The True Value of Words

Obviously, I’ve been having fun lately, beating the crap out of VCAA Version 2022 (which may or may not resemble VCAA Version 2025).  This post is also on exams and exam writing, at least in part, but not in the same crap-beating manner. It’s more of a puzzle for me.

I like writing. I like words. Even if I’m not good at them. I tend to get too cute and too adjectivey, and I’m more dictionary-ish or Seuss-ish than natural with word choice. But if I whack myself with Strunk and White a few times I can usually get the words to work well enough. The maxim is, Show, Don’t Tell. I just always find myself tempted to tell, if not yell. I really use too many words. Like “really” in the previous sentence; it is only there for emphasis and isn’t really needed.

Writing exams well is difficult. It is made much harder when the questions are poor and micromanaged into tiny fragments, but even writing a good question so that it is clear and correct is hard work. There is almost always a trade-off between precision and clarity. Each word should be chosen with the greatest care, and no superfluous words should be used. This post is about one such word.

First, a little pondering about a different word:

True

What does “true” mean? What is the purpose(s) of using the word?

Of course we all have a good working sense of what “true” means, just as we have a good working sense of what “cat” means. That it is difficult to describe these senses doesn’t mean that we don’t understand.¹ But now I want us to do a little work, to try to put the sense in words.

To begin, we can consider how the word is used. For example, we have,

Kamala Harris is a true idiot.

Here, “true” is simply being used for emphasis, like “really”. Not necessarily the same is,

The true value of words is as a path to imagination. 

Here, “true” is still being used in a mostly rhetorical manner but there is a suggestion that there may also be a false value one might consider. The true-false dichotomy is getting to the proper meaning of “true”, but of course this dichotomy is insufficient in itself to explain what either word means. Still, we’re getting there, and an example of a statement that gets more directly to the meaning is,

It is true that CAS poisons everything.

This is more directly declaring something to be “true”. Can we do more that? Not a whole lot. But a bit.

Formal logic takes truth as a primitive notion. We have truth tables and whatnot, where the truth of (P v Q) is defined in terms of the truth of P and the truth of Q, and so on. But the Boolean games do not tell us what “true” means, only some of how it works.

The logic and philosophy of language is a crazy world, and it is not my world.² I know a little, however, and I’ve always been fascinated by one of the great Alfred Tarski‘s famous contributions: Tarksi figured out what “true” means.

In a long 1931 Polish paper,³ Tarski tried to make precise the notion of truth underlying the monumental program on mathematical logic being undertaken by Gödel and others. This was encapsulated in a memorable statement, made near the very beginning of the paper. In English,

“it is snowing” is a true sentence if and only if it is snowing.

It doesn’t appear very deep, does it? But in a sense that is all there can be: declaring a sentence to be true undoes/inverts the act of quoting the sentence. It was the simple foundational concept in Tarski’s long and difficult and important paper. It is why Tarski’s statement is now so famous.⁴

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That was a long introduction to a much shorter discussion, of a different word:

Value

What does “value” mean? What is the purpose(s) of using the word?

As with “true” and “cat”, we have a good working sense of the common English usage of “value”. If we want to pass the buck there are synonyms for general usage, such as “benefit” or “importance”, and we can stop there. But “value” generally has a summing up sense to it, where the “value” is almost or exactly an estimable quantity. That slides into “value” as referring to monetary value, and from there we slide into the mathematical usage.

We commonly talk about the value of a function, when there are lots of numbers around and then “value” is used to focus on a specific number:

The maximum value of f is 7.

The value of f(x) when x = 3 is 5.

Of course the second sentence could be phrased more simply as f(3) is 5 (or, of course, f(x) equals 5, or f(x) = 5). There can be stylistic arguments for avoiding such phrasing. In general, mathematical sentences are best begun with English words: launching straight into symbols tends to feel abrupt and to be more difficult to read. But the important point is that “the value of” is not contributing to the meaning of the second sentence. We could just as well write,

When x = 3, then f(x) is 5.

(The “then” is there now, also for stylistic reasons, to not have the x = 3 and the f(x) separated only by a comma.)

Now what about the following sentences?

Given that g(0) = 5, the value of g(2) is 7.

The value of Pr (X < 0.5) is 15/16.

In order, the algorithm prints the values 3, -3, 2.

The value of Pr (X < 12) is closest to 0.337.

The value of the definite integral is 4k.

Is there any value in any of these uses of “value”? I cannot see that there is.

These mathematical examples all come from the 2024 Methods Exam 2 (with answers filled in),⁵ and there are plenty of others.⁶ It is a standard form and, except for stylistic reasons, or when required in phrases such as “maximum value”, I cannot see the purpose. It feels akin to writing

The value of 3 is 3.

It is like Tarski’s snow example except here there is nothing to “invert”: the “value” in the above sentence is not undoing anything. The value of a number is simply that number and so why not, when possible, just say the number?

Again, and for once, I’m not having a big go at VCAA. The sentences quoted from the Methods exam are all fine and sensible, even if they may be better without the “value” framing. But it seems to me they would be better without the “value” framing. So why have it?

In general, I can see very little value in “value”.

 

1) This permits me to digress, to give Ambrose Bierce’s definition of “cat”: A soft, indestructible automaton provided by nature to be kicked when things go wrong in the domestic circle.

2) It was my brother Jeff’s world.

3) Tarski’s work was translated into German in 1935 and into English in 1983.

4) There is a surprising wrinkle: very few people know the “famous” statement as I’ve translated it above. In English, Tarski’s snow statement is almost always given as,

“snow is white” is true if and only if snow is white.

That’s what I thought Tarski wrote (in Polish) until I started writing this post. But I had never looked at Tarski’s paper, and I hunted down the English translation to check the context in which the famous snow line appeared. I was very much surprised to find the line as quoted above, referring to “it is snowing”.  Not that the version of the statement matters whatsoever for Tarski’s purposes, of course, but it was like being told that your fondly remembered Shakespearean quote wasn’t precisely what Shakespeare wrote. I was curious. I hunted down Tarski’s original Polish paper and confirmed that Tarski’s original statement was as translated: the “it is snowing” version. (In Polish, “śnieg pada” jest zdaniem prawdziwem wtedy i tylko wtedy, gdy śnieg pada.)  It seems that the “snow is white” version is due to Donald Davidson, who used it in an influential 1967 article. It is amazing what one finds out when one embarks upon a silly blogpost and begins factchecking. 

5) Presumably the answers are correct. Since only the language rather than the mathematics is the issue, I just grabbed the answers from VCAA’s marking guide. 

6) Many of the usages are of the clearly sensible “maximum value” type, but many others are not.