Inductive and Deductive ExamplesDate: 11/21/2001 at 12:39:48 From: Mehrdad Khademi Subject: LOGIC 7th grade: Is this question in logic valid? Hi Dr. Math, I have a math background but had only one course in Logic as an undergrad a long time ago. My son's Life sciences teacher gave them a question in Logic, and I am pretty certain that: a) The statement is not a sound argument, and b) as a result of (a) it cannot be proved inductively or deductively. The question was: Read the following situation and then show how you would solve the problem with deduction and then with induction. Bob wants to figure out what his teacher wants for his birthday, but he cannot ask his teacher directly. How does he pick the perfect present? I told my son that the statement is not a sound logical argument, because: a) The conclusion of the argument that he wants to prove is not an objective statement (e.g., an opinion, recommendation, etc.). b) Even if the statement were a valid argument, it would be a "non sequitur" (arguments whose conclusion cannot follow from the premise(s)). If we take there to be either 2 simple premises: [1] Bob wants to figure out what his teacher wants for his birthday and [2] He cannot ask his teacher directly (or take them as one combined premise), then the conclusion will not follow (and is not objective) from the premise(s). My son got full credit for his absurd answer to the question (certainly not logical statements). But I don't want him to learn logic incorrectly. I would like to point this out to his teacher and provide him with a few simple situations that do make sense for 7th graders as sound logical arguments. First Request: I appreciate your comments/modifications about my remarks. Second Request: Could you please provide me with few English statements that the teacher can use in his class to introduce his students to logic and to demonstate the use of deductive and inductive reasoning (proofs) for these English statements (not puzzles or mathematical statements, since I do have plenty of those myself). Thanks a lot, Mehrdad (Parent of a 7th grader) Date: 11/30/2001 at 14:30:17 From: Doctor Achilles Subject: Re: LOGIC 7th grade: Is this question in logic valid? Hi Mehrdad, Thanks for writing to Dr. Math. This is an interesting question that required some thinking. You are certainly right that the question does not, by itself, give enough information to either induce or deduce an answer. However, I think that it does provide an avenue for learning about those concepts. Let me first start by proposing a few other (fairly simple) example variations on the question that I feel would demonstrate induction and deduction (I changed the teacher to a female for ease in pronoun use): -------Start of Deduction variant ----------- Read the following situation and then show how you would solve the problem with deduction. Bob wants to figure out what his teacher wants for her birthday, but he cannot ask his teacher directly. How does he pick the perfect present? These are the things Bob knows: 1) She has a "Far Side" calender that she always laughs at. (premise) 2) She drinks coffee every morning from a plain mug. (premise) 3) She owns no decorated mug. (premise) 4) All people who have "Far Side" calendars like any other "Far Side" items. (premise) 5) All people who drink coffee like to drink from decorated mugs. (premise) 6) All people who do not have something they like would like to get that thing. (premise) 7) The perfect present to get someone is something that that person would like. (premise) -------End of deduction variant----- (It is left to the student to fill out the appropriate conclusions from these premises). A similar induction variant can be created with appropriate inductive premises. The way the question as presented differs from my variant is that the student is left to come up with his/her own premises AND then make a deductive conclusion from those premises. I believe that if a student is able to generate premises from which one could validly infer a statement of the form: "The perfect birthday present for Bob's teacher is x." where x is some potential present, AND THEN that student were able to go through the premises and make the proper valid inferences to lead up to the valid conclusion, that student will have learned about deduction and demonstrated his/her knowledge of the subject. The way that these should be graded, in my opinion, is: a) Did the student set up premises that allowed for a valid inference to be made? b) Did the student clearly show the steps in making that valid inference? A similar grading criterion can be used for the inductive variant. I.e. did the student start from a collection of facts to induce a conclusion? Was the inductive reasoning explained well, etc. One other note: I have a fair amount of practice doing logic, so my premises were many and elaborate. Someone just learning may not have as many or as elaborate premises (although it is still important that the inference be valid). I don't know what your son turned in, so I'm not sure what logical holes, if any, it might have had. If you would still like to find examples of logical statments, I'd suggest you search our archives at: http://www.mathforum.com/mathgrepform.html or look at our Logic area in the archives: http://mathforum.org/dr.math/tocs/logic.high.html If that's not helpful, you can write back and I'll try to come up with some for you. I hope this has been helpful for you. If you'd like to discuss this issue some more, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ Date: 12/02/2001 at 15:28:25 From: Mehrdad Khademi Subject: Re: LOGIC 7th grade: Is this question in logic valid? Hello Dr. Achilles, Thank you so much for your compehensive answer. I don't think most 7th graders will come up with the premises that you came up with. (Actually I know none of them did, but most of them got full credit!) I came up with a conclusion to your variant statement, which I wanted to run by you. I will refer to your statements as P1, P2,..,P7. from P1 to P7 one can infer: The perfect gift for Bob's teacher is either a "decorated mug" or a "Far Side" item. This is a logical inference and the conclusion is: Bob should pick either a decorated mug or a far side item for his teacher's birthday. I think both the "logical inference" and the conclusion are necessary. Do you agree with me or not? 2) On the inductive version of your