This has a lot in it: it’s more of a coven than a WitCH. We couldn’t see what else to do.

As with our recent PoSWW, this WitCH comes from the Logic and Proof chapter of *VicMaths*, Nelson’s Specialist Mathematics Year 12 text. This is a new VCE topic, for which the summary from VCAA’s study design (Word, idiots) is,

This summary is also given as the prompt of Nelson’s chapter. Then, the extended excerpt here is Nelson’s introduction to the chapter, together with a few of the associated exercises + answers. (The textbook then continues with its coverage of “conjecture” and so forth.)

Readers who lose sight of land may wish to refer to a solid introduction to argument to get their bearings.

## EXERCISES

## ANSWERS

**2. (a) deductive reasoning, (b) neither, (c) inductive**

**4. E**

**7. A**

**9. D**

I’ll start with a quick and easy one…

A logical conclusion is either valid or invalid. The concept of a conclusion being “true” is nonsense.

You can draw logical conclusions that are valid but not true from untrue premises.

That was one of the few places I thought the text was ok: I interpret “conclusion” as meaning a statement/declarative-sentence/proposition, and thus is truth-valued. The way you are using the phrase “logical conclusion” I would instead use “argument” or “inference”.

Perhaps. Perhaps not.

Consider the following from Lewis Carroll:

1. There are no pencils of mine in this box.

2. No sugar plums of mine are cigars.

3. The whole of my property, that is not in this box, consists of cigars.

A logically, deductively valid conclusion from (1) and (3) is “All my pencils are cigars”.

From this and premise (2) it is deductively valid to conclude “no pencils of mine are sugar plums”.

This does not make the final statement true or false. It is simply VALID, logically, by the principle of deduction.

Maybe it depends on your logical philosophy. I’ve always thought of logic as a tool for Mathematicians, rather than it being truth-dependent in and of itself.

I agree that statements (1), (2), (3) do not make the statement “no pencils of mine are sugar plums” true. If that last statement is true, it’s because of the way the world is, not because of any arguments we make. Nevertheless, that statement has a truth-value – it is true if none of LC’s pencils are sugar plums, it is false if he has at least one pencil that is a sugar plum. I would not describe the statement as valid / invalid, but I would describe arguments as valid / invalid.

Or do you think of logic purely syntactically, so that arguments are valid / invalid if and only if they conform to a valid schema, and rather than speaking of propositions as true/false, we should only speak of them as provable / not provable from a given collection of statements?

This is getting quite deep – I love it!

Ultimately it comes down to your definition of “truth”. There is the pragmatic approach: something is true if it is useful for it to be defined as true (common in many areas of knowledge). Then there is the coherence approach: something is true if it is not contradicted by any existing ideas that we hold as true (common in areas of knowledge such as Mathematics). There is also the correspondence approach: something is true if it corresponds to an observable phenomenon in the “real world”, but then we have to decide what it means to be observable (are electrons observable?)

Carrol’s* argument behind what he called “symbolic logic” is that you can create new facts from existing ones, so logic was somehow separate to truth.

(*) Carrol, not Dodgson because these were published under his pseudonym.

Of course, none of this really matters because I don’t think the author of this textbook chapter planned their arguments properly at all. I reject both their premises and their conclusions!

RF, this is a terminology issue, not a logic issue. Arguments may be valid or invalid; statements may be true or false. A conclusion of an argument is, in particular, a statement and is to be considered true or false.

Hi SRK, RF is using the standard terminology from mathematical logic. He has taken a set of statements and used theorems that follow themselves from a set of axioms to deduce another statement. His use of the word “valid” is correct; his proof is using permitted steps and thus those three statements infer the last.

None of this has anything to do with truth value. In fact the separation of truth value from logical deduction (inference) was an important (early) leap, and one that stars in the first week of a logic subject (or it should). Unfortunately I can see the consequences of this not being the case in Marty’s post here, which is really a complete disaster.

