How do we approach a new proposition?

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Rian
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Re: How do we approach a new proposition?

Post by Rian » Mon Nov 05, 2018 4:18 am

Hey Og, I'm not around much anymore, but wow! people sure do need to be taught to think more nowadays! The things you hear that just make you want to pull your hair out! Logical fallacies, assumptions that people refuse to realize are assumptions - it's nuts. I haven't studied this subject as in-depth as you have, but I've definitely studied it and it really has made me a much better thinker.

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Mon Nov 05, 2018 7:49 am

Rian wrote:
Mon Nov 05, 2018 4:18 am
Hey Og, I'm not around much anymore, but wow! people sure do need to be taught to think more nowadays! The things you hear that just make you want to pull your hair out! Logical fallacies, assumptions that people refuse to realize are assumptions - it's nuts. I haven't studied this subject as in-depth as you have, but I've definitely studied it and it really has made me a much better thinker.
I'd like to see the basics of logic taught in grade schools -- not the abstract mathematics of it, but the simple breakdown of propositions into valid or invalid. At best, we sometimes see this taught to high school students, and even then it's usually as a kind of a "no wrong answer" kind of a class.

This is fundamental to effectively learning anything else -- science, critical thinking, writing, politics -- so one would hope that more people would want to see a solid foundation built on it. I am constantly reminded of Feynman's CalTech commencement speech about Cargo Cult Science.
EGO TE ABSOLVO, and there's nothing you can do about it.

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Tue Nov 06, 2018 11:54 am

So, I'd like to talk about the Hidden Premise.

A hidden Premise occurs when we take for granted that a single premise leads to a conclusion. As we know, there must be a general premise and a specific premise in order to apply modus tollens or modus ponens. Statistics are a good place for hidden premises to conceal themselves, so let's see if we can capture one in the wild.

"83% of people who quit smoking do so before the age of 25. Therefore we should concentrate on smoking cessation campaigns aimed at this age group."

What's the hidden premise?

..

..

..

It's "If people do not quit smoking, then they are unable to do so (i.e. too old and too addicted)" and while that may or may not be true, it may also be true that the statistic is telling us that we've concentrated 83% (or more) of our smoking cessation efforts on persons under 25. We need more information than just the bare statistic if we are to make a valid General Premise, and from that draw a valid conclusion.

Let's try another one: "95% of college graduates do not believe that Australia exists."

Here, not only is a premise hidden, but the conclusion is not explicitly stated. It is only implied that "Therefore Australia does not exist." Why would we not state that explicitly? Well, when stated, we see that it is an appeal to dubious authority (the fact that someone is college educated does not make him or her an authority on Australia, Existence, Logic, or the Box Jellyfish (I prefer the kind in jars, myself).

So what's the hidden premise?

..

..

..

"If college graduates do not believe a proposition, then it is not true." Right-o, the Appeal to dubious authority in so many words.

Now, you may recall that earlier, we said that your premises or axioms lead to your conclusions. So let's say that we also assume that Australia does exist -- yes, that's silly, but bear with me: Then we get a different implied conclusion when we say that 95% of college graduates do not believe in Australia: Therefore our colleges are doing a horrible job at educating our children! Only 5% believe in the true *coughcough* proposition that Australia exists!

This illustrates two things: First, that sneaky SOBs can hide what they truly mean by using hidden premises, and thus make the false conclusion appear true; and secondly, that we must be clear on our axioms if we are to draw valid conclusions. Beware the hidden premise. Avoid it like you would a jabberwock.
EGO TE ABSOLVO, and there's nothing you can do about it.

Rian
Posts: 203
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Re: How do we approach a new proposition?

Post by Rian » Fri Nov 30, 2018 7:01 am

Og3 wrote:
Mon Nov 05, 2018 7:49 am
Rian wrote:
Mon Nov 05, 2018 4:18 am
Hey Og, I'm not around much anymore, but wow! people sure do need to be taught to think more nowadays! The things you hear that just make you want to pull your hair out! Logical fallacies, assumptions that people refuse to realize are assumptions - it's nuts. I haven't studied this subject as in-depth as you have, but I've definitely studied it and it really has made me a much better thinker.
I'd like to see the basics of logic taught in grade schools -- not the abstract mathematics of it, but the simple breakdown of propositions into valid or invalid. At best, we sometimes see this taught to high school students, and even then it's usually as a kind of a "no wrong answer" kind of a class.

This is fundamental to effectively learning anything else -- science, critical thinking, writing, politics -- so one would hope that more people would want to see a solid foundation built on it. I am constantly reminded of Feynman's CalTech commencement speech about Cargo Cult Science.
I think it would be great to start to teach logic in elementary school! You could make it kind of fun, and it would give students a good start on good thinking. If you introduce a subject like that for the first time in high school, they'll probably tune it out. You could start out with things like:

The president of the United States must be 35 years of age or older.
Donald Trump (gag!) is 35 years of age or older.
So, Donald Trump is president of the United States.

Get them to agree that it's true that Trump is president, then substitute another name in (of a person who is older than 35) and then ask them to find out why it doesn't work. Try to hit some of those things that at first glance make sense, and then show the problem with it; things like that. Then get out some flannel graphs and do Venn diagrams - great stuff!

