# Logical literacy is essential to mathematical fluency.

October 12, 2010 5:50 PM Subscribe

And in the US we proceed to beat it out of children via geometry. Because the best way to educate people is to take a pure abstract concept and chain it to a fairly intuitive concrete system, thereby ensuring nobody loves the subject.

posted by pwnguin at 6:06 PM on October 12, 2010 [2 favorites]

posted by pwnguin at 6:06 PM on October 12, 2010 [2 favorites]

Took this in college and loved it. It's one of the things that rekindled my interest in programming (doing proofs was like an abstract form of programming).

posted by treepour at 6:17 PM on October 12, 2010 [1 favorite]

posted by treepour at 6:17 PM on October 12, 2010 [1 favorite]

When I was a CS undergraduate, every CS freshman took a Programming Logic class, using this book. Covers natural deduction, first order logic and then applies them to programs with Hoare logic. A computer engineering buddy of mine took the class a couple times, but apparently just couldn't cope with logic outside of NAND gates.

posted by pwnguin at 6:29 PM on October 12, 2010 [2 favorites]

posted by pwnguin at 6:29 PM on October 12, 2010 [2 favorites]

Yeah, took Logic ! and II in college and they were awesome.

Also took geometry in HS and loved it. Best. Math class. EVAR. I have no idea what the problem is supposed to be with teaching abstracts via concretes.

posted by DU at 6:59 PM on October 12, 2010

Also took geometry in HS and loved it. Best. Math class. EVAR. I have no idea what the problem is supposed to be with teaching abstracts via concretes.

posted by DU at 6:59 PM on October 12, 2010

pwnguin, I don't quite get how geometry ruins someone's understanding of logic.

posted by RobotHero at 7:01 PM on October 12, 2010

posted by RobotHero at 7:01 PM on October 12, 2010

To really do proofs in logic, you need to forget their translation into English words. Later you can map propositions to sets, 'or' to 'union', 'and' to 'intersection', 'implies' to 'is a subset of'.

posted by Obscure Reference at 7:06 PM on October 12, 2010

posted by Obscure Reference at 7:06 PM on October 12, 2010

pwnguin has discovered a truly remarkable proof that geometry ruins one's understanding of logic, which his comment is too small to contain.

posted by sebastienbailard at 7:14 PM on October 12, 2010 [17 favorites]

posted by sebastienbailard at 7:14 PM on October 12, 2010 [17 favorites]

I loved taking a logic class in college, and did well in it. That said, nowadays if I so much visualize a ∉ my brain suddenly gets cloudy and.. so... tired, gonna take a nap now

posted by not_on_display at 7:15 PM on October 12, 2010

*[head hitting desk]*what where am I?posted by not_on_display at 7:15 PM on October 12, 2010

"Hoare logic" is that spelled wrong?

That said... my wife, the math teacher loved this link...thanks.... Me, i just got confused...

posted by HuronBob at 7:18 PM on October 12, 2010 [1 favorite]

That said... my wife, the math teacher loved this link...thanks.... Me, i just got confused...

posted by HuronBob at 7:18 PM on October 12, 2010 [1 favorite]

Pairing it with geometry leads some students to question the point in all the formalism when it just

OK, I don't actually think the fundamental problem here is the pairing with geometry, though I do believe many students react like I just described. I think the fundamental problem is that proofs and showing your work often

I got a pretty decent public education, but I have to say this wasn't part of it, it wasn't really until I got to university that I learned this, and even there, it wasn't so much taught to me as sortof absorbed. Early in my freshman year I realized I didn't

I often wondered when I was thinking I might become a high school math teacher if it might actually be best to avoid teaching any formulas before spending some time wrestling with problems a given formula might solve, and then working to tease the formula out of what might already be known.

Many kids don't like that much either, though, so maybe it's not that much better than pairing logic/proof with geometry.

posted by weston at 7:22 PM on October 12, 2010 [1 favorite]

*looks right*, and the proofs just become silly hoops the teacher wants you to do, like showing your work.OK, I don't actually think the fundamental problem here is the pairing with geometry, though I do believe many students react like I just described. I think the fundamental problem is that proofs and showing your work often

*are*more or less things you do to satisfy the teacher, when their real value has little to do with that. The real payoff is when playing around with the formal system can help you discover implicit facts about the system that you didn't already know and satisfy*yourself*(as well as anybody else you may need to convince) about the truth (and limits) of these facts you're discovering/deriving.I got a pretty decent public education, but I have to say this wasn't part of it, it wasn't really until I got to university that I learned this, and even there, it wasn't so much taught to me as sortof absorbed. Early in my freshman year I realized I didn't

*have*to remember all of the various rules for derivatives or the formula for integration by parts or various trig integration tricks; all I had to do was remember the definitions and some basic rules of the system and I could re-discover/derive them when necessary.I often wondered when I was thinking I might become a high school math teacher if it might actually be best to avoid teaching any formulas before spending some time wrestling with problems a given formula might solve, and then working to tease the formula out of what might already be known.