Yes, the a priori difference between validity as the correct form of an argument and validity as truth-preserving is certainly important.

Maybe I’m misunderstanding something, but I thought the idea of logical consequence as truth-preservation was pretty standard (ie. from Tarski).

I also would have thought that logical-consequence-qua-truth-preservation is the best way to introduce the idea to novices, since it’s easy to understand logical consequence in propositional logic via truth-tables.

Yes, for propositional logic it is totally standard (and IMO good) to teach truth tables early. However it is important to understand that when we teach the truth table for “A and B”, we are not saying anything about the truth or otherwise of the statement. We are *defining* a binary operation in the logic.

Proof and the “following of truth” in propositional logic is more advanced, and comes after truth tables. It should be explicitly postponed + avoided before that in my view. Proof is a valid sequence of deductions. The conclusion of a proof, or the truth value of any part of the proof, is not related to it being valid. Validity is the important aspect from a logic point of view. If it is a useful proof or not, that’s an application of the result (to whatever the proof is about).

This is not just pedantic logic speak. For instance, a famous story involves a student at Harvard who proved in their dissertation (after, it is alleged, years spent unsupervised) remarkable properties of non-constant functions that are greater than 1-Holder continuous. Unfortunately, as the story goes, it was pointed out to him that there are no such functions, and so he had proved many remarkable properties of nothing at all.

Logically, that student may have created many wonderful valid (trivial) proofs. Practically speaking, that isn’t particularly useful.

In a lot of ways, systems of logic and the sentences in them are just simplified mathematical models for (real world) deductive reasoning. I wouldn’t teach this, but in practice things like “if it is raining, I take my umbrella”, “I have my umbrella” and then asking: is it raining? Is not a question for mathematical logic. I think it just generates confusion to even talk about these kinds of real-world fuzzy deductions.

Anyway. This is all to say that unless you want confused student logic (if taught at all) should be taught very carefully and with very careful language.

This can get muddied (completely wrecked) by natural language constructions, self-reference, and other things that aren’t part of propositional logic. For those more complex things, it is important to separate the notion of truth value (or any value) of a statement from the statement itself as a syntactically correct sentence in the logic. As a syntactically correct sentence in the logic, we can ask what might be inferred from it, just as we can with propositional logic, but the result might not be particularly meaningful (just as with propositional logic).

I don’t know if this helps at all, but I hope so.

Q9 – if the question said “the dog ONLY barks when the postman arrives” then I would be more comfortable describing the conclusion as valid.

It the current state, I do not believe any of the answers are necessarily correct.

Again, you should not be referring to the proposed conclusions as “valid”. The question is asking which statement can be “inferred”.

I may have missed it, but i cannot see that the text excerpt ever uses the terms “valid” or “invalid”. I see (pseudo)definitions for “logical argument” and “inductive reasoning” and “deductive reasoning”.

If I’ve missed the fact that the text never uses the word in an exercise in logic, then I have missed a decent pile of elephant crap.

For me, logic is about validity, not necessarily truth.

The above extract is disappointing.

But here is one of my favorite examples of a valid syllogism due to Smullyan.

Everyone loves my baby.

My baby loves only me.

Therefore, I am my own baby.

Cute, Terry, but I would say “Therefore I am my baby”.

“disappointing”.

I think Q4 deserves its own post… I’ll refrain from commenting any further for a few days though to give others a chance.

Pretty much every line deserves its own post.

Is this what happens when Logic disappears from syllabus for a quarter of a century?

This topic, if properly run/taught is nothing but fun.

This is a totally perverted kind of logic. This is not how logic is normally taught at an introductory level.

Yes and no. It depends what you mean by “logic”. If you click on the “introduction to argument” link I included, you’ll see some good, and standard within its world, stuff.

I think you have to distinguish between:

a) formal logic (propositional, first order, etc)

b) mathematical proof

c) philosophical logic and critical thinking and the like

Of course these overlap, but their emphases are way different, and they’re not necessarily great introductions for each other.