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Fri Nov 30, 2018 9:08 am

That sort of thing would be a good start. It could tie in with English lessons (because we see that "All presidents are 35 years of age or older" is the same as "If he/she is the president, then he/she is at least 35 years old" really say the exact same thing, etc). And we can tie it to math (though we really don't want to get that the null set is the union of the set of all red things and the set of all non-red things, etc., until much later).

If I knew an elementary teacher who wanted to write a book, I might collaborate on a textbook...
EGO TE ABSOLVO, and there's nothing you can do about it.

Rian
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Re: How do we approach a new proposition?

Post by Rian » Sat Dec 01, 2018 9:45 am

I agree the null set in the way you stated it would be a bit much for elementary schoolers!

I think using manipulatives such as colored flannel would be a really good tool, like "let the big green circle stand for all dogs, and the little blue one stand for dalmatians. Put the blue circle inside of the green circle, and you can see that while all dalmatians are dogs, there are dogs that are not dalmatians." Start there, then add in other breeds, then maybe intersect with cross-breeds. Then set up an entirely new scenario and just give them words and let them see if they can represent it with the flannel pieces.

My daughter and I were talking today and she told me that she is SO grateful that I went over logical fallacies and analysis with her! Things like the false dilemma, the "No true Scotsman", appeal to authority, etc. And also that I taught her to look for unstated premises/assumptions/definitions - those are so important, because the argument can be sound but the conclusion can be incorrect if there are incorrect premises/assumptions/definitions. She then told me that one way she deals with people who try to throw arguments at her that she has a feeling they are just repeating and don't understand, is to say "I don't quite understand what you're saying - could you please explain it in a little more detail?" and then they can't, because they never understood it in the first place! They just got used to how so many people won't argue with you if you speak loudly and use big words! :D

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Tue Dec 04, 2018 1:14 am

Rian wrote:
Sat Dec 01, 2018 9:45 am
I agree the null set in the way you stated it would be a bit much for elementary schoolers!

I think using manipulatives such as colored flannel would be a really good tool, like "let the big green circle stand for all dogs, and the little blue one stand for dalmatians. Put the blue circle inside of the green circle, and you can see that while all dalmatians are dogs, there are dogs that are not dalmatians." Start there, then add in other breeds, then maybe intersect with cross-breeds. Then set up an entirely new scenario and just give them words and let them see if they can represent it with the flannel pieces.

My daughter and I were talking today and she told me that she is SO grateful that I went over logical fallacies and analysis with her! Things like the false dilemma, the "No true Scotsman", appeal to authority, etc. And also that I taught her to look for unstated premises/assumptions/definitions - those are so important, because the argument can be sound but the conclusion can be incorrect if there are incorrect premises/assumptions/definitions. She then told me that one way she deals with people who try to throw arguments at her that she has a feeling they are just repeating and don't understand, is to say "I don't quite understand what you're saying - could you please explain it in a little more detail?" and then they can't, because they never understood it in the first place! They just got used to how so many people won't argue with you if you speak loudly and use big words! :D
I've been seeing that on internet forums for a long time. It used to puzzle me when I would state a concept in my own words -- Kant's Categorical Imperative, perhaps -- and then have someone argue with me that it was really [textbook-recitation-of-KCI]. What I finally realized was that they had memorized the words without ever understanding the concepts behind the words.

It was a bit like the example with Polarization of light that Richard Feynman used in his book* "Surely you're joking Mr. Feynman." He was talking about teaching in Brazil, and that if he asked for a definition, the entire class could recite it in unison, verbatim, perfectly. But if he asked them to apply it, they were utterly stumped. Lesson learned: Knowing what it means is better than reciting it.

Teaching her to ask "What do you mean by that?" or"Give me an example?" would also be great tools.

For children especially, the two Raymond Smullyan books "What is the Name of This Book" and "The Lady or the Tiger?" are great learning tools.

There is a game built into windows that can also be a great tool, called "Minesweeper." It requires finding patterns and making deductions based on incomplete information: If there are three mines within one square, then this one cannot be a mine and those two must be mines.

To really put together a course, though, you'd have to organize and stratify the information...
____________________________
* Okay, it was actually a collection of his essays, transcribed lectures, etc., and he didn't intend for it to be a book... But it makes a great one.
EGO TE ABSOLVO, and there's nothing you can do about it.

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Fri Dec 07, 2018 6:40 pm

So, a quick review of syllogisms:

General Premise(GP): If X, then Y

Modus Tollens:
Specific Premise (SP): X is true
Conclusion: Y is true.

Modus Ponens:
Specific Premise (SP): Y is false
Conclusion: X is false.

We cannot make a method where we say Y is true, because that tells us nothing about X. We cannot make a method where we say that X is false, because that tells us nothing about Y. In these cases we say that the conclusion "Does not follow."

Also, if the conclusion is unrelated to the premises, that is also a Non Sequitur ("does not follow") (Or even make sense).

Questions, comments, War Stories?
EGO TE ABSOLVO, and there's nothing you can do about it.