Many kids don't like that much either, though, so maybe it's not that much better than pairing logic/proof with geometry.

posted by weston at 7:22 PM on October 12, 2010 [1 favorite]

Tony Hoare is a brit computer scientist that is perhaps the most unsung computer scientist, he was well ahead of his time and his ideas are becoming really important when cpus have more than a couple cores.

posted by sammyo at 7:35 PM on October 12, 2010

posted by sammyo at 7:35 PM on October 12, 2010

"

posted by kenko at 8:04 PM on October 12, 2010

*f*is the function that squares its argument and adds two." and "∀x(*f(x)*=*x*^{2}+ 2)" aren't really equivalent; the first defines the function*f*in terms of what it does to its argument (it tells you how to determine*f*'s extension) and the second states an equivalence between the extension of the function*f*(otherwise known) and a different expression. So it's misleading to say that the second (universally quantified) formula is more often written simply as "*f(x)*=*x*^{2}+ 2".posted by kenko at 8:04 PM on October 12, 2010

Ham-fisted treatments of logic, like this one, suitable for neither beginner nor expert, serve only to irritate the ignorant and fellate the knowledgeable.

posted by libcrypt at 8:14 PM on October 12, 2010 [8 favorites]

posted by libcrypt at 8:14 PM on October 12, 2010 [8 favorites]

*Ham-fisted treatments of logic, like this one, suitable for neither beginner nor expert, serve only to irritate the ignorant and fellate the knowledgeable.*

Bless your heart.

posted by jjray at 8:21 PM on October 12, 2010 [6 favorites]

jjray...you made my night! :)

posted by HuronBob at 8:24 PM on October 12, 2010 [1 favorite]

posted by HuronBob at 8:24 PM on October 12, 2010 [1 favorite]

*Bless your heart.*

Watch it, sarcasto. I will crush you with a chain of ultrapowers of the reals after caving in yr skull with a generic object.

posted by libcrypt at 8:46 PM on October 12, 2010 [1 favorite]

One of the reasons I was so unhappy in Philosophy was because of the ... obsession that most Philosophers today seem to have with symbolic logic. I'm HORRIBLE at it. I can write code like no tomorrow, but doing proofs? Forget it. Set theory I can do - but put it into symbolic logic of any type and my brain just Dies. And when my interest was in eastern philosophy, and political philosophy? It was useless to me, but it was insisted upon.

So, I understand the necessity of being logically literate, and understanding argument forms and proofs. On the other hand, I consider it to be over rated in many fields.

posted by strixus at 8:54 PM on October 12, 2010

So, I understand the necessity of being logically literate, and understanding argument forms and proofs. On the other hand, I consider it to be over rated in many fields.

posted by strixus at 8:54 PM on October 12, 2010

This is great.

I do find that a lot of interesting work in the field of logic and reasoning is unnecessarily obtuse, or maybe I should say "plain badly written", because it's only discussed by a the same handful of people for decades on end and as a result is a form of short hand for the illuminati.

posted by joost de vries at 8:57 PM on October 12, 2010

I do find that a lot of interesting work in the field of logic and reasoning is unnecessarily obtuse, or maybe I should say "plain badly written", because it's only discussed by a the same handful of people for decades on end and as a result is a form of short hand for the illuminati.

posted by joost de vries at 8:57 PM on October 12, 2010

joost, I think this is true for any small field, not just logic.

(If "small field" has some technical meaning I didn't intend it.)

posted by madcaptenor at 9:24 PM on October 12, 2010

(If "small field" has some technical meaning I didn't intend it.)

posted by madcaptenor at 9:24 PM on October 12, 2010

*This is great.*

Can someone say why this is any good at all? I'm inclined to agree with libcrypt; it seems pretty useless to me.

posted by kenko at 9:32 PM on October 12, 2010

When I was an undergraduate this (required) class was "taught" in such a way as to ensure that 50% of the class failed. Every semester. That's not education; it's hazing. So perhaps you'll forgive my lack of enthusiasm for pointlessly proving things already known to be true.