I think what the text is doing here is trying a quick (c), as an introduction to (b), which the curriculum document is kind of pretending is (a).

A very very painful read, Here is a non-exhaustive list of complaints.

Box 1 (“language of logic”)

Conclusions ≠ inferences. Conclusions are statements (and thus are truth-valued). Inferences are collections of statements, some of which play the role of premises and others the role of conclusions, such that the premises support the conclusions (in some sense of “support” to be made precise depending on the kind of reasoning at issue).

One can make an inference or “draw a conclusion” not based on facts. (As RF points out above, valid arguments do not require true premises).

It is poor to give an example of a deductively invalid argument before giving examples of deductively valid arguments.

Box 2 (worked example 1)

Mapping natural language onto propositional logic is not trivial. This example is a mess, and it’s probably not worth picking on individual flaws with each question / answer. Worked Example 1 should be a sequence of sentences with a clear logical structure where students can clearly identify the logical form of the premises and conclusion, and thus determine the validity of the argument.

Box 3 (inductive / deductive reasoning)

“Our reasoning is logically true but we cannot be certain that our conclusion is correct”. Any student first coming to logic would be completely confused by this. So our reasoning can be “logically true”, but that does not guarantee a “correct conclusion”. What could be the point of logic, then?

Box 4 (more on inductive / deductive reasoning)

The first example is poor, since it is often called abduction (or perhaps inference to best explanation). A more paradigmatic example of induction would be better, something like “Every swan we’ve seen so far is white, so the next swan we see will be white as well”.

The phrase “valid example” is very poor in this context.

Deductive and inductive reasoning are not “opposites”. Deductive reasoning is not limited to arguments of the form “All Fs are G, x is an F, therefore x is a G”.

Maybe I’ll come back to complain about the rest in a little while (and another glass of wine). Yuck.

Agree with all of that. Will add the following:

1. A valid deductive argument needs a minimum of two premises to form a conclusion. It is pretty simple to determine if a deductive argument is valid or not; it is one of the first skills I would assume is taught in any logic course.

2. An inductive argument can start with only one premise and still reach a valid conclusion. Such is the nature of inductive reasoning. Many philosophers have argued that this is one core weakness of inductive reasoning; but if we think within the rules rather than about the rules, then you can argue that a conclusion is valid based on inductive logic, provided you begin with a specific example and conclude a generality.

3. Words (or lack thereof) are very important when it comes to determining the logical validity of a conclusion (as illustrated by Glen, above)

I find it concerning that inductive reasoning (I’m assuming by this it is meant “going from specific to general”) is here at all. The principle of mathematical induction, yes, that would be nice. But I’d guess they probably shouldn’t try to get all the way to first order logic in high school, that takes about a year (two subjects at least) at university if done right. Propositional logic should be more than enough. The leap in complexity from propositional to first-order is quite large and I’d be very worried that there would be available staff to teach that. Propositional on the other hand, while tricky in some places, could be taught.

(Even though I’m a long-time fan of logic and studied it with intent to become a logician before I turned to analysis, I actually don’t like the idea of Propositional logic being in the curriculum. Thought I should add that disclaimer.)

Two problems:

1) Content is being written by non-experts.

We see safe, traditional content written by ‘experts’ that is riddled with mistakes. So Blind Freddy could see what was going to happen when non-experts started writing on abstruse content. Whether that content be in textbooks, resources or exams questions.

2) Because it’s abstruse, it will only ever be treated in a half-assed way. Teachers are expected to teach in two weeks material that could easily take a semester to teach properly.

This is the logical outcome when non-experts have a thought bubble about including such content in secondary school curricula.

And this was all done (at the expense of dynamics and statics) to make Specialist Maths allegedly more attractive to students. There is nothing attractive about the excerpt that is the subject of this blog.

I agree. The entire new “logic” topic is an aimless disaster. There is no point to the propositional stuff, which is done explicitly at year 11, and is thus officially examinable, but then disappears in year 12.