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Tue Jan 01, 2019 9:56 pm

Alright, so now let's talk about evidence.

First, I have encountered a few people who are foggy on the difference between evidence and proof, or of proof and of a proof.

Evidence is anything that seems to pertain to a question. If I ask myself whether 2 + 2 = 4 in base 10 integers, then the peano axioms are evidence of this (but not proof of it). I can use the peano axioms to construct "a proof" of the question ("does 2+2=4") and if that proof is solid then if forms "proof" (i.e. strongly suggests a reasonable conclusion) that 2+2=4.

For simplicity, when we refer to "A Proof" we will call in "An Argument."

By "An argument" we mean a series of propositions which lead to a reasonable inference (lead to "proof").

Now a reasonable inference can be defined as meeting any of several standards:
1. Geometric proof. By this we mean that the inference must be necessarily be objectively true given the assumptions.
2. Beyond a shadow of a doubt. By this we mean that the potential for error is so low that no one could possibly harbor any kind of doubt.
3. Beyond a reasonable doubt. By this we mean that no reasonable person, using logic, could make a reasonable inference other than this one.
4. By Preponderance of Evidence. By this we mean that the sum total of the accumulated evidence points primarily to one conclusion.
5. By clear and convincing evidence. By this we mean that more evidence supports one conclusion than another.

Very few things can be defined to Geometric certainty, and even Geometric certainty hinges upon the axiom set that one chooses to endorse. You can find posts on this board where Moonwood has discussed axiom sets at some length, if you wish to delve into the induction/deduction problem.

Beyond a shadow of a doubt is almost as difficult a standard for anything at all. Above I said I was asking if 2+2=4 in base 10 integers, because if I didn't specify, one might argue that 2+2=10 in base 4 integers, or that 2+2=5 for high values of 2 (that is, 2.3 rounds to 2, but 2.3+2.3 = 4.6, which rounds to 5). Of course you understand that the latter example requires an ambiguous definition, and thus is sophistry.

So the most common and most useful standard for "proof" is "beyond a reasonable doubt." If a hypothetical reasonable person, with no prejudices and no biases, were to look solely at the evidence before us, would he necessarily reach this conclusion (that 2+2=4), or would there be room for him to accept that 2+2=12, 2+2=0, or 2+2=sqrt-1? Of course he would say that 2+2=4, beyond any REASONABLE doubt.
EGO TE ABSOLVO, and there's nothing you can do about it.

Og3
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Re: How do we approach a new proposition?

Post by Og3 » Tue Jan 01, 2019 10:34 pm

So, to give an example of reaching a conclusion based on a standard of proof:

I sat on a jury in a criminal trial. The question we were to decide was whether the defendant was guilty of DWI (Oz: "Drink Driving") in an incident that occurred on a certain date. A question, you will recall, is a thing to be determined.

The defendant and two others were found at the scene of an accident (car vs. sign-post) and the defendant stated then that he had been driving. He was given a breathalyzer and was above the legal limit. A second car was called out with a second breathalyzer, and the defendant also failed that test.

Evidence was given at some length about how breathalyzers work and how they are calibrated. Please note (digression alert) that this evidence, in and of itself, did not PROVE anything about the question. It supported the reasonable inference that the breathalyzer tests were correct and accurate. Evidence supports an inference, it does not of itself PROVE that inference.

When we retired to the jury room, we all immediately agreed that beyond any reasonable doubt, and for that matter beyond a shadow of a doubt, that the defendant and his friends had all been drunk, and that the crime of DWI ("drink driving") had been committed. We stopped short of geometric certainty.

But the crux of the matter came to whether he had been driving. Someone had obviously been driving (it was beyond a shadow of a doubt that the car didn't teleport on top of the traffic sign by itself). But the defendant stated that one of his friends had been driving. He stated that his mother had given him strict instruction not to let anyone else drive her car, but that he had been unable to drive, and had allowed a slightly more sober friend to drive instead.

When asked by the officer, he thought that his mother would be even more angry if he had allowed her car to be wrecked by someone else than if he had done it himself. So he stated to the officer that he had been driving.

Evidence was given about the number of opportunities he had had to recant this statement before being placed in a cell; he stated repeatedly that he had been driving. Now, again, this is not "proof" per se: It is evidence that supports the inference that he was driving.

His primary evidence that he had allowed a friend to drive consisted of his testimony, his fear of his mother's wrath, and a medical condition called nystagmus that is aggravated by alcohol. Again, not proof, but in support of an inference that he was not driving.

So we debated the question of whether he was driving. We used the court-mandated standard of a reasonable doubt. Could a reasonable person say he was driving? Yes. Could a reasonable person say he was not? Well... Yes. So we had to err on the side of presumption of innocence, and we let him go.

By Clear and Convincing Evidence? GUILTY
By Preponderance of Evidence? GUILTY
By Reasonable Doubt? POSSIBLY NOT GUILTY.
By Shadow of Doubt? DRUNK off his patoot.

The reasonable inference was that he was guilty; the correct verdict by reasonable doubt was Not Guilty. So let it be written, so let it be done.
EGO TE ABSOLVO, and there's nothing you can do about it.

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