Logicians are perhaps of rather limited use to the world (as I’m occasionally reminded by Mrs Logic Matters). But they can be tolerably helpful when you are in danger of inadvertently confusing use and mention, or if you want to avoid getting into muddles about variables, and so on.
Consider this:
A set is merely the result of collecting objects of interest, and it is usually identified by enclosing its elements with braces (curly brackets).
No: what gets surrounded by curly brackets in forming an expression identifying a set are expressions designating the elements, not the elements themselves. (And odd to say that sets are usually identified this way, using lists enclosed in curly brackets, when that only works for finite sets!)
Or how about this:
A property is a statement that asserts something about one or more variables. For example, the two statements “x is a real number” and “$latex y \in \mathbb{R}$ and $latex y \notin \mathbb{N}$” are clearly properties that assert something, respectively, about x and y.
Ok: it makes sense to say e.g. that
“$latex \pi$ is a real number” asserts something about $latex \pi$,
because $latex \pi$ is a denoting term. But it doesn’t express any complete claim to say that
” x is a real number” asserts something about x
if “x” is left as a dangling free variable. And we can’t tidy up by imagining there is a governing quantifier, as you can’t quantify across quotes. Anyway, a property isn’t a statement of any kind (even if we allow open sentences with unbound variables to count as statements) — properties are what are expressed by open sentences, or are their semantic values, or some such.
How about this?
[An example of a compound sentence is]
$latex P \land Q$ (means “P and Q” and is called conjunction)
What convention on quotation marks is in play which would make it right to have the quotation marks this way round? And, being pernickety — but why not? — it is certainly not the case that the wff is called “conjunction”! It is a conjunction.
Quite rightly, any logician would balk at write each of the above. Not so the mathematician Daniel Cunningham in his brand new book Set Theory: A First Course (CUP, 2016). Those quotations are all from the first six pages.
This seems a huge pity as the book later promises well as an introductory set theory text. I’ll report back in due course on the real content, once the book gets going. But it really is worth talking to your local friendly logician to avoid silly foul-ups like these.
I agree that the word “usually” is not appropriate. It should be replaced with the word “sometimes”. This correction is now listed in my errata sheet, which is available on my web page: http://math.buffalostate.edu/~cunnindw/
As stated in the preface, “my primary goal was to produce a book that would be accessible to a relatively unsophisticated reader.” In particular, I do not assume that the reader is well-versed with logical notation, and so in Chapter 1, my intent is to allow the reader to review the logical notation that they typically have seen in a discrete mathematics course and/or an introduction to proof course.
Perhaps, I should have made it more clear that conjunction is to be thought of as an operation, rather than a compound sentence.
The expression {x\in A : P(x)} is read as “the set of all elements x \in A that satisfy the property P”. Thus, this set is identified by “enclosing its elements by braces.” I believe that this latter expression is more accessible to students, new to mathematics, than the telling students that “what gets surrounded by curly brackets in forming an expression identifying a set are expressions designating the elements.”
Peter, thank you for taking an interest in my new book, and I look forward to your future comments, concerning my book.
Oh, wow. I’m betting the very distinguished reviewers simply skimmed the introductory fluff until they got to the real stuff. It is to be hoped the students do the same. An fun exercise for students would be to fix all the errors economically and charitably.