This is a continuation of the previous WitCHes, here and here, on the Logic and Proof chapter of VicMaths, Nelson’s Specialist Mathematics Year 12 text. Again, we’ve stuck to the highlights, and we’ve resisted the temptation to include some (pretty weird) exercises.
Been going over six years and we’ve only hit the sixth Witch?
Very droll. There are 10 types of people that will understand your comment, Craig.
Please tell me that this is a joke.
It is. A quantifier, a predicate and a negation walked into a bar …
Criminal incompetence or criminal insanity? Not sure, but it’s an incredible inditement on our society if this book is used in even just one Australian high school.
Hi, Grant. Of course I’m not defending the book, which is poor throughout. But it must be noted that, in the case of this chapter, a good dose of the criminal incompetence lies with VCAA.
The Logic and Proof chapter of this book is on VCAA’s new curriculum material. And, VCAA rushed in the new curriculum, against all common sense, and despite strong advice, and pleas, for VCAA to not do it. The chapter here is obviously a rush job, and was obviously given only a cursory check. It is almost inevitable that there would be a screw-up of this sort, and of course VCAA’s own “logic and proof” (and other) resources are also error-strewn and generally poor, or worse.
Haste makes Wasteland.
But this is not just a “screw-up” and there are no (correctable) “errors” in the text. It is like two comedians pretending to talk to each other in Italian by repeatedly uttering the word “spaghetti” and making stereotypical hand gestures. It wouldn’t even make any sense to say that they “screw up” the subject or point out “errors” in their arguments. They (in case of the comedians: intentionally) do not even understand what the words (in case of the book: symbols) they use mean.
Which leads me to a bad guess: Could this be a text created by an AI? I mean, it looks very much like cargo cult: Combining words and symbols in a way that “looks like” a text on mathematical logic, with at the formal syntax being correct (i.e. nothing like “P(x→(Q¬ ∀”), but with no understanding of its meaning.
Yes, of course. The text is inexcusable. But I still think it’s possibly “screw-up” in the sense that more time would have permitted more time for decent vetting.
It does raise a good question though – why did this publisher not hire an actual logician to write/proof read this chapter?
Maybe all the logicians were busy at the time.
(Makes as much sense as any other explanation)
I think that’s part of the problem. I assume these publishers function pretty much on autopilot, which means that the text can’t go *too* far wrong. But, the logic was too far away from the traditional topics, and so autopilot rammed the thing into a cliff.
Maybe, or even probably, that makes sense (I’m afraid of using the quantifier “logical” here…)
But surely if teachers said enough is enough and refused to use a crap textbook with a class then change would be forced.
Or are the alternatives just as bad?
What makes you think that the teachers know any more than the textbook writers?
OK. Fair point.
When a textbook requires re-writing because new content must be added and old content deleted, publishers will generally ask the original writers to do the job. If the original writers say no, the publisher will get a new writer(s) and the new edition will be published with the attribution to the original authors plus the new author.
The publisher generally has no idea what expertise is required, it will assume that a secondary school textbook can be written by secondary school teachers. It would not have entered the publishers head that a chapter on logic should be written by a logician. So if the original writers say “Yes, we’ll write chapters for logic. No sweat.”, then the publisher will trust that the job is done and done competently.
Logic is not a traditional area of secondary school mathematics and I would posit that most mathematics teachers never studied it at university (a very few might have done a unit of discrete mathematics and been taught *some* logic in that unit). Most mathematics teachers will therefore be teaching it ‘out-of-field’ and since it is mathematics teachers that are writing the textbooks, I would posit that most of the textbook writers are writing about it ‘out-of-field’. They probably thought they could self-learn it and then write all about it.
It is reasonable to think that publishing deadlines left little time to do a decent writing job, even assuming that the writers were competent in logic to start with. Furthermore, the study design offers little insight as to how much detail is needed, and so the writers would have felt compelled to throw everything including the kitchen sink into what they wrote. As Marty has already said, the ultimate blame lies with the VCAA for bulldozing this topic onto the study design (razing dynamics and statics in the process) despite strong advice and pleas not to by people who knew what would happen.
And teachers are forced to teach logic in 2 weeks, when it would take a term to teach it properly.
The result is an inevitable mess. The mistake the writers made was agreeing to write the chapters on logic. To quote the great philospher Harry Callahan, “A man’s got to know his limitations.”