posted by LastOfHisKind at 9:39 PM on October 12, 2010

posted by LastOfHisKind at 9:39 PM on October 12, 2010

I agree with libcrypt... this page reads like a bit of a mess to me.

posted by painquale at 9:43 PM on October 12, 2010

posted by painquale at 9:43 PM on October 12, 2010

I don't think it's a mess, just a whirlwind tour of what you might come across if you're reading logic arguments. My perspective is that of someone who enjoyed Logic in college but hasn't dealt with it much since then. The site was a nice reminder/refresher and might make reading proggit a bit easier in the future.

posted by mad bomber what bombs at midnight at 9:56 PM on October 12, 2010

posted by mad bomber what bombs at midnight at 9:56 PM on October 12, 2010

I took a formal logic class in college (required as a philosophy major) and unexpectedly adored it even though I'd always hated math. Doing proofs felt mostly like spending time mapping how my brain worked: laying out on paper over and over again how I always tried a more complicated solution first instead of seeing the easy one, how I got distracted by strange possibilities, how horrible I was at admitting I just needed to start over. I came out of the class with a more organized brain than I had gone in with, which was always what I liked most about normal philosophy classes anyway.

Also, I had an absolutely fabulous logic professor, who said something in a discussion about truth tables that I've always liked:

"How do you evaluate counter-factuals? How do you evaluate what would have happened if you had done something differently? How do you evaluate statements like 'The elephant in my closet is trampling my clothes' when the elephant doesn't exist?"

posted by colfax at 10:08 PM on October 12, 2010 [5 favorites]

Also, I had an absolutely fabulous logic professor, who said something in a discussion about truth tables that I've always liked:

"How do you evaluate counter-factuals? How do you evaluate what would have happened if you had done something differently? How do you evaluate statements like 'The elephant in my closet is trampling my clothes' when the elephant doesn't exist?"

posted by colfax at 10:08 PM on October 12, 2010 [5 favorites]

There are all sorts of little things that bug me about this page. The treatment of a quantified expression as a definition (as kenko mentioned), the fact that he demonstrates the soundness of modus ponens using syntactic transforms that haven't themselves been proven sound, calling the language that proofs are conducted in "a second metalanguage" when it's not a metalanguage at all, portentous pronouncements about logic being the calculus of truth (?), etc.

He seems to think this will help young mathematicians interpret theorems and conjectures. He admits that they are presented in "crisp, standardized English" (and he takes this to always be a notational variant of symbolic first-order logic, which I find pretty dubious). I'm not a mathematician, so I don't know how often statements of theorems trip people up, but I really doubt that the formal exposition of first-order logic on this page will help interpret them. Are De Morgan's laws, presented symbolically, ever going to be helpful for this purpose? There's something kind of ironic about the page: he says that every field constructs proofs in its own hard-to-understand mathy language, so he's going to teach how to read the preambles of math papers. And then he goes ahead and teaches a hard-to-understand mathy language!

posted by painquale at 10:08 PM on October 12, 2010 [3 favorites]

He seems to think this will help young mathematicians interpret theorems and conjectures. He admits that they are presented in "crisp, standardized English" (and he takes this to always be a notational variant of symbolic first-order logic, which I find pretty dubious). I'm not a mathematician, so I don't know how often statements of theorems trip people up, but I really doubt that the formal exposition of first-order logic on this page will help interpret them. Are De Morgan's laws, presented symbolically, ever going to be helpful for this purpose? There's something kind of ironic about the page: he says that every field constructs proofs in its own hard-to-understand mathy language, so he's going to teach how to read the preambles of math papers. And then he goes ahead and teaches a hard-to-understand mathy language!

posted by painquale at 10:08 PM on October 12, 2010 [3 favorites]

*How do you evaluate counter-factuals? How do you evaluate what would have happened if you had done something differently?*

He was hinting about modal logic.

posted by libcrypt at 10:15 PM on October 12, 2010

I wonder what Gödel would have to say about all this?

posted by AElfwine Evenstar at 10:27 PM on October 12, 2010

posted by AElfwine Evenstar at 10:27 PM on October 12, 2010

I'm neither a beginner nor an expert, and I both enjoyed it and followed it. Perhaps it wasn't written for either of those categories of individual, and the ham-fistedness is on the part of one who assumes it must be, or should be.

posted by perspicio at 10:33 PM on October 12, 2010

posted by perspicio at 10:33 PM on October 12, 2010

*I wonder what Gödel would have to say about all this?*

Curse Paul Cohen! Curse him and his evil counter-Platonist techniques! GCH is dead; long live GCH!

posted by libcrypt at 10:54 PM on October 12, 2010

*a lot of interesting work in the field of logic and reasoning is unnecessarily obtuse, or maybe I should say "plain badly written"*

Try Mel Fitting's book called First-Order Modal Logic (or really anything by Fitting). It's fantastically well-written, even funny in parts, and does a good job of presenting modal logic using tableau proofs, my personal favorite proof system.

