Entries tagged with logic

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I was pleasantly astonished to see Peter Norvig comment on my recent post about the Stanford online courses - with 160,000 students competing for his attention, that's dedication! I completely take his point that I'm only a small part of the intended audience, and what works best for me is probably not what works best for the students they really want to reach.

Nevertheless, the course has sped up in the last couple of weeks to the point where I'm finding it pleasantly stretching. Today I tweeted:

I take it back: #aiclass is covering the Oxford 1st-year logic course in <1wk. The good bits of the syllabus, anyway :-)

I want to expand on that remark a bit. There were actually three first-year Oxford logic courses:

"Introduction to Symbolic Logic", taught in the first term to all Philosophy students (including those like me who were studying Maths and Philosophy), and also a few others like Classics students. This covered propositional logic, first-order predicate logic, proof tableaux, and endless pointless arguments about the precise meaning of the word "the" and whether or not the symbol → accurately captures the meaning of the English word "if". This is the course I was talking about on Twitter. I found it unbearably slow-paced, but I remember a couple of folk who'd given up maths years before and couldn't handle being asked to do algebra again. "The alarm is sounding and Mary called" was fine, but "A & M" was apparently unintelligible to them.

According to a legend which was told to me by the Warden of New College and is thus of unquestionable veracity, a class of ItSL students were once sent to the Wykeham Professor of Logic's graduate-level logic seminar due to a scheduling error. The WPoL walked in, saw the expected roomful of youngsters, and started "Let L be a language recursively defined over an alphabet X..." The poor undergrads, still in their first week (and quite possibly at their first class) of their undergraduate careers, must have started to entertain grave doubts about their ability to handle this "Oxford" place. When the WPoL was eventually informed of the mistake, he is supposed to have meditatively said "I thought they seemed rather ill-prepared..."

"Elements of Deductive Logic", taught in the second term to those studying (Maths|Physics) and Philosophy. This bumped the mathematical content up a notch, covering (for instance) completeness and consistency theorems for the languages that had been introduced in ItSL. There was also a rather handwavy treatment of modal logic, and lots more philosophical wrangling about what it all meant and how relevant it was to the broader philosophical project. This was one of the most intense courses I did in my four years as an undergrad.

There was an introduction to symbolic logic buried somewhere in the first-year computer science course - without the philosophy, I'm assuming.

Maths students were expected to pick the basics of logic up in the course of learning real analysis, as is traditional.

I always thought they'd have been better splitting ItSL and EoDL into two more evenly-sized courses - perhaps one heavily mathematical one, to be shared with the CS students (and perhaps incorporating some digital electronics), followed by a purely philosophical one to be taken by the (Maths|Physics) & Philosophy students. Meanwhile the algebra-phobes could do a single course more tailored to their level. I'm guessing that resource availability ruled this idea out, though.

There was also a two-term second-year maths course called "b1 Foundations", which was set theory, logic (up to, IIRC, the Löwenheim–Skolem theorem and Skolem's paradox) and some computability theory. This was compulsory for Maths & Philosophy students, but I didn't take it because I'd given up philosophy by then and was sick of the whole thing. In light of my subsequent career, this was probably a bad decision.

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logic

Magical realism

Intellectually, I know that "if I buy a new X, then the old X that I lost will show up" is unreliable magical thinking.

I just wish it would stop working.

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Contrapositive 34

You've probably already heard of Rule 34 of the Internet:

If it exists, there is porn of it on the Internet. No exceptions.

Now, as any mathematician can tell you, the statement "if X then Y" is equivalent to its contrapositive, "if not-Y then not-X". For instance, "if Socrates is human, then he is mortal" is equivalent to "if Socrates is not mortal, then he is not human".

[Pause to digest that for a second if you haven't thought about contrapositives before.]

Hence, Rule 34 is equivalent to its contrapositive:

If there is no porn of it on the Internet, it doesn't exist. No exceptions.

At DrMathochist's suggestion, I'm going to refer to this equivalent statement as Contrapositive 34, though we could just think of it as a re-statement of Rule 34: it's true if and only if the original Rule 34 is true.

Note that Rule 34 is not equivalent to its converse, which states "If there is porn of it on the Internet, it exists". That one's probably false. Links NSFW, obviously. This means that we can't, for instance, construct existence proofs by writing pornography featuring the thing whose existence we wish to prove ("'Come here, you sexy thing,' said the set whose cardinality was greater than that of the integers but less than that of the real numbers"). Bummer. However, Contrapositive 34 does have some interesting consequences:

While the amount of pornography on the Internet is mindbogglingly large, it's still finite; hence, there are only finitely many things in existence.
Since all Internet pornography, when you get right down to it¹, consists of finite strings of bits, the set of all possible Internet pornography is countable. Hence, there are only countably many possible things.

Edit: Conor McBride points out that this is a sexed-up version of Richard's Paradox. Which makes me wonder about the extent to which pornography could be reduced to a formal symbolic language - but that way lies madness.

Will Strinz suggested that you could build an accurate database of all things by searching for Internet pornography ("mining pornspace", as he called it). Unfortunately this doesn't quite work, since the converse of Rule 34 is false; you could, however, build an accurate database of things that might exist. If your search engine were powerful enough, you could try building a database of things that are believed with high confidence not to exist - generate search queries somehow (taking sets of dictionary words and adding "porn" would be a good starting point), and anything that returns no hits probably doesn't exist.

You might have a hard time explaining your multi-petabyte porn collection to your grant committee, though.

¹ *snigger*

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While I'm talking about philosophy....