A logician would have done a better job but I think the odds would have been stacked against them to do a good job.
I think the best outcome would have been that nobody wrote chapters on logic …
What greatly concerns me is that the writers of this material might have influence on exam questions. Logic and proof replaced dynamics and statics, ostensibly to attract more female students to Specialist Maths and to increase Specialist numbers in general. My money is on the topic of this blog being the negation of that quantifier.
Incidental anecdote: I was talking to a (female) student the other day whose eyes welled up with tears of sadness when I said that dynamics and statics had been deleted from the Specialist Maths SD. We chatted for a while, I told her about the halcyon days of Pure and Applied Mathematics in the 60’s and 70’s and how moments of inertia (torque) were included in the dynamics and statics topic in Applied Mathematics. She was genuinely upset and asked several times why this content was deleted. I had answers which cannot be stated here. She does not intend to do Specialist in 2024 (maybe she wouldn’t have even if dynamics and statics had not been deleted, but its deletion sealed the deal).
Whilst I doubt the writers of this section will have any influence on exam questions, your conjecture that enrollment will grow as a result of the curriculum changes is easily enough to test… maybe we can even make a game of it:
H0: enrolments will not change. H1: enrolments will increase.
What p-value do you want for the critical area?
(Negative answers are not acceptable)
*Ahem* It’s not *my* conjecture, RF. It was a reason I heard from someone apparently ‘in the know’.
As for the p-value, I suspect it will be close to 1 (think about it … Look at H1 …)
The data will undoubtedly tell the story.
Amendments accepted BnB.
I’m not sure enrolments will increase, at least not more than would be expected by natural variation from year-to-year (the trend seems to be downwards anyway…)
As to data telling the story… that depends on who writes the story, surely.
Enrolments will almost certainly decrease. I have never said otherwise. That’s why I say the p-value will almost certainly be close to 1. (The p-value will only be ‘small’, and Ho rejected, if enrolments (significantly) increase. If enrolments decrease the p-value will be ‘large’).
The data is pure – it will show either an increase or a decrease in enrolments over the lifetime of the study design. And naturally there will be people who try to defend poor decision making with lies, damned lies and statistics.
Yep, right, forgot the direction of H1 for a second there… (oops!)
Enrolments will decrease, but will the
of decrease itself decrease?
(I’m guessing you could go as far as the third derivative before finding a positive message to sell… but maybe not).
I think the third derivative accurately captures the situation.
I suspect there will be a ‘plummet’ followed by a plateau. I’ll be happy to eat humble pie if enrolments are the same or higher in 5 years time.
For my students in Years 7 and 8 next week.
Attachment this time.
4-logic
VCAA does specify use of the quantifiers *for all* and *there exists*, but I’m not sure the use of the symbols
and 
do anything but make the text more confusing in this case.
I have a few nit-picks on this section, some annoy me more than others, but one that is repeated is the use of negation.
The negation of “all” is quite simply “not all”. How you interpret “not all” is a different matter. “Some” and “at least one” I would argue are subsets of “not all” that are really not necessary in this case. “Not all” is easy to understand, so leave it there and move on…
Worked example 14b makes no sense. For all
, OK fine… but to disprove this you have to show there is no value of 
at all that satisfies the statement FOR AT LEAST ONE VALUE OF 
. Simply showing it doesn’t work for a specific pair 
is not disproving anything.
Thanks, RF. Very much yes to your comment on negation: NOT means NOT.
You’re correct about 14(b), that their “counterexample” is insufficient to prove anything. But note that the statement is in fact true: see annoyed’s comment below.
They need to cut it. I love logic, but this is poison. Get rid of it. It is doomed.
“They” have not had a history of listening to reason.
Edit: whatever latex functionality this blog supports seems to not recognise my backslashes… eg
don’t render correctly. If an admin could fix that, it’d be appreciated.
Running remarks –
you can fool 
.
——
‘You can fool some of the people all of the time’ ironically has two interpretations: using the text’s own notation the other is
The statement
is simply false, and claiming it is true “since 
” is meaningless.
The textbook uses
to be the statement ‘
is a complex number’: the trivially true 
does not ‘read’ 
.
The statement
is different to the tautology* 
‘
‘
‘ is not a logical ‘statement’ you can meaningfully assign a truth value to. The statement 
does not mean what the text thinks it does.
