tag:blogger.com,1999:blog-4987609114415205593.post1663430486908704015..comments2020-07-13T20:26:53.037+01:00Comments on M-Phi: Problems with reductio proofs: cognitive aspectsJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-4987609114415205593.post-50431871204712540392017-06-30T05:41:13.858+01:002017-06-30T05:41:13.858+01:00We as a whole are conceived similarly, yet we are ...We as a whole are conceived similarly, yet we are not quite the same as each other after that.one of us has a tendency to be a space traveler ,one of us winds up being an artist, while one turns into a homeless person. The distinction is made by training and this crevice must be filled by it.<a href="http://www.professionaltyper.com/how-we-work/" rel="nofollow">typing help online</a><br />Anonymousehttps://www.blogger.com/profile/09223272712961116915noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-52792082136382732015-07-15T12:29:28.775+01:002015-07-15T12:29:28.775+01:00Agreed! (See point above about pedagogical implica...Agreed! (See point above about pedagogical implications.) Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-52116738936246949292015-07-15T12:28:19.285+01:002015-07-15T12:28:19.285+01:00In my experience teaching intro to logic, students...In my experience teaching intro to logic, students do find reductio proofs a bit puzzling. But you are right, it is related to the kind of explanation they get (see my point above on how adopting a dialogical conception could have some positive pedagogical implications).<br />And thanks for the reference to the distinction between perfect and imperfect information; I hadn't thought of it in this context, will think more about it!Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-71992406267234574872015-07-15T12:28:19.150+01:002015-07-15T12:28:19.150+01:00In my experience teaching intro to logic, students...In my experience teaching intro to logic, students do find reductio proofs a bit puzzling. But you are right, it is related to the kind of explanation they get (see my point above on how adopting a dialogical conception could have some positive pedagogical implications).<br />And thanks for the reference to the distinction between perfect and imperfect information; I hadn't thought of it in this context, will think more about it!Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-55369446331323518392015-07-15T12:14:07.328+01:002015-07-15T12:14:07.328+01:00Thanks!Thanks!Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-81884868440293834352015-07-15T12:13:54.877+01:002015-07-15T12:13:54.877+01:00Yes, it's pretty clear that Maria hasn't b...Yes, it's pretty clear that Maria hasn't been much helped by how reductio has been explained to her... The general point of the paper is to argue for the dialogical conception, which at the end I note also has some practical pedagogical applications. The hypothesis would be that, if students are told that a reductio proof is like a dialogue with an opponent, then many of these issues and confusions would not arise. However, the last step, the 'act of faith' remains problematic to my mind, not just a matter of modes of presentation. The problem remains that we need to be sure that the enumeration of cases is exhaustive.Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-34104568027104013582015-07-15T12:11:00.479+01:002015-07-15T12:11:00.479+01:00An account of logical pluralism is also one of the...An account of logical pluralism is also one of the things that the dialogical perspective of logic and deduction that I endorse can deliver. But more specifically (and this is coming up in a post soon), the last step presupposes that the enumeration of cases is exhaustive, something that doesn't seem obvious at all in a number of circumstances, perhaps including portions of mathematics.Catarinahttps://www.blogger.com/profile/03277956118114314573noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-40481533899351562412015-07-15T07:51:36.750+01:002015-07-15T07:51:36.750+01:00Can't help but think that educator's own d...Can't help but think that educator's own difficulty in explaining the proof adds an "illusory" perception of complication. There is something resembling the writing of antiquity in a typical educator's account, that wordiness... It should flow more naturally. I swear I could get students to give every-day reductios of basic propositions. The most well known case might be the alibi, which even 5 year olds can do. Why should "I was somewhere else" be proof that you didn't commit a crime? Is the answer easier because we are used to reasoning about people and their doings?<br /><br />The idea is not that we KNOW something's true, but we're wondering if it's true. Often students are told briefly "Prove P." I like when authors phrase it "P? Prove your result." Perhaps that could fill the "first gap," the discomfort of assuming a lie; you're not 'supposed' to know it's a 'lie.'<br />When you write it out, it seems long and involved, but the narrative structure is so common that Popper thought to found science on it ("falsification"). Perhaps emphasizing an experimental frame of mind could be a move towards filling these gaps.Paul S.http://metatheton.