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Logical Literacy

matt.might.net

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Re: Logical Literacy

#21

Post by mattmight »

nene wrote: post #13
Someone wrote: post #11

Earlier quoted context omitted.

I also think it is unfortunate that his first theorem is incorrect (n=1 cannot be written as the product of two distinct integers) That may cost the author some of the readers for whom this text is the most educational, i.e. the ones that have trouble grasping the logic.

Even if it's not meant to be true, I find it even more disturbing that the symbolic and English formulations doesn't seem to quite match up. (IMHO the symbolic formulation doesn't state that n must be integer.) It's like the source code and comments being out of sync - you don't know which one is wrong.

Another great point.

I fixed the broken theorem, and I added a blurb on the "universe of discourse."

Re: Logical Literacy

#22

Post by Jach »

gacba wrote: post #6

It's unfortunate that this kind of logic isn't emphasized more in mathematics or science in a pure form. I'm pretty sure in high school I had one semester of this kind of logic associated with geometry. That was it until I hit discrete structures in college. Failure to understand this logic might be part of the reason there's so much magical thinking about things relating to pseudoscience (e.g. paranormal activity, U…

I think a stronger base in real probability theory (as opposed to simply frequency analysis) would be more beneficial, especially if you follow Jaynes' route and treat it as an extension to logic. As a commenter noted above, "implication" is easy to grasp, except it's not, because logical implication is not necessarily causal or does not even have to make real-world sense, whereas one of the goals of Bayesian Probability Theory is that it should have a qualitative correspondence with common sense.

Re: Logical Literacy

#23

Post by nitrogen »

omaranto wrote: post #20
astrofinch wrote: post #18

This is a great example of how awesome teaching yourself using the Internet can be. If you read and understood that in 30 minutes then you learned it in about one quarter of the time we spent covering this stuff in my discrete math class--and I'm supposedly going to a really good college.

I don't see why you mention your college is good seemingly surprised it took longer than 30 minutes to cover this: wouldn't you expect a good college to have good teachers and wouldn't you expect a good teacher to go slowly enough that all or most students understand?

I suspect there is an optimum speed at which the greatest number of students are able to understand, and I suspect that it isn't the slowest speed possible.

In late elementary and junior high school I recall getting in trouble for being ahead of the class, particularly in computer-related classes and assignments. A faster pace would've allowed students like me to benefit from the class, but may have left slower students behind.

Re: Logical Literacy

#24

Post by tommorris »

gacba wrote: post #6

It's unfortunate that this kind of logic isn't emphasized more in mathematics or science in a pure form. I'm pretty sure in high school I had one semester of this kind of logic associated with geometry. That was it until I hit discrete structures in college. Failure to understand this logic might be part of the reason there's so much magical thinking about things relating to pseudoscience (e.g. paranormal activity, U…

I'm not sure logic exclusively helps in stopping pseudoscientific and magical thinking. Really a more holistic approach is required:

Logic: to see how to reason deductively.

Epistemology and/or philosophy of science: to see why common meta-beliefs about knowledge and discovery are wrong.

Direct exposure to either a scientific or logic-driven discipline of study or practice: doesn't matter too much if this is a history book or the demands of a compiler. You have to put the reasoning in practice.

Developing a high-quality bullshit detector relies on much more than just an introduction to logic textbook. It is a bit like immunization: you have to encounter bad reasoning in a safe environment and build up your own defenses. (That's all IMHO of course.)

Re: Logical Literacy

#25

Post by astrofinch »

sid0 wrote: post #19
astrofinch wrote: post #17

Earlier quoted context omitted.

It makes perfect intuitive sense if you're used to the way the word "if" is used in logic.

It makes perfect intuitive sense if you're used to the way the word "if" is used in classical logic. Remember that it's you who has bent your intuitive sense to fit how material implication works. Most people are not going to think it makes intuitive sense, and the fault doesn't lie with them -- it lies with the logic. Logic is like UI: it's Not the User's Fault he doesn't find implication intuitive.

Logic is not like UI. UI is a translation layer between the way humans think and the way a machine works. Logic isn't a translation layer. It's more like a theoretical machine.

The English terms we use to talk about logic could be considered similar to UI. But I don't think that using the word "if" (instead of making up a new word like "garply") was obviously the wrong design decision for the inventors of English logical terminology. "If" is the word in the English language that most closely maps to the concept the terminology designers wanted to describe.

Re: Logical Literacy

#26

Post by astrofinch »

omaranto wrote: post #20
astrofinch wrote: post #18

This is a great example of how awesome teaching yourself using the Internet can be. If you read and understood that in 30 minutes then you learned it in about one quarter of the time we spent covering this stuff in my discrete math class--and I'm supposedly going to a really good college.

