Witch 98: Contrapositive Definite

Having finished going through the Specialist Exam 1 sample questions, we started on Exam 2. We didn’t get far:

UPDATE: (28/02/23)

We shall spell out, in painfully long detail, why the above question is wrong and why it is so inexcusably bad. Grab a cuppa.

Part 1. First of all, a couple “minor” nitpicks.

(a) Punctuation matters. The sentence to be contrapositived should have been written as,

For all integers n, if n2 is even then n is even.

The second comma in VCAA’s sentence is unnecessary, and it greatly confuses the natural construction of the sentence and its meaning. These people simply do not know how to write.

(b) Definitions matter. The natural negation of “even” is not “odd”; the natural negation of “even” is “not even”. See here.

Yes, we are aware that no one gives a damn, but these things matter: if “not even” means “odd” then stick to it; if “not even” means “2k+1” then stick to it. In either case there is something to be proved, or at least statedly and consciously assumed; VCAA clearly couldn’t give a stuff, and everyone is following suit. Which is telling. The topic here is proof, for Christ’s sake.

Part 2. The main point, as noted by Alasdair, is VCAA’s statement has no contrapositive, and so the question is stuffed.

The contrapositive of an implication  \ \boldsymbol{P\, \Longrightarrow \, Q\ }  is the statement  \ \boldsymbol{{\sim \hspace{-4 pt} Q}\ \Longrightarrow \ \sim  \hspace{-4 pt} P}.  A statement not of the form of an implication has no contrapositive.

VCAA’s statement is of the form \ \boldsymbol{\forall n\in\Bbb N,\ P\, \Longrightarrow \, Q\ }. It is the quantification of an implication. It has no contrapositive.

Part 3. VCAA is permitted to make stuff up, but not without telling anybody.

Could VCAA have defined the “contrapositive” of a quantified statement so that their question above made sense? Yes, they could have. But they didn’t. If VCAA is going outside the clear and accepted definitions of standard logic, they have to tell someone.

Part 4. It is stupid to make stupid stuff up.

This is important. It is a really, really bad idea to extend “contrapositive” to quantified statements in the manner VCAA has (implicitly). It screws up the natural manner in dealing with logical operations. Consider, for example, the statement,

All VCAA exams are screwed up.

What is the negation of that quantified statement? Why? Compare this to VCAA’s contrapositiving of the quantified statement in the question above.

Part 5. What is the damn point of the sample question?

Seriously, what is the purpose of this question? If the question was intended to test students’ understanding of “contrapositive” in a slightly trickier setting, then of course VCAA has done nothing but trick themselves. If, however, the question was intended to test a bread and butter understanding of “contrapositive”, then just word the damn thing to be simpler (and correct):

For n an integer, consider the statement:

If n2 is even then n is even.

The contrapositive of this statement is:

Or, if you want the “integer” as part of the statement:

Consider the following statement:

If n is an integer and n2 is even, then n is an even integer.

The contrapositive of this statement is:

The second formulation is clumsier, and one might prefer slightly different wording. But, unlike VCAA’s statement, either formulation works. Or, just choose a damn sentence without needed or implied quantification.

Part 6. Finally, “Proof by contrapositive” is not really a thing: the term is unnecessary and confusing.

It’s a side point, but the expression “proof by contrapositive” has really been bugging us, ever since Red Five needlessly tangled themselves on this post. (Needless to say, it not RF’s fault and we’re not picking on him.)

How would one prove (for all integers n) that,

If n2 is even then n is even?

Yeah, you could reasonably call it “proof by contrapositive”. But it differs not one iota from a proof by contradiction.

A “proof by contrapositive” is always going to be a proof by contradiction, and that is what it should be termed. To do otherwise simply introduces needless jargon, and it encourages doomed expeditions to determine the contrapositives of statements with no contrapositives.

