Dominance of connectives: Why do we teach this?

These were two actual exercises given to students I have been tutoring for a college algebra class:

I have been working very hard to convince my students of the importance and utility of learning formal logic, something which I do think more non-STEM majors would benefit from understanding, but I really cannot imagine ever needing something like this. These are just horribly written symbolic expressions that no one should ever use and seem to exist just for the purpose of tormenting lower-division math students. Can anyone provide a justification for including this in math curricula?

• There is no justification. This is just horrible. Sometimes you have to call a spade a spade. Commented Mar 7 at 11:56
• This question is more suitable for law.SE. My guess would be: fraud, if the students pay for this, embezzlement of public funds otherwise. Commented Mar 7 at 12:15
• I have not seen a logic chapter in any cvollege algebra texts I've perused. I agree. This is awful. Commented Mar 7 at 15:34
• The ‘typesetting’ of the negation symbol is horrible and seems to indicate a precedence which I think is not the intended one. Commented Mar 9 at 13:41
• We invented parentheses for a reason. This is just the pedantic version of "What is 6÷2(3+1)? 99% will fail" meme. Commented Mar 15 at 23:33

This shouldn't be taught, and those exercises are pointless.

There is clearly no intrinsic value in introducing and memorizing precedence of operations. If there is a point, it either is that we would frequently encounter expressions involving so many parenthesis that parsing them becomes difficult otherwise; or it is that a particular convention regarding precedence is so common that without knowing it we struggle reading third parties' mathematics.

When it comes to stuff like multiplication binding stronger than addition, both reasons apply. [However, there are still plenty of situations where just putting in a few more parenthesis just makes it easier to read.]

However, no such reason applies to putting some precedence between $$\leftrightarrow$$ and $$\rightarrow$$. I'm a logician by trade, and I'm unaware of any claimed convention when it comes to these two. Authors I'm reading seem to manage just fine to use parenthesis or other signifiers to clarify their meaning without relying on such conventions.

When I see exercises like this, I often find that it teaches students to make assumptions about symbolic statements that may not be there - in a real world situation, if a statement is ambiguous, I would expect the response to be asking for it to be provided in a way that's explicit... considering that's the point of reducing something to a symbolic expression.

Looking into it, the only reason it would be something that could be useful is as put in this explanation of logical connectives by Bruce Ikenaga - when translating a sentence into a symbolic expression, it's uncommon to use parentheses, even though in mathematics it's rare to not make the statement explicit by grouping. So in some applications, it may be a useful thing to know but I honestly can't see it as a useful thing to memorize and test on. I'd guess that the push to memorize the precedence is similar to expecting students to memorize things most people tend to use a reference for (and now look it up online) - curricula is still written as if students won't have internet access in their pockets.

• Even without internet in your pocket, you would usually have a reference handy. I would think any C programmer of the older generations would have their Kernighan-Ritchie within arm's reach, and know both by heart and by actual wear exactly where the table of operator precedences is. Commented Mar 7 at 22:08
• I agree, especially having grown up around someone who taught rhetoric. There's no field in which you ever would have had to memorize 99% of what you're required to memorize for a test. Commented Mar 7 at 22:48

I think we all agree that there is a commonly accepted precedence to, say, arithmetic operators, and that it's common to write things like $$3x^2$$ without parentheses and expect it to be understood. (In fact, I routinely have college students who cannot correctly evaluate $$3(5)^2$$ and need correction.)

Similarly, for any other type of operator (relational, logical, etc.) it makes sense to have a defined order of operations and communicate that. This is defined and exercised in standard texts such as Rosen's Discrete Mathematics and its Applications (Sec. 1.1):

We will generally use parentheses to specify the order in which logical operators in a compound proposition are to be applied... However, to reduce the number of parentheses, we specify that the negation operator is applied before all other logical operators.... Another general rule of precedence is that the conjunction operator takes precedence over the disjunction operator... Finally, it is an accepted rule that the conditional and biconditional operators... have lower precedence than the conjunction and disjunction operators...

