Logic LawsDate: 03/04/2003 at 20:36:25 From: Jessica Subject: Logic laws We are learning about logic. I do not understand the laws of inference, simplification, disjunctive inference, and disjunctive addition. Date: 03/06/2003 at 23:11:23 From: Doctor Achilles Subject: Re: Logic laws Hi Jessica, Thanks for writing to Dr. Math. For a "crash course" in symbolic logic, see the Dr. Math FAQ: http://mathforum.org/dr.math/faq/symbolic_logic.html The "crash course" doesn't use the same terms to refer to all of the laws that you're asking about. I do recommend that you read it to give you background in logic and show you a systematic way to think about logic. With regard to the specific laws you asked about: 1) "inference." This refers to what the crash course calls "->elim" or what some people call "modus ponens." The rule is this: If you have: (A -> B) And you have: A Then you can conclude: B This follows almost directly from the meaning of "->" or "if..., then." The statement (A -> B) means that if A is true, then B must also be true. It doen't say anything about whether A or B actually is true, it just gives a relationship between them. However, if we also know A that is, we also know that A is true, *then* we can conclude that B is true as well. 2) "simplification." This is called "^elimination" in the crash course. The law is this: If you have: (A ^ B) Then you can conclude: A And/or you can conclude: B The reason is fairly easy to see from the definition of "^" or "and." We know for starters that (A ^ B) is a true statement. This means that *both* A and B *have* to be true. So we can conclude A is true from that, or if we prefer we can conclude B is true, or if we like we can make both of those conclusions. 3) "disjunctive inference". This is also sometimes called "disjunctive syllogism." The rule states: If you have: (A v B) And you have: ~A Then you can conclude: B [As a side note, the rule also works if you have (A v B) and you have ~B, then you can conclude A.] This one is a little trickier (it's one of the advanced rules in the crash course). Let's take this step by step. We start off with (A v B) This means that *either* A is true, or B is true, or both are true. So there are three possible ways that (A v B) could be true: First, A is true and B is false. Second, A is false and B is true. Third, A is true and B is true. So far we have no idea which of these three is actually the case. But then we see that we also have ~B This means that B is false. Let's review are three possible scenarios: First, A is true and B is false. (This one still may be correct.) Second, A is false and B is true. (Since we just found out that B is false, this *cannot* be correct.) Third, A is true and B is true. (Since we just found out that B is false, this *cannot* be correct.) So the only scenario that can possibly be correct is: A is true and B is false. We can take that and make a logical statement out of it: (A ^ ~B) And by simplification we can get from that A So A must be true if we start with (A v B) and ~B. 4) "disjunctive addition." This is called "vaddition" in the crash course. It's a fairly short rule, but it seems like cheating so it is hard to feel comfortable using it. If you have: A Then you are allowed to conclude: (A v B) It doesn't matter what B is. B could be something completely new that you just made up. B could be something that appears as part of another line in the same problem. B could even be something that you know is false! How can that work? How can we go around just willy-nilly adding letters to the end of sentences? Well, let's think about this. We started off with A So we know that A is true. No matter what we do with the rest of the problem, we can be sure that A will always be true. What if we know already that B is true? Well, in that case (A v B) will come out true because the disjunction of 2 truths will be true. What if we know already that B is false? Well, in that case (A v B) will still come out true because the disjunction of a truth with a falsehood still comes out true. What if we don't know whether B is false? What if we saw it earlier, but never could prove one way or the other? What if we just made B up? In that case, we don't know whether B is true or false, but *either way* (A v B) will be true (we just proved that in the first two cases above). Basically, as long as we start off with A and we know that A is true, we can be sure that (A v B) will be true, no matter what B is. One final note: I have used A and B for every example, this (of course) works for any sentence letters, it also works for more complicated sentences so if you have something like: [(A v B) -> (B ^ C)] and (A v B) then you can do inference on it and conclude (B ^ C) I hope this helps. If you have other questions or you'd like to talk about this some more, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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