I don't see why you mention your college is good seemingly surprised it took longer than 30 minutes to cover this: wouldn't you expect a good college to have good teachers and wouldn't you expect a good teacher to go slowly enough that all or most students understand?

Well I'd figure that a good college would have smart students and with smart students a teacher could go fast.

In any case, even if the teachers are making the right move by going slowly, that doesn't change the fact that teaching yourself using the internet is awesome for the justification that I gave.

Re: Logical Literacy

#27

Post by sid0 »

astrofinch wrote: post #25
sid0 wrote: post #19

Earlier quoted context omitted.

It makes perfect intuitive sense if you're used to the way the word "if" is used in classical logic. Remember that it's you who has bent your intuitive sense to fit how material implication works. Most people are not going to think it makes intuitive sense, and the fault doesn't lie with them -- it lies with the logic. Logic is like UI: it's Not the User's Fault he doesn't find implication intuitive.

Logic is not like UI. UI is a translation layer between the way humans think and the way a machine works. Logic isn't a translation layer. It's more like a theoretical machine. The English terms we use to talk about logic could be considered similar to UI. But I don't think that using the word "if" (instead of making up a new word like "garply") was obviously the wrong design decision for the inventors of English log…

> Logic isn't a translation layer. It's more like a theoretical machine.

More precisely, it is a model for human thought. The way logic is similar to UI is that logic's philosophical success is a function of how well human thought maps to it and how intuitive theorems in the logic are, just as UI's success is a function of how intuitive it is. The last hundred years of discourse on philosophical logic have been focused on one thing -- to formalize the human notion of implication better than material implication does. That's why logicians came up with non-classical logics of the modal, counterfactual (which captures the very intuitive notion of mutatis mutandis), fuzzy and intuitionistic varieties.

Re: Logical Literacy

#28

Post by astrofinch »

sid0 wrote: post #27
astrofinch wrote: post #25

Earlier quoted context omitted.

Logic is not like UI. UI is a translation layer between the way humans think and the way a machine works. Logic isn't a translation layer. It's more like a theoretical machine. The English terms we use to talk about logic could be considered similar to UI. But I don't think that using the word "if" (instead of making up a new word like "garply") was obviously the wrong design decision for the inventors of English log…

> Logic isn't a translation layer. It's more like a theoretical machine. More precisely, it is a model for human thought. The way logic is similar to UI is that logic's philosophical success is a function of how well human thought maps to it and how intuitive theorems in the logic are, just as UI's success is a function of how intuitive it is. The last hundred years of discourse on philosophical logic have been focus…

OK, but has anyone managed to construct the real numbers with intuitionist logic? Maybe different logics are like different building materials, each with their own advantages and disadvantages.

Re: Logical Literacy

#29

Post by sid0 »

astrofinch wrote: post #28
sid0 wrote: post #27

Earlier quoted context omitted.

> Logic isn't a translation layer. It's more like a theoretical machine. More precisely, it is a model for human thought. The way logic is similar to UI is that logic's philosophical success is a function of how well human thought maps to it and how intuitive theorems in the logic are, just as UI's success is a function of how intuitive it is. The last hundred years of discourse on philosophical logic have been focus…

OK, but has anyone managed to construct the real numbers with intuitionist logic? Maybe different logics are like different building materials, each with their own advantages and disadvantages.

> OK, but has anyone managed to construct the real numbers with intuitionist logic?

I think it should be possible to do so. As long as you don't need the law of the excluded middle you should be fine.

> Maybe different logics are like different building materials, each with their own advantages and disadvantages.

I see where you're coming from, but I think logic is more than just a tool.

Re: Logical Literacy

#30

Post by sid0 »

sid0 wrote: post #27
astrofinch wrote: post #25

Earlier quoted context omitted.

Logic is not like UI. UI is a translation layer between the way humans think and the way a machine works. Logic isn't a translation layer. It's more like a theoretical machine. The English terms we use to talk about logic could be considered similar to UI. But I don't think that using the word "if" (instead of making up a new word like "garply") was obviously the wrong design decision for the inventors of English log…

> Logic isn't a translation layer. It's more like a theoretical machine. More precisely, it is a model for human thought. The way logic is similar to UI is that logic's philosophical success is a function of how well human thought maps to it and how intuitive theorems in the logic are, just as UI's success is a function of how intuitive it is. The last hundred years of discourse on philosophical logic have been focus…

Sorry, that's more precisely ceteris paribus -- "all else being the same".
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