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Consider the following problem:

  • Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach.

Obviously this statement is wrong. Maria could have gone to other place and bought an ice-cream. You don't need any math tool to arrive at this conclusion, all you need is reasoning.

However, several adults (18~50 years old) with difficulty in math, also have a really hard time to solve/understand such kind of problems. For them, learning math is the same as memorizing rules and formulas. Anything different than that (ex: reasoning) is extremely painful.

So, is it possible to make such students correctly answer GMAT style questions?

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    $\begingroup$ After writing this argument on the board and finding that some people have trouble seeing that it's incorrect reasoning, try marking through "buys ice-cream" in both places and writing above each of these marked-out phrases something like "breathes". Also, to REALLY get them to pay attention (and give them needed practice), perhaps put something like this on every in-class graded assignment for 1 or 2 points extra credit (out of 100 points total, so that grades won't be all that much different on what you're really testing them on). $\endgroup$ Nov 16, 2015 at 18:25

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You can say that this is "just reasoning", but the truth is that this is a specific application of basic logic, in particular the implication (if/then) relation. I have a colleague with a PhD in logic who says, "Implication is tricky!" when I bring this up. And I do think that it's a major problem that schools don't teach basic logic as a high-school (or earlier) requirement; it really puts all their later coursework on a foundation of shifting sand without that.

If I had complete dominion at the community college where I teach, then personally I would mandate a 1-credit seminar in basic logic (at least drills with and-or-not-if/then statements) for incoming students. At times I've tried to find an hour in my basic math classes to work on this, but unfortunately at the moment other priorities take precedence for that time.

Some blog articles I've written on this subject:

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    $\begingroup$ Perfect! Without logic there's no foundation. It's a tricky subject, but it's an important part of math. However, if this is the foundation the students have and they don't have much time to solve that, how to improve their math scores in exams? $\endgroup$
    – Mark Messa
    Nov 13, 2015 at 21:25
  • $\begingroup$ @Mark Messa: It's a real problem. To summarize, my answer is "teach logic". Perhaps you can do a better job than I have of finding time in the curriculum to make that happen. $\endgroup$ Nov 14, 2015 at 3:18
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    $\begingroup$ I agree with you, the best solution would be teach them logic. Unfortunately, the problem is that such kind of students consider logic too difficult (some even complaint about headaches) and are not interested in learning it. They really prefer only the shortcuts (ie: memorization). If you insist with logic, they just give up and move to another course. $\endgroup$
    – Mark Messa
    Nov 14, 2015 at 9:58
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    $\begingroup$ I might qualify that as a good thing; it raises the level of discourse in the class and you can devote more time to the dedicated students. Consider reading Clark's article "The Cooling-Out Function in Higher Education" if you haven't already. $\endgroup$ Nov 14, 2015 at 15:13
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    $\begingroup$ Hi @DanielR.Collins, the links in this answer are broken. Any chance that the blog entries are still available somewhere? Edit. found them at madmath.com :) $\endgroup$
    – BKE
    Dec 15, 2021 at 21:00
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There is a paradox here. Implication IS tricky. Part of the reason is it's REALLY HARD to avoid implicit assumptions in human scenarios. The only absolute way to avoid assumptions is to translate the English into meaningless symbols (p, q) and rigorously apply logic rules. The rules themselves are tricky. You pretty much need to prove them once to know they are correct, then apply them by rote from then on. Your example is on the cusp what most people can handle in their head. For anything much more complex, most of us will need to manipulate symbols using logical rules to "reason correctly."

In your example, some of your students will intuitively see the correct premise and conclusion. Others will need some help. All of us will need some help if the scenario gets complicated enough. (Or if it is sneaky enough to trick us into making unwarranted assumptions. It happens.) I'll probably lose points for politicizing this, but consider the recent debate on gay marriage. There were some not irrational arguments along the line of "If you want to have children, then you should be married." But there was an implicit assumption of bi-implication. Not once did I hear anyone--on either side of the debate--point out that this is not the same thing as saying "If you want to get married, then you should have children." So this sort of logical error is very easy to make, every by well-trained people.

I agree with Daniel and his blogs. The only solution is formal training in logic. If only to make people aware of how EASY it is to make mistakes.

If you're looking for the most bang for your teaching buck, what I have personally found the most useful is this:

  1. What implication is: An if-then statement. Show how to identify the premise and the conclusion.

  2. Show the differences between the INVERSE, CONVERSE, and CONTRAPOSITIVE. My first exposure to these three conditionals was a real eye-opener. It did more than anything else in my life to avoid the kind of error shown in your example.