18 Replies to “Witch 98: Contrapositive Definite”

  1. Strictly speaking, the term “contrapositive” should only be applied to a propositional statement: “if P, then Q”, for which the contrapositive would be “if not Q, then not P.” But when a predicate is included, as here, the waters become murky. The question should simply have asked for the contrapositive of “if n^2 is even, then n is even”. There is, as far as I know, no fully accepted definition of the contrapositive in predicate logic. You can certainly define the negation of “for all n, if P(n) then Q(n)” as “there exists an n for which it is not true that if P(n), then Q(n)”, and work from there. We can start by rewriting the “if, then” implication in standard Boolean form (“if P, then Q” is equivalent to “not P or Q”) – all this being in the context of first order logic.

    The negation we want then, is “there exists n for which P(n) and not Q(n)”, and so the contrapositive would be the negation of that: “There does not exist an n for which n^2 is even and n is odd”. This is, however, not one of the options given.

    Oh, dear.

    It will be interesting to see the examiners’ report. With sloppiness such as this, it’s hardly surprising that students, when they arrive at university, are more confused than enlightened.

    1. Thanks, Alasdair. Note that these are sample questions for the new curriculum. So, there is not and will not be a companion examination report.

      1. It’s really the blind leading the blind isn’t it? Or maybe the idiots have taken over the asylum. It’s just so incredibly depressing. Our students deserve better than this.

        Can we mount a sort of counter-attack? I mean, use the wisdom from these pages to demonstrate the inadequacies of the proposed curricula, the teaching materials, and the assessments? Each one is worse than the others!

      2. There are no sample answers except for multiple choice questions either.

        Something tells me there never will be sample answers.

        Which is an issue in and of itself.

          1. I know that is sarcasm laced with truth, but it really does speak to a major issue.

            There is an inherent assumption that readers of these questions share the same understanding of key terms. The assumption is made by VCAA and is flawed.

  2. Another logic problem (R. Smullyan, What is the name of this book? p. 106):

    An old proverb says: “A watched kettle never boils.” Now, I happen to know that this is false; I once watched a kettle over a hot stove, and sure enough it finally boiled. Now, what about the following proverb?

    “A watched kettle never boils unless you watch it.” Stated more precisely, “A watched kettle never boils unless it is watched.”

    Is this true or false?

  3. Hi Marty – my background didn’t include any formal study of mathematical or predicate logic, so I’m going to have to rely on what others say.

    Contraposition is not negation where the universal quantifier gets flipped to an existential quantifier. A nice example playing with the statement “You can fool some of the people all of the time” is here https://www.csm.ornl.gov/~sheldon/ds/sec1.7.html

    I think the exam question is ok. In fact, it is exactly the example of proof by contrapositive used in the PD run for VCE teachers by Dr. Anthony Morphett (UMelb) – see the attached extract of the marked-up lecture slides.

    There is a Math Stackexchange answer that summarizes it nicely

    An implication is equivalent to its contrapositive (in first order classic logic); meaning that they always have the same truth value for the same atoms. Thus when one is universally true, so too will be the other: ∀a (P(a)→Q(a)) ⟺ ∀a (¬Q(a)→¬P(a))

    If I’m missing something in my understanding here, I’d appreciate a bit more of a hint!

    Logic and proof lecture slides – Contrapositive – Mon 24 Jan

    1. Sigh.

      1) The exam question is very not ok.

      1) Yes, I’ve read the stackexchange discussion(s), but they are making it up as they go along.

      2) Morphett’s slide is misleading, “Dr” notwithstanding.

      3) As Morphett correctly writes, the contrapositive of an implication \boldsymbol{P\,  \Rightarrow \, Q} is the statement \boldsymbol{\sim Q\,  \Rightarrow \, \sim P}.

      4) The contrapositive of a not-implication is “does not compute”.

      5) A statement of the form \boldsymbol{\forall n\in\Bbb N, (P(n)\,  \Rightarrow \, Q(n))} is not an implication, and therefore has no contrapositive.

      6) One might reasonably define the contrapositive of a statement such as in (5), as intended by VCAA. But that is not standard, it must be made explicit that that is what one is doing, VCAA has done nothing of the sort, and that is not what Morphett has even claimed to have done.

      7) One might also reasonably define/accept the label “proof by contrapositive” as Morphett uses it in his slide. But “proof by contrapositive” is not the same as claiming “the contrapositive” of a universal statement is a thing.

