What words/phrases can grammatically specify groupings to remove ambiguity in statements?

Question

It is very common for research involving mathematics to be primarily written in English (including proofs) because both correctness and comprehensibility are valued in this kind of writing. Any English statement written without appropriate wording and adherence to grammar can be ambiguous. Hence, appropriate wording and adherence to grammar is very important when writing about mathematics. I'm used to writing code and formulae where parentheses are used to specify groupings that remove ambiguity. However, parentheses do not work like this in the English language.

Hence, when writing articles involving mathematical statements, I try to use certain approaches to wording/phrasing that grammatically specify groupings and therefore remove ambiguity (within reason). But there are still tricky scenarios where I cannot think of the right wording/phrasing to do this whilst ensuring grammatical correctness. I am looking for solutions in these scenarios.

I will first give an example and a generalisation of how this grouping-based ambiguity can arise in any statement (regardless of mathematical content), I will then give examples of solutions/approaches I am already aware of for combatting this ambiguity and also point out the limitations of these approaches that are leading to the tricky scenarios where I'm getting stuck.

Ambiguous statements

The statement

I will go to the supermarket if and only if I have run out of food and I am in the town centre.

could be interpreted to mean either

I am in the town centre, and I will go to the supermarket if and only if I have run out of food.

or

I will go to the supermarket if and only if it is the case that I have run out of food and that I am in the town centre.

These two interpretations have different meanings. Only the second interpretation specifies that going to the supermarket is dependent on being in the town centre. Hence, it can be important to try to remove these kinds of ambiguities whenever possible. In mathematics, this is extremely important.

The example above can be generalised to any written statement. Consider two statements that I will name a and b. Compound statements of the form “[...] if and only if a and b”, “let [...] such that a and b”, “[...] if a and b” and others can be interpreted as having more than one meaning. I can introduce code/formulae-