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Topic: Is logic part of mathematics - or is mathematics part of logic?
Replies: 8   Last Post: Jul 7, 2013 5:19 PM

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Robert Hansen

Posts: 7,702
From: Florida
Registered: 6/22/09
Re: Is logic part of mathematics - or is mathematics part of logic?
Posted: Jul 7, 2013 5:19 PM
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On Jul 7, 2013, at 1:17 PM, Joe Niederberger <niederberger@comcast.net> wrote:

> This has gotten very confused. Wayne was making a joke.
> Propositional logic is the same in CS as it is in math,
> and the fallacies are the same.


It was a joke Joe.


>
> But, to get back to the point that Robert was making -
>
> You can start with a simple proposition like,
>
> "If it rains today, we (definitely) will go to the movies".
>
> Most kids will understand that readily, and if they are keen to get to the movies they will notice when it starts to rain and press their parents with the logical conclusion that they should be off to the movies.
>
> That's unreflective use of logic.


That isn't logic, reflective or not. I don't know why people make this mistake. If I drop a ball it falls down. I drop a ball. It falls down. Is that logic? No. I liken this to the "ants doing trig" story. Just because what ants do can be described with trig doesn't mean that the ants are doing trig, reflective or not. When you tell a kid "if it rains then we are going to the movie" the kid EQUATES rain with movie. That by itself is simply not logic.


> Now suppose you ask one of them the hypothetical "if we don't go to the movies, what does that mean?" I think very few would pipe up and say "it means it must not have rained!" (But some might.)

Ok, now we are getting somewhere, maybe, but probably not. IF a 10 year old pipes up and says "Since we didn't go to the movies it must not have rained BECAUSE you said if it rained we would go to the movie." then I would start feeling the love, but I would still want more evidence. However, most 10 year olds will say "We didn't go to the movie BECAUSE it didn't rain." That is NOT logic. That is equating rain with movie. That is cause and effect. Do you see the difference?


> You can get probably more followers of your logical line of thought if you simply state (rather than try to get them to "discover" this): "And - if we do not go to the movies, it can only be if it does not rain today." Your kids might have to chew on that a while, but if they are the attentive kind, and think about it, they will see its connected to what you said before.

I don't think you have brought them to the water yet. They are going to chew on it forever unless you have some more parts to this lesson plan. They are going to think that they didn't go to the movies for a bunch of reasons (car broke down, school night, etc.). This is part of the problem with introducing "propositional" logic too early. They are not mature enough yet to understand "propositional". It is going to take some work for them to understand what you mean by "it can only be if it does not rain today".

> And so on, and so on, if you can keep their attention, you can go over *all* the cases:
>
> 1. it rains; you go to the movies
> 2. no movie; so must be that no rain
> 3. no rain; what's that mean?
> 4. we went to the movies; so what's that mean?
>
> 3 & 4 are tricky, are they not?


They are tricky because people don't think that way.:) Take away analogy and equating and most people don't even understand what the hell you are asking.

If you win the lottery you will be rich...

1. Sam won the lottery, is Sam rich? Yes
2. George is not rich. Did George win the lottery? No
3. Paul didn't win the lottery. Is Paul rich? Who Knows?
4. Jane is rich, did Jane win the lottery? Who Knows?

Maybe a tad better but still the same problem. If Jane can be rich without winning the lottery, then why can't George be not rich and win the lottery? You say that Jane could be rich from something other than the lottery then maybe George gave his winnings away. This subtle difference kills it for most. Also, #3 and #4 are not questions with an answer.


> One could go over these very carefully, and also make careful and exhaustive note of what was *not* said.
> The original statement was *not*:
> "If it rains, we *may* (or may not,) go to the movies".
> Etc. etc. etc. through many variations.



It might be better to frame it this way...

Which of the following statements are logically correct?...

1. Sam won the lottery, therefore Sam is rich?
2. George is not rich, therefore George did not win the lottery?
3. Paul did not win the lottery, therefore Paul is rich?
3a. Therefore Paul is not rich?
4. Jane is rich, therefore Jane won the lottery?
4a. Therefore Jane did not win the lottery?


> I would say that is all leading to a reflective, attentive, and sustained use of whatever latent logical ability the individuals possess natively.

It looks like it to me and you. I think you should try it with some live children.:)


> After all that is very firm (at what point exactly? I don't know.), then one might proceed to the notion of "implication" as an abstract connective, with its associated truth table etc.
> Before one ever got to implication though, double negatives, the difference between inclusive and exclusive or, etc. should have been given this same level of careful attention.


Kids actually get double negatives, inclusive, exclusive etc. almost as soon as they get language.

I think getting the abstract notion of "implication" is critical for any of this to pan out.

Joey will be fine with "If Jane is rich, then can we say that Jane won the lottery?" because Joey will think "No, Jane could be rich for other reasons." But that thought process works with or without the sense of logic because Joey is thinking it through using common sense. We will not know if Joey actually gets "logic" until we start asking him questions like...

If A then B. B therefore A?

And bigger questions made up of these little questions.

Plus, this doesn't account for the fuzzy logic which is, by definition, the predominate part of mathematics while you are learning it.

We would have to use a lot of examples and tests, which is what I thought mathematics was for.


Bob Hansen



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