Similarly, the statement ‘for every natural number
there exists a natural number 
such that 
‘ is different to the suggested statement 
, which instead asserts that every positive integer is a square. The text incorrectly swaps n and m, so both statements are written are actually false. The second statement does not claim that ‘all natural numbers can be written as the square of a natural number’: that is what the first statement says.
The solution to (b) of the worked example makes no sense, not least because **the statement is true**: if y = 1, then the implication
is tautologically* true. The statement 
is fundamentally different from the ‘there is a y value greater than 2 such that x = y^2’.
The exam hack is lovely advice; if the negation of ‘all’ is ‘some’ then clearly the negation of ‘all horses are red’ is ‘some horses are red’.
Please correct me if I’ve said anything glaringly wrong.
* you know what I mean
Huh. Someone’s actually doing the proper WitCH work.
Thanks, A. I think there are TeX errors. You can tidy, or I’ll try to do later.
I hope people doing the proper WitCH work (work a competent vetter should have done) send their invoice to the publisher. The publisher can retrospectively deduct the fee from what the vetters got paid (*).
Alternatively, in the next edition the publisher can acknowledge this blog as providing the appropriate errata for the numerous errors.
(And as for getting a logician to do the writing, only a magician could have pulled off that trick. Even then …)
@annoyed: “The exam hack is lovely; if the negation of ‘all’ is ‘some’ then clearly the negation of ‘all horses [are] red’ is ‘some horses are red’.”
That’s pure gold!!
In fairness to the writer(s), the addition of ‘exam hacks’ is most likely the editor’s edict. (For example, I am aware of some commercial organisations that demand that written solutions to their trial exams include ‘exam tips’. It’s a poorly disguised attempt to fool the weak-minded into thinking they’re getting a point-of-difference ‘inside information’ **).
* At least, as far as I know, VCAA exam writers and vetters don’t get paid for their work.
** Snort
It is difficult to know where to start with WiTCH work in this chapter Marty…
There are lots of little errors, which are not really little when taken in context because they speak to a rather poor (if any) understanding of the topic.
There are heaps of books on logic out there which are neither difficult to find nor read. Before teaching this stuff (even though I have studied logic at university – it was many, many years ago) I sought out these books, read them, worked through the exercises and then looked at the textbook my school uses.
To be fair, Cambridge is OK with a lot of this, but not without issues of its own.
The point? Writing coherent material for use with anyone but yourself takes time. For some reason (I’d bet the time between VCAA announcing logic was to be taught and the intended start date was the major factor) time and care does not seem to have been taken here.
And thus we have a coven of WiTCHes.
[End Rant]
Yeah, I know it’s overwhelming. I wasn’t really criticising, I was just surprised that someone began the hard work of itemising.
Hi annoyed. I believe I’ve fixed the LateX (simply by pasting in the email of your comment). I suspect it’s a weird browser or word processor thing at your end, not a Latex issue with this site.
Thanks again for the very hard work, annoyed. Here are my thoughts on your thoughts.
*) “Some of the people …”, you’re right, it’s officially ambiguous, although I’d say the text’s interpretation is standard and more natural. So, I wouldn’t call it an error as such; maybe an opportunity missed (although just shutting up entirely and getting on with the maths would have been preferable).
*) Yes, the “x is a complex number implies x = ±i” is nuts. (Very technically, I wouldn’t say the statement is false, since we don’t know what x is.)
*) Yes, the use of P(x) in the complex number thing simply does not do what the text claims.
*) Yes, the polynomial P(x) is plain wrong in the same manner.
*) Yes, x + y = 3 cannot be assigned a truth value, although it is a fine statement about x and y. Just like, for example, “Ben and Jerry are brothers”. We just cannot determine the truth value of these statements unless we know something about x,y, Ben and Jerry.
*) Yes, their translation of “for all values, y there is an x …” is something out of Monty Python.
*) Yes, the first “squares” statement is false, and presumably the issue is, as you suggest, the accidental swapping of m an n in the quantification.
*) Yes, the second “squares” statement doesn’t state what the text claims: it states that all natural numbers are the same square.
*) Yes, Example 14(b) is a very bad error, that may be easily overlooked. The confusing of “If P then Q” with “P and Q” seems to underly the whole section, including the introductory paragraph.
*) Yes, on the horses. It brings to mind the induction proof that all horses are the same colour, but not in a good way.