Of course there is even less point to the inductive stuff excerpted above.

Thanks very much, SRK. That’s why I pay you the big bucks.

I think “inference” may be used ambiguously in general, both as a form of argument and the (tentative) conclusion of such an argument.

Thanks to everyone who has commented so far. I’ve been reading the comments, but I’ve had no time to take part. Tomorrow …

I have often introduced my students (Years 7,8) to logic by asking them to solve Sudoku puzzles. I make it clear that solving these problems is an exercise in logic and therefore in mathematics. The students enjoy this, and they get the idea! Usually about 10% of my students have never met a Sudoku puzzle before.

When I ask a student “How did you get 4 in that square?”, I hear explanations like this. “The number has to be 2 or 4 because there is no other alternative. But it can’t be 2 because we already have a 2 in that column. So it must be 4.”

I tell the students not to guess, but a few ignore my suggestion and eventually get into trouble. Then they realise that they have to think.

Students get considerable satisfaction in solving just one of these. And, may I say, these are real-world problems. I hear statements like this: “My grandmother solves these.”

OK, I think I’ve given others a fair chance…

Q4. The question says “…will be true when…” It does not say “…will be true when…”

This means that all of the options could be correct, for suitably chosen values of

Now, is a multiple choice question with five correct answers better than one with none?

I’m tempted to say yes, by a margin of

Thanks, RF. Yeah, the wording isn’t great, although I’m not against implied quantifiers. My greater problem with Q4 is that it’s nuts.

Oh, I’m not finished with Q4 and its issues…

…but that was my first thought when reading it.

I’m a bit late to this party but will offer my 2 cents’ worth.

Question 4: Statements 1 and 2 can (and should) be re-written as: a > b and c > b. The only information that can be deduced/inferred/exorcised/whatever from this is that both a and c are greater than b. That’s it, nothing else: with the information provided it is impossible to determine a relationship between a and c. That is, a could be > c, or it could be < c, or it could be = c. For the conclusion that a ≥ c to be valid then one of the options (i.e. A, B, C, D, or E) must provide information which supports such a conclusion. None do. The alleged ‘correct’ answer E does nothing of the sort. For |a – c| = 0 then a – c = 0 (the absolute value of a – c is absolutely irrelevant) hence a = c. Concluding that a ≥ c from this is not possible since, at best, we can only determine that a = c. The answer should be F: logically, it is not possible to deduce that a ≥ c hence the conclusion is invalid/wrong/false/whatever. The presence of Question 4 in the textbook is likely to create a great deal of confusion in the absence of a competent and vigilant teacher, but could be a good teaching opportunity for the right class with the right teacher.

We can’t conclude a ≥ c from a = c?

I have a problem with the > part not the = part. From the information provide, a ≤ c is just as possible as a ≥ c. So, no, I don’t believe that we can conclude that a ≥ c purely from a = c. All we can conclude is that a = c, nothing more.

From the statement “Felix is a cat” can we conclude that “Either Felix is a cat or Felix is a dog”?

In this day and age Felix can identify as whatever Felix likes – who are we to judge Felix? Identity issues aside, the statement “Felix is a cat” leaves no wriggle room for Felix being a dog. Cats and dogs are two different species (and ‘cat’ is a species-level definition); well, at least they are in my universe. If, however, “Felix is a mammal”, then “Felix is a cat” or “Felix is a dog” would, in the absence of further information, be possibilities. Stating that “Felix is a cat” removes all doubt as to Felix’s species. Felix, on the other hand, could be a member of the genus Felis in which case it is possible that Felix is a cat but Felix may also be a snow leopard but cannot be a macacque. Regardless, I’m now concerned about Felix being roped into this without its/his/her/they’re consent.

Very funny. Nonetheless, and ignoring that Felix may be species-fluid,

You can, but what purpose would it serve?

(Rhetorical question – most of this section serves no point)

Well, to understand the rules of propositional logic, if nothing else.