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    $\begingroup$ Could the most common logical error be assuming bi-implication based on an example of implication? For example, "beach = ice cream" or "marriage = babies"? Perhaps the reason I find the distinctions between Inverse, converse, and contrapositive so compelling is that they show how implication does necessarily imply equality (bi-implication). I wonder if there is something about our educational system that primes students to see equality when none is there. Except for a little greater-than and less-than exercises in first grade, virtually all they work with are equations. $\endgroup$ Nov 18, 2015 at 0:25
  • $\begingroup$ I too am frequently irritated by how many people read a stronger statement that isn't there into something, and do not know the provenance of this tendency. (In programming, 'if' branches trigger if and only if the condition is met, but I doubt this is responsible for many of their mistakes or bizarre cognitive biases.) I try to convince them of their misapprehension using examples such as 'if you do X then you are damned; is it reasonable to assume that you are not if you don't?', but often I still have to resort to the lengthier 'if X, then Y. If not X, then I guarantee nothing.' ... $\endgroup$ Nov 18, 2015 at 6:24
  • $\begingroup$ ... to be clear when I would have liked to simply say 'if X, then Y'. I think there is an unspoken tendency to think in terms of equations rather than inequalities (which implication is an instance of in the sense of truth values), given how if I have 5 apples and am asked whether I have 4, I -- yet somehow very few others whom I know -- would find it honest and in fact obligatory to answer 'yes' (indeed, would 'no' be any more honest?).... $\endgroup$ Nov 18, 2015 at 6:24
  • $\begingroup$ I also hypothesise that the Gricean maxims may also play into this, even as unreasonable as it is to assume omniscience, that the speaker always knows the exact conditions under which something will happen as opposed to some convenient sufficient condition. $\endgroup$ Nov 18, 2015 at 6:24
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This is a subtle issue. It goes to the heart of the difference between math and physics.

That A implies B does not entail that B implies A. One encounters frequently the errant reasoning that it does even among engineering students in the university (yesterday a student told me that because a matrix was diagonalizable it must be symmetric).

However, that A implies B and one has observed B provides evidence for believing A. This statement is somehow the basis for the scientific method.

In the example given in the original post, that Maria bought ice-cream today does provide some evidence that she might have gone to the beach today.

The difference is that between deductive reasoning and inference.

Inferential reasoning is more common, more natural, and more powerful (why does anyone believe Euclid's axioms?).

I quote from V. I. Arnold (Translation of the V. I. Arnold paper “From Superpositions to KAM Theory” (Vladimir Igorevich Arnold. Selected — 60, Moscow: PHASIS, 1997, pp. 727–740)), who in his inimitably polemical manner explains the issue better than I can anyway:

Now it became possible to apply the techniques developed in the problem of adiabatic invariants. As soon as I accomplished that, Kolmogorov suggested that I should submit the paper on perpetual adiabatic invariance to ZhETF, the main physical journal in the USSR. A few weeks later, M. A. Leontovich (who was, as far as I remember, a deputy to the editor-in-chief of ZhETF) invited me to his home (near the Atomic Energy Institute of the USSR Academy of Sciences) to discuss the manuscript. Having fed me, as usual, by boiled buckwheat and calling me, as usual, “Dimka” (M. A. called me in such a way until his death), Mikhail Aleksandrovich explained to me that the paper could not be published in ZhETF due to the following reasons.

  1. The manuscript contained the words “theorem” and “proof” forbidden in ZhETF.
  2. The manuscript claimed that “A implies B” while every physicist knew examples showing that B does not imply A.
  3. The manuscript used the unintelligible terms “Lebesgue measure”, “invariant tori”, “Diophantine conditions”. Mikhail Aleksandrovich therefore proposed that I should rewrite the paper.

Now I realize how right he was in defending a physical journal from the Bourbaki-like mathematical jargon. For instance, indeed, while claiming that “A implies B” the author must point out explicitly whether the converse holds, otherwise any reader not spoiled by the mathematical slang would understand the claim as “A is equivalent to B”.

The issue is that mathematical deductive reasoning is rarely if ever applicable outside of mathematics where it should be replaced by the Bayesian paradigm of what Polya called plausible inference, of which it is an extreme case.

The moral for teaching is that the difference between necessary and sufficient conditions is not something to be passed over lightly and that confusion in regards to it is not necessarily evidence of stupidity. When one teaches that all symmetric matrices are diagonalizable one must remind students that not all diagonalizable matrices are symmetric. Moreover, all orthogonally diagonalizable matrices are symmetric ...

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