P | Q | If P, then Q (same as P --> Q) ------------------------------------------------- | | F | T | T | | F | F | T | | I call it HALF of a truth table because it only has the "F" cases for P. An example of such a false P sentence is "The moon is made of green cheese." The truth table just says that if the first part is something like that, it doesn't matter what the second part is. Here are two examples that illustrate the two rows of that truth table. IF the moon is made of green cheese, then you will find a $20 bill on the sidewalk sometime this week. IF the moon is made of green cheese, then you will NOT find a $20 bill on the sidewalk sometime this week. I capitalized IF because it is a "big IF"; the first part of those two sentences is not going to happen. To prove either of those two sentences false in a court of law, you would have to convince the jury about the green cheese, and only then consider the second part. You will have to actually wait out the entire next week to find out about what really happens concerning the twenty dollars. BUT, no matter what you happen to find on the sidewalk, both of those above sentences are True. They are True, because they didn't really promise anything, and they didn't promise anything because the P-part is false. Think about it. It may take some time to sink in." As for the "Quantifiers" part of your question, notice the similarity to the word "Quantity," which means "how many." Logic is not usually interested in specific quantities like "113" of "several dozen". The important quantities to concentrate on are ALL and SOME : 1. ALL, which usually involves sentences starting out like "For all" or "For every," or which could be written that way. Consider the sentence "Every police officer wears a badge." You could also say, "For every police officer, that officer wears a badge." In symbols, that could come out "For every police officer P, B(P)" where I am using B(P) to stand for P wears a badge. By the way, the sentence we are talking about is probably false, because of undercover detectives and the like. It's sort of a predictable thing that whenever a mathematician or logician is turned loose without much supervision, very soon some symbolism is introduced (grin). Like, how many times have you heard a math teacher start working on a word problem by saying "Let X be the unknown"! 2. SOME, which usually involves sentences starting out like "For some" or "There exists a," or which could be written that way. Consider the sentence "Some police officer enjoys listening to Mozart." You could also say, "There exists a police officer, such that that officer enjoys Mozart." In symbols again, "There exists a police officer P such that M(P)" where I am using M(P) to stand for P enjoys Mozart. The ALL-type phrases are called Universal Quantifiers because they are claiming something is true for all things in a certain collection (or universe) of things, as all the police in Brooklyn, or all flute players studying at the Eastman School of Music. The SOME-type phrases are called Existential Quantifiers because they are claiming that something exists within a group, such as a course in Japanese among the offerings at your local High School. One more thing. When you negate (say "It is false that ...) any Universally quantified sentence, you get an Existentially quantified sentence, and vice versa. An example. Saying that "All police wear badges" is False, is the same as saying "Some police do NOT wear badges" is True. I hope this helps. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 09/08/2004 at 14:41:21 From: Jay Subject: Why is "false implies true/false" always "true"? Why are the logical statements "false implies true" and "false implies false" always considered "true"? I've read the previous note, but could you please give a more "formal" explanation? Thank you, Jay Date: 09/08/2004 at 15:54:44 From: Doctor Schwa Subject: Re: Why is "false implies true/false" always "true"? Hi Jay, Great question! More formally, I'd say "implies" means the same as "subset" in set theory. That is, when you say if it rains, then the ground gets wet you mean the set of times when it rains is a subset of the set of times when the ground gets wet. So, since the empty set is a subset of any set, a false statement implies any statement. I hope that helps clear things up! - Doctor Schwa, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 09/09/2004 at 05:29:20 From: Jay Subject: Why is "false implies true/false" always "true"? That's exactly what I wanted to know. Thank you very much! |
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