I'd like to talk about a couple of simple thought experiments invented by the mathematician and philosopher Gottlob Frege, which for me really nail the relationship between mathematics and the physical world, and at the same time raise some fundamental questions in the philosophy of mathematics.

Suppose you have a vessel containing 2cc of liquid. To this, you add a further 3cc of liquid. How much is in the vessel now? Well, 2+3 = 5, so obviously it's 5cc. But the liquid you added wasn't the same as the liquid that was in there: the two underwent a chemical reaction, emitting some gas, and so the volume of the liquid in the vessel is actually less than 5cc. Or suppose you have five things, and you then add two more half-things. You've got six things, right? Except the "things" were pairs of boots, and the half-things were individual boots, but both individual boots were left boots. You have twelve boots, but only five pairs.

In both cases, you have some physical system (liquid in a vessel, pairs of boots in a rack) that you're trying to model using some mathematical formalism (whole numbers and addition, fractions and addition). In each case, it turns out that the model isn't a very good one, as it incorrectly predicts the behaviour of the system. But this doesn't mean that 2+3 is not equal to 5, or that 5 + 1/2 + 1/2 is not equal to 6! Nor does it mean that there's no mathematical model that would fit - in both cases, it would be pretty easy to construct one. It just means that we've chosen the wrong models. Standard arithmetic still works fine, it just happens to be the wrong thing in this case.

( SCIENCE! )

( PHILOSOPHY! )

Frege's story's rather a sad one, as it happens. He did pioneering work in the philosophy of language and in logic (he's got a solid claim to be one of the three great logicians of all time, along with Boole and Aristotle, and without his work much of modern mathematics would be literally unthinkable). In 1903 the culmination of his life's work, the Basic Laws of Arithmetic (Grundgesetze der Arithmetik), was on the verge of publication: in it, he claimed to show that arithmetic (and thus all of mathematics) could be derived from the obvious truths of logic. As it was about to go to press, he received a letter from Bertrand Russell informing him of what is now known as Russell's Paradox. This blew the entire enterprise out of the water. Frege inserted a preface to the effect of "This doesn't work any more, but I hope you find it interesting", then went off and had a nervous breakdown.

¹ If you can get hold of a copy of his (sadly out-of-print) Foundations of Arithmetic, I strongly recommend it, if only for the chapter contra Mill.

Current Music: Four Tet - everything ecstatic

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Sliding Definition Ploy

Closely related to the No True Scotsman fallacy is Sliding Definition Ploy¹. It goes like this: "for the sake of argument", give a slightly-odd definition to some common term, like Mind, or Freedom, or Justice, or Beauty. Deduce some consequences from that definition (ideally, this stage should take a while, to overflow your audience's input buffers). Claim that these deductions tell you something about the ordinary meaning of the term you started with. Better yet, don't claim it; just proceed as if they do.

SDP is, unfortunately, endemic to philosophy, or at least was when I last took a look. In philosophy, one of the big problems is that all the really interesting ideas you want to talk about (Truth, Justice, Ethics, Vision, Desire, the Soul, Mind, Body, even simple things like "looking"²) are ill-defined. The other big problem, of course, is that you can't do experiments on most of these things (even if you're the sort of philosopher who believes we can trust our senses and the sort who's prepared to accept the validity of the scientific method, which is by no means all of them). So you're left with pure reasoning. Mathematicians can get away with relying on pure reasoning, because they're working with abstract things that are precisely defined. Sorta³. So, in order to get any traction at all on their problems, philosophers often have to provide definitions of common terms. This allows the unscrupulous philosophers⁴ to insert a Sliding Definition Ploy or two.

SDP is also common in arguments about politics: we're all in favour of Liberty, Justice, etc, but these terms are all ill-defined, and by choosing definitions carefully, it's easy to show that your opponent is opposed to any given Good Thing.

zompist has written more about this in one of his rants, in which he also elaborates on the linguistic problems with definitions.

There are lots more standard logical fallacies: Wikipedia has a list (Googling will reveal several others), and here's infidels.org's guide to logic and fallacies, which also has what looks like a good section on what logic is, what it isn't, and what it's useful for.

[By the way, I am no longer eating salted porridge. I am now about to tuck into a sausage sandwich. An organic venison sausage sandwich, no less, filled with sausages from the farmers' market at Queen's Park yesterday :-) ]

¹ The term is due to John Puddefoot, AFAIK.
² I know a philosopher who recently wrote a paper on the difference between "looking" and "watching". Or possibly "watching" and "seeing". Or something like that. It's subtle stuff.
³ At the risk of being accused of SDP myself, this is probably the best definition of mathematics we have.
⁴ A depressingly large subset, from what I can see.

Current Music: Four Tet - everything ecstatic

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The No True Scotsman fallacy

Happy Easter, everybody! Today, I'm going to talk about two of my favourite logical fallacies. This post got a bit long, so I've split it into two: the second part, about Sliding Definition Ploy, is here.

My favourite, for the name if nothing else, is the No True Scotsman Fallacy¹, which canonically goes like this:

Hamish: No true Scotsman puts sugar on his porridge.
Jock: But my cousin Angus MacAlasdair from Glen Coe puts sugar on his porridge.
[Beat.]
Hamish: No true Scotsman puts sugar on his porridge.

[We're assuming that Angus MacAlasdair's Scottish nationality is above reproach, apart from his unnatural and perverted breakfast habits.]

( Discussion and further examples )

By the way, porridge really is a lot tastier with salt on it. In fact, I'm eating some salted porridge now. Mmmm...

¹ That's its name. Really!