And for all the people in this thread who have mentioned enjoying an intro to logic course as an undergraduate, you're easily prepared for and very likely to enjoy the puzzle books by Raymond Smullyan, a logician who studied under Alonzo Church (inventor of the lambda calculus, also known for the Church Turing Thesis). Smullyan's books appear to be fun pastimes but in fact end up teaching you quite a bit about formal logic.

I wonder what Gödel would have to say about all this?

I wonder what Gödel would have to say about all this?

You don't actually have to wonder! Gödel, who is mentioned a few times at the end of the linked article, wrote lots of stuff down, including defining the limits of the field with his incompleteness theorems. Smullyan's book Forever Undecided is a very clever and accessible introduction to the theorems for those who have the interest but not the background to handle Gödel's original work.

Unless by "all this" you meant metafilter, in which case I have no idea, except to say that I'm pretty sure it's inconsistent, and thus is likely to contain a statement of its own consistency.

posted by tractorfeed at 11:50 PM on October 12, 2010 [5 favorites]

This gets all my math nerd love.

posted by Mental Wimp at 12:49 AM on October 13, 2010

posted by Mental Wimp at 12:49 AM on October 13, 2010

*"You know the day destroys the night, Night divides the day."*

Dig that without logical literacy. If you can.

posted by klarck at 2:18 AM on October 13, 2010

I always get so excited when I see one of these overviews. I think, "Yum yum, the building blocks of language! Mmmm!" I get to "P if and only if Q," and all is well. "YUM! Conditional existence!" Says my hungry mind. Things go along smoothly like this for awhile, just feeding off boiled-down statements.

Then, one more line down the page, semblances of language I use (like "If I pour this milk, the cereal gets soggy!") vanish and I'm left squinting at mathy-looking things which seem to say, "Not (P or Q) is the same as NotP and NotQ. Surely you follow this if you've followed the other things we were just talking about?"

And my mind panics. It tells me, "You got no hands and yer eyes itch like crazy dude!" And I go and play with my Magic Magnet Kit. Who cares how they work? They look tasty!

posted by gorgor_balabala at 4:52 AM on October 13, 2010 [2 favorites]

Then, one more line down the page, semblances of language I use (like "If I pour this milk, the cereal gets soggy!") vanish and I'm left squinting at mathy-looking things which seem to say, "Not (P or Q) is the same as NotP and NotQ. Surely you follow this if you've followed the other things we were just talking about?"

And my mind panics. It tells me, "You got no hands and yer eyes itch like crazy dude!" And I go and play with my Magic Magnet Kit. Who cares how they work? They look tasty!

posted by gorgor_balabala at 4:52 AM on October 13, 2010 [2 favorites]

*Not (P or Q) is the same as NotP and NotQ*

It's not so bad: if you are not (black or white), that is the same as (you are not black) AND (you are not white).

posted by Jpfed at 8:12 AM on October 13, 2010

This is bullshit ther is not one mention of Dr. Spock in the hole thing

posted by everichon at 8:36 AM on October 13, 2010 [1 favorite]

posted by everichon at 8:36 AM on October 13, 2010 [1 favorite]

What, no mention of Godel, Escher, Bach yet? As long as you're not in a terrible hurry, it's a wonderful introduction to symbolic logic.

posted by anigbrowl at 9:56 AM on October 13, 2010

posted by anigbrowl at 9:56 AM on October 13, 2010

*He admits that they are presented in "crisp, standardized English" (and he takes this to always be a notational variant of symbolic first-order logic, which I find pretty dubious).*

This seems like the real deep failure of the piece, if it's supposed to be a guide to the practical question of how to write for student mathematicians. I mean, it's obviously true in some trivial sense that logic is the language in which mathematics is often expressed and evaluated (though why he thinks "metalanguage" is the word for this, rather than "language," totally escapes me). But it's just as clearly not the case that this logic is mostly practiced or published in "mathy" symbolic notation (or even in an English form which clearly and directly reduces to it). And the piece says very little about the role of English-language/natural-language

*writing*here (while much of what it does say is ambiguous and/or fishy). A sociological and/or theoretical and/or just pragmatic take on the role and scope of written "math English" would be a really interesting thing to see, but instead this piece just handwaves its way out of the only real hard question in its neighborhood, the one about the role of all that natural language

*alongside*the notation.

posted by RogerB at 10:22 AM on October 13, 2010

*Can someone say why this is any good at all?*

As a programmer, the value is in proving high level multithreaded code correct, assuming the compiler is proven correct, the OS is proven correct, and the computer has no signficiant errata or reorders instructions. Which might happen some day before I'm dead. That's the real value I see, in being able to prove, through the myriad of possible interactions, that these concurrent systems are safe.