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-67641677748049771282015-07-14T17:54:04.995+01:002015-07-14T17:54:04.995+01:00This is a bit tangential to the post, but I wonder...This is a bit tangential to the post, but I wonder about the *pedagogical* gap that is behind some of the student's confusion. One thing I noticed is that both students from your examples weren't majoring in math or philosophy, so perhaps their exposition to the subject was a bit too quick? Do you also experience this type of confusion when teaching undergrads, Catarina? <br /><br />Also, one thing that occurred to me is if the distinction between perfect and imperfect information couldn't be of help here. It seems to me that part of the problem is that the student's are trying to force a situation which trades on imperfect information into a situation which trades on perfect information. Generally speaking, when attempting a proof we have only imperfect information available to use (those are encoded in the premises, axioms, and rules of inference); in particular, in the case of RAA, we don't know the status of the premise we're trying to prove. Of course, outside the proof environment, we do know this status, but *this* piece of information is not available *inside* the proof environment (I got this distinction between perfect-imperfect information from Kaye's *The Mathematics of Logic*, which gives as a helpful example König's Lemma).Daniel Nagasehttps://www.blogger.com/profile/09389957277629676271noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-41271147906778418132015-07-14T14:16:53.380+01:002015-07-14T14:16:53.380+01:00Perhaps some readers may be interested in this dis...Perhaps some readers may be interested in this discussion by mathematicians of the use of reductio in mathematical proofs: http://mathoverflow.net/q/12342/1946. Anonymoushttps://www.blogger.com/profile/03016500743689022122noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-89572557013064399672015-07-14T13:53:15.639+01:002015-07-14T13:53:15.639+01:00My instinct when reading what Maria has to say is ...My instinct when reading what Maria has to say is to want to tell her that her conceptual problems could be avoided if she expressed essentially the same proof in a different way. The less confusing version goes like this. Suppose that a and b are two numbers and that ab=0. (Note, I am not assuming any kind of impossible imaginary world here -- it's perfectly possible for two numbers to have product zero.) Now suppose in addition that b is not zero. In that case, we can divide both sides by b and deduce that a = 0/b = 0. So if b is not zero, then a must be zero. <br /><br />From here we have to get to the conclusion that one of b and a is zero. In other words, we have to accept a deduction of the form (not p implies q) implies (p or q). If we use the formal definition of "implies", we might say that the meaning of (not p implies q) is (not not p or q). So the one step that remains is the idea that not not p is the same as p. While some people may find that philosophically suspicious, I don't think it is counterintuitive.<br /><br />Actually, when discussing what's counterintuitive and what isn't, I prefer not to take this last step and go back to the deduction of (p or q) from (not p implies q). I think this falls into Dennett's category of types of reasoning we go in for the whole time. For example, if I say, "If Djokovic doesn't win the US open, then Federer will," it's absolutely clear that I'm claiming that either Djokovic or Federer will win the US open.<br /><br />This doesn't invalidate what you are saying, since there are other examples of arguments where reductio ad absurdum is used in a more essential way. I just think that Maria's conceptual difficulties with this particular example may result more from a not very well expressed proof than from fundamental philosophical problems with the proof itself.<br /><br />Another thing I'd like to say to Maria is that the rules that apply in the absurd world are the same as the rules that apply in the actual world. Her difficulty here seems to me to be bound up with the word "blatant". In order even to think about how to prove that there are no non-trivial zero divisors, you have to get out of your head the idea that it's just plain obvious. I think her problem is mainly the familiar one that the statement seems too obvious to need a proof, which again is not a fault with reductio ad absurdum. Someone with more mathematical experience will think of examples like the integers mod 6, where you do have non-trivial zero divisors. Simple examples like that help to dispel the idea that there's nothing to prove. It would be interesting to see how these students feel about reductio proofs of statements that they find less obviously true in the first place. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-14447600323771759782015-07-14T00:02:22.696+01:002015-07-14T00:02:22.696+01:00This is very interesting. It occurred to me that i...This is very interesting. It occurred to me that if one is the sort of logical pluralist that I (sometimes) am---holding roughly that different consequence relations are relevant in different contexts---then reductios may not always be valid outside of classical mathematics. The thought then is that, especially in the case of `the gap at the end', we have noticed that this argument form fails in other contexts, and thus are not convinced that it should apply in the mathematical case either<br /><br />Apologies if this is something you will be covering in subsequent posts.Aaron Thomas-Bolducnoreply@blogger.com