As an analog, teaching computer programming in a C-like language, there is necessarily a particular precedence applied by the compiler to translate any expression, and this precedence list is at least 15 levels deep. Again, the C-type language's precedence of the logical operators not-and-or are the same as outlined by Rosen above. It's likewise common to test students on their awareness of this fact.

For a place where this knowledge is needed, I just grabbed the first book from my shelf in reach, Casella/Berger's Statistical Inference (1990). On page 2 we see the first symbolic definition for "containment":

$$A \subset B \iff x \in A \implies x \in B$$

Now, which symbol creates the higher-precedence grouping, the logical equivalence, or the implication? If a student doesn't know, then they can't make sense of this definition, or any others.

We'll take another example from, say, Manin's A Course in Mathematical Logic. Manin expresses the Axiom of Choice this way (Sec. 2.4, p. 47):

$$\forall x (\lnot x = \emptyset \implies \exists y (\text {"y is a function with domain of definition x"} \land \forall u (u \in x \land \lnot u = \emptyset \implies \exists w (w \in u \land " \in y"))))$$

Note in the second-innermost group in the latter half, there are symbols for logical and, not, and implication -- not separated by any parentheses, so the reader is expected to understand a certain precedence to parse this (and other definitions). In Sec. 1.2 (p. 8) he notes, "We shall not specify precisely when parentheses may be omitted; in any case, it must be possible to reinsert them in a way that is unique or is clear from the context without any special effort." But in Sec. 1.3 he presents a formula that is fully, formally parenthesized, and gives the challenging exercise to find the open parenthesis corresponding with a particular closing parenthesis near the end (so demonstrating that expecting to fully parenthesize formulas is not viable).

As a further analog from computer science, we can look to the Backus-Naur form for specifying formal languages. In this notation, conjunction (indicated by juxtaposition) binds first, then disjunction (indicated by a vertical bar "|"), and then finally the replacement symbol (indicated by "::=", or in other notations an implication-style arrow). Of course, that's the same precedence common to all the prior examples. Parentheses are not used to separate these groupings (we can look to the C syntax as an example). This notation is widely used to specify languages such as Java, LISP, SQL, XML, and so forth.

The only way to confirm that students have digested these facts is to test them on it. Yes, one might say that the test questions are unrealistically convoluted. But the only other option is to just regurgitate one of the small number of previously-seen, very-simple, real-world examples (and everyone moans about rote memorization, right?).

One may as well complain about why ball players do weightlifting (when weightlifting isn't in a ball game), or why musicians practice scales (when scale-playing isn't in any performance). Practice is practice, and we practice on artificially difficult tasks so that things are easy when they count in a more elaborate context.

• The right arrow was a double arrow? (I would have expected a single arrow there.) I studied logic. I don't remember memorizing operator precedence, and I don't think it's relied on much. Commented Mar 7 at 15:37
• The only reason I even understood that you meant it to be parsed as $$A \subset B \iff (x\in A \implies x \in B)$$ rather than as chaining transitive relations is because I already know the definition of $A\subset B$, so if this symbolic expression was meant to teach me the definiton of $A\subset B$, it would fail. In fact if I saw someone write it the way you did, I would assume they made a mistake!
– Stef
Commented Mar 7 at 19:10
• This expression only makes sense because there is an implicit quantification there. It should be $$A \subset B \iff \forall x, x \in A \implies x \in B$$. Commented Mar 7 at 20:59
• (Anecdotally: I am a logician by trade and this is the first that I hear that there is supposed to be some blanket rule that $\implies$ binds more tightly than $\iff$.) Commented Mar 7 at 21:42
• If there were a standard precedence for operators between languages, this might follow, but there isn't. This actually highlights how teaching something like this leads students to make mistakes, assuming there's a general convention. Furthermore, the context here isn't discrete math, which tends to be more code-oriented, but college algebra, which is a general education course. Forcing rote memorization of something like this to only have to unlearn it later is bad pedagogy and discourages students who struggle with it. Commented Mar 8 at 0:02