variant. One trivial way is to change all the "ALL"s in your premises P4, P5 and P6 to something like "MOST." Is there a better way to do it? 3) I went to the sites that you recommended and could not find any other English example (I don't mean puzzles). If you can forward me some more English examples I will much appreciate it. 4) In your opinion is it a good idea to make some universal assumptions as accepted truth (So that 7th graders do not have to come up with a lot of premises) and use those, and a "lax" form of rules of inference to come up with questions that are closer to a sound argument than the question the teacher gave them? An example of this could be: Given: Defn D1: Good grade means a grade above 3.75 (on a 4.0 scale). Defn D2: A good college is one of the top 100 colleges in the USA. Axiom A1: Studying hard produces good grades. Axiom A2: A college degree with a good grade (an honor degree) will produce a successful person. Show how you would solve this problem using "deduction": If you study hard, you will be successful. I will use Ij for inference #j. Note that I1 is not a sound argument and I may need your help on this. I1: For Good grades mean that you will get into a good college. I2: By axiom A1 you will get good grades in a good college. I3: For good grades mean you will get an honor's degree from a good college (I know this is neither an axiom nor a valid inference). I4: By Axiom A2 An honor's degree implies you'll be a successful person. 5) I think the question above is perhaps better than something like the following, which is a valid logical inference. But please note that to a 7th grader it is like proving the "obvious." Use deduction to show that: Given premises: P1 : No women are men P2: Some doctors are women Deduce "Some doctors are not men". I1: All women are nonmen. Applying Obversion to P1. I2: Some doctors are nonmen Follows from P2 and I1 I3: Some doctors are not men. Apply Obversion to I2 Thanks a lot for and looking forward to your response. Mehrdad Khademi Date: 12/03/2001 at 16:45:56 From: Doctor Achilles Subject: Re: LOGIC 7th grade: Is this question in logic valid? Hi again Mehrdad, Thanks for writing back. I should note that a simpler deduction could be created. Just a simple syllogism: A) For all people who have a desk, a pen is the perfect gift. B) Bob's teacher has a desk etc. 1) You actually spotted a logical hole in my premises. I should've added: There exist mugs decorated with "Far Side" cartoons. 2) Changing the "all"s to "most" would make it inductive. Another thing is just to say something like: Bob has given all of his teachers apples for Christmas, and each time it has been the perfect gift. Induce from there that that will hold true for this teacher as well. 3) (See below) 4) I chose to use universals in my premises because they lend themselves much more easily to elementary deductions. I think that they are good ways to show students just learning about logic the idea of deduction. They also make very clear the idea that in a valid deduction it is IMPOSSIBLE for the premises to all be true and the conclusion to be false. Generalizations (e.g. most, some) are perhaps better for induction. However, it is possible to deduce conclusions from non-universal statements (almost always generalized, or non-universal, conclusions). I think your grades example needs another two axioms: A3: Studying hard will get you into a good college. A4: Good grades will get you an honors degree. Then, however, the inferences are perhaps too easy. 5) This brings up an interesting question about teaching introductory logic to 7th graders. Most of the examples I had in mind would seem like "proving the obvious" to a 7th grader. There are really a couple of issues here: First, in 7th grade, one is usually only given examples that take one or two steps. Second, deductive logic is designed so that each individual step is always obvious. From that you can easily deduce that: 7th grade logic examples will appear to be obvious. There are, however, some classic rules and logic examples that are easy for 7th graders (most appear obvious), but they are the foundation for more involved logic. Finally, I owe you a few examples (the first few are obvious, the later ones I hope are not quite so): I) The syllogism A) All men are mortal. (major or universal premise) B) Socrates is a man. (minor or particular premise) C) Socrates is mortal. (conclusion) II) Modus ponens (this seems a lot like a syllogism, but the logical forms are different) A) If it is raining, then it is wet. B) It is raining. C) It is wet. III) Modus tollens (this one is less obvious, and it is a good exercise to have someone explain WHY it is a valid inference) A) If it is raining, then it is wet. B) It is not wet. C) It is not raining. IV) Fallacy of affirming the concequent (this is INVALID; having someone explain why is a good exercise) A) If it is raining, then it is wet. B) It is wet. C) It is raining. V) DeMorgan's rule's (this one is tricky to do formally, but an informal (i.e. in prose) proof would show a good grasp of mid high- school level logic): Given: notA and notB Prove: not(A or B) Given: not(A or B) Prove: notA and notB Given: notA or notB Prove: not(A and B) Given: not(A and B) Prove: notA or notB VI) Learning how to deduce from an "or" statement (here and elsewhere I use an "inclusive or" which means "A or B" is true if A is true or if B is true or if both are true). Given: (P and Q) or (Q and R) Prove: Q Rule: Given an "or", assume the left and derive something. Then, assume the right and derive THAT SAME THING AGAIN. Then, you are entitled to keep what you got from both sides. Steps to take: a) P and Q (assumption) b) Q (simplification of a) c) Q and R (assumption) d) Q (simplification of b) e) Q (for keeps, now that we've proved it from both halves of the "or") Figuring out why this is valid shows what I would call "mastery" of logic for even a high-schooler. As far as induction goes: Good examples can be found in any classic science conclusion. Take any relation F = ma, PV = constant, etc. from classic science and start with observations such as: Force mass acceleration 1 1 1 2 2 1 2 1 2 etc. or: Pressure Volume (of a gas) 1 1 2 0.5 3 0.33333 etc. I hope all of this is helpful to you. If you'd like to talk about it some more, write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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