The other application is in proving that programs terminate or meet specifications. But they're mostly academic in application, because few programmers study this and the few that do generally hate it. And for the most part programmers seek to write "obviously correct" programs where termination isn't even a question. Every once in a while though, management comes to you with some kind of scheduling or allocation problem, and proving equivalence to NP-complete problems demonstrates why you'll need way more hardware than was planned for, or that they'll have to be satisfied with a merely "close enough" answer. Then you go back to writing mundane SQL queries.

posted by pwnguin at 12:21 PM on October 13, 2010

I understand the value of logic, pwnguin. I've

posted by kenko at 1:16 PM on October 13, 2010

*taught*logic. I mean: what is the value of*this piece*.posted by kenko at 1:16 PM on October 13, 2010

kenko: "

False?

posted by Cogito at 3:09 PM on October 13, 2010 [1 favorite]

*I understand the value of logic, pwnguin. I've*"*taught*logic. I mean: what is the value of*this piece*False?

posted by Cogito at 3:09 PM on October 13, 2010 [1 favorite]

The value of this piece is that it allows those of us who took classes in logic to recall the experience fondly or to lament the difficulties we had with it. It's also a good review of material. As with any linkage provided here, the value will ultimately depend on the audience, rather than depend on some absolute scale of intrinsic worth.

If that's fellating the knowledgeable, I'd like some more please.

posted by Roger Dodger at 6:39 PM on October 13, 2010

If that's fellating the knowledgeable, I'd like some more please.

posted by Roger Dodger at 6:39 PM on October 13, 2010

*I mean, it's obviously true in some trivial sense that logic is the language in which mathematics is often expressed and evaluated (though why he thinks "metalanguage" is the word for this, rather than "language," totally escapes me). But it's just as clearly not the case that this logic is mostly practiced or published in "mathy" symbolic notation (or even in an English form which clearly and directly reduces to it).*

Yeah, that's what bugged me most about this piece, but he's hardly alone in this. The idea that FOL is hugely important because it "underlies" standard English or the language of thought is what's driving him, I think, and it's a really common misconception. Intro textbooks don't really go into philosophy of logic, so when they ask the student to translate English sentences into FOL, they often present the translations as revealing the "deep structure" of thought, as if they were all continuing Russell's

*Principia*. And it's also really common to motivate the study of logic by saying something about how FOL is at the root of thought or good reasoning or something like that. (I've definitely fallen into this trap... it's hard to make logic exciting for students who didn't realize they were taking what is, for all intents and purposes, a math course. I end up accidentally making big embarrassing claims about how studying formal logic helps you look directly into the heart of the universe or whatever).

All of this is nonsense. FOL is a one formal system of many possible systems, and to learn it is to learn what implies what given certain axioms and rules of inference. That's it. If there is a logic (semantics) of natural language or of thought, it will look far different than FOL---probably some form of lambda calculus. Translating English sentences into FOL is an art, not a science. I'm a fan of Gilbert Harman's position: logic is the study of implication, and this is completely separate from the study of inference and good reasoning. Logical systems are tools or maps we that we can deploy to help us reason well; they are not intrinsically about reasoning any more than systems of geometry are.

I'm also pretty skeptical of the idea that taking formal logic courses makes you a better reasoner in everyday life. We don't expect calculus courses to do wonders for one's everyday reasoning ability, and formal logic courses are not too far removed from these. There have been a few empirical studies that show that students who have taken a logic course are better at exactly one thing: taking logic exams. It helps students on the LSAT, anyway.

posted by painquale at 11:52 PM on October 13, 2010

*Translating English sentences into FOL is an art, not a science.*

I'd always thought this was due to the imprecision of language, ameliorated to some extent by the context and shared history between the communicants. That discourse is fraught with misunderstandings supported my belief. I have always assumed that everyone was trying to use reasoning (formal logic) to arrive at whatever goals from whatever assumptions, but some weren't very good at it (see "logical fallacies" for more info). This seems to me to be distinct from the logic of the mechanics of language itself, which is not quite the same as the purpose of language.

posted by Mental Wimp at 9:56 AM on October 14, 2010

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