      8) For Morphett to wordlessly slip from the correct definition of “contrapositive” to an unexplained “proof by contrapositive” is, at best, misleading.

      9) See (1).

  4. I’ll just chime in again: the contrapositive can only be applied to a first order implication. If you have a predicate, such as \forall x:P(x)\to Q(x) then replacing the implication with its contrapositive: \forall x:\neg Q(x)\to\neg P(x) produces a statement of equal truth value. But that is not the contrapositive of the entire predicate statement. To call my second statement here the “contrapositive” of the first is a gross misuse of terminology.

    See the last paragraph at https://math.stackexchange.com/questions/3774879/

  5. Thanks Marty and Alasdair for the responses.

    I guess you need to be fairly disciplined in the use of language in logic – and the contrapositive only applies to conditionals and not to predicated conditional statements.

    So Morphett’s (sorry for the Dr bomb, Marty) use of Proof By Contrapositive could be interpreted to be proving the contrapositive for arbitrary integer n, hence showing the statement true for all integer n. He never actually stated that the contrapositive of ∀n (P(n)→Q(n)) is ∀n (¬Q(n)→¬P(n)).

    Actually, the stackexchange answer I linked to also didn’t explicitly talk about the contrapositive of a predicated conditional. Even the link by (Dr 😁) Sheldon linked previously and cited in the wikipedia article, talks about the “Contrapositive form” of “Universal Conditional Statements” and not the contrapositive of

    This all said, what is the danger extending the use of contraposition to ∀n (P(n)→Q(n)) and ∃n (P(n)→Q(n)) statements? What makes it such a “gross” as opposed to minor misuse of terminology? It seems unambiguous and innocuous – but I am inexperienced in these waters.

    And what would be a better and more standard way of talking about “the contrapositive” of ∀n (P(n) → Q(n)) ? Is there a simple rephrasing to save the question? As testing the confusion around how the ∀ and ∃ swap in negating a predicated statement and how the implication “swaps” in the contrapositive seems like a reasonable idea for an exam question.

    1. Simon, I’ll write more tomorrow, but the short answer to your final question is that it is entirely irrelevant to the issue at hand. The issue at hand is that VCAA’s question is stuffed.

    2. Hi Simon, I’ve posted a long update on this thoroughly idiotic question. I’ve briefly address your final question, in Part 4.

      1. Thanks for the updates Marty – they are appreciated – and do help to clarify things for me and hopefully others.

        OK – so I think I can summarise your complaints as VCAA does not communicate clearly, which is especially important in mathematics and doubly so in logic and triply so in exam questions on mathematical logic. And clear communication requires following the conventions of the community and specifying when conventions are broken and when new definitions/assumptions are made.

        I think some of my confusion might have been similar to some middle-school students that have trouble separating “equation” from “expression”… A quantified statement is quite different from just a statement, even if some statements seem to beg for quantification. In your re-worded example For n an integer, consider the statement: If n² is even then n is even the statement doesn’t care if its true or false for any particular (group of) n; it’s just a statement hanging out.
        But my brain keeps wanting to say “for any integer n” instead of “for n an integer” – but the former is actually saying that the statement that follows it is true for the specified set of n, not just defining the domain of possible n.

        To demonstrate that the language and ideas around this are slippery, let’s interpret your “All VCAA exams are screwed up” statement the way you intended, as a quantified one:
        “for all exams, if they are made by VCAA then they are screwed up” [∀x (P(x) → Q(x) ⇔ ∀x(¬P(x) ∨ Q(x)) ]
        I’d say the negation is :
        “There exist exams that are made by VCAA and are not screwed up” [∃x(P(x) ∧ ¬Q(x)) ]
        and the contrapositive (of the statement inside the quantifier) is:
        “for all exams, if they are not screwed up then they are not made by VCAA” [∀x (¬Q(x) → ¬P(x)]
        which seems like a reasonable equivalent statement.

        But, I think that many readers would also be ok to interpret your “All VCAA exams are screwed up” statement as a simple implication:
        “if an exam is made by VCAA then it is screwed up” [ P → Q ⇔ (¬P ∨ Q) ⇔ ¬(P ∧ ¬Q) ]
        Then the negation is: “an exam made by VCAA is not screwed up” [ P ∧ ¬Q ]
        and the contrapositive is “if an exam is not screwed up then it is not made by VCAA” [ ¬Q → ¬P ]

        There is definitely an implied quantification in the second interpretation – it doesn’t just feel like a logical statement hanging out and not caring about its truth value.

        PS. In the update, the link to the Red Five comment should use the URL https://mathematicalcrap.com/2023/02/07/witch-94-what-are-the-odds/#comment-20306

        1. Thanks, Simon. I agree with you, that there can easily be a conflict between the rigidity of formal logic and the casualness and/or accepted implications of natural language. Yes, there is an implied quantifier in my “VCAA exams” statement. But an implied quantifier is not the same as an actual quantifier.

          I think you’ve done a very good job of setting it all out, although I’ll write a bit more below. But the critical take-aways here, which you’ve mostly noted but which I want to hammer, are:

          (a) VCAA screwed up.

          It is simply madness for VCAA to have a topic on logic/proof, where a fundamental aspect is that We Must Be Very Careful With Our Definitions And Our Assumptions And Our Form, and then VCAA to simply assume everybody will be on the same page. With other topics this is bad, but with proof it is a disaster.

          (b) VCAA screwed up.

          Even on VCAA’s own terms, there is absolutely no point to the sample question we’re discussing. The only reasonable purpose of proof as a VCE topic is to prove things; it is not to fuss over pedantic detail of definitions. It is just nuts to have teachers and students worry about the contrapositive of a quantified implication.

          (c) VCAA screwed up.

          The role of “contrapositive” is being way overplayed in this topic. What is fundamental is proof by contradiction. Having a sense of contrapositive (and converse) is worthwhile, and part of a natural introduction to the topic, but the notions of “contrapositive” and, worse, “proof by contrapositive”, are not required or even helpful in the practical application of proof by contradiction. It is implicit in what is going on, but there is no benefit to the explicitness, and in practice will mostly just clutter and distract.


          Now, a bit more on implied quantifiers and the like.

          First of all, I don’t think the confusion is quite the expression/equation = noun/sentence confusion. In our case here, we’re confusing two different types of sentence structures. But there are similarities in the confusion.

          A fundamental aspect of playing with mathematical expressions is the outer form, the first layer to attack. Think of differentiating a nested function, (x + log x)/(x + 5) or whatever. Your first reaction, and what you want the first reaction of your students to be, is “Oh, that’s a quotient”. Everything else can and must wait. You might choose to rewrite the function but that doesn’t change the form in which the function is provided.

          Similarly, with formal logic, the outer layer, where you begin the attack, is critical. If the statement is of the form “For all x, ….” then your first reaction, and what you want you want the first reaction of your students to be, is “Oh, that’s a quantified statement”. Everything else can and must wait. And the fact that one can rewrite the statement (to be the negation of an existential statement) doesn’t change the form in which the statement is provided.

          Now, to the implied quantifier thing. To use another analogy, for me the adding/subtracting of a quantifier is like students filling in the hole for a function like f(x) = (x2 – 9)/(x + 3). It is natural, but it is not legal. And it shouldn’t be legal. It screws things up.

          Consider the following three statements:

          (1) If Simon is nitpicking then Simon is doing his job as a commenter.

          (2) If Joe Blogs is nitpicking then Joe Blogs is doing his doing as a commenter.

          (3) If a person is nitpicking then they are doing their job as a commenter.

          All three statements are in the form of (and thus are) implications. You seem to be tempted to regard the third statement as “really” a quantified statement. But what about the second? Do you want to think of it as “really” a quantified statement? One might, I guess, but it seems more strained to do so.

          I think “implied quantified” statements should be treated in the same way as expressions such as 2C + 4. Here, at least so far, C is not a number to be quantified or anything: it is simply a number, which we don’t happen to know, and which we’ve doubled and then added 4. Similarly, Joe Blogs and “a person” are not (yet) people to be quantified, each is simply a person, which we don’t happen to know, and about whom we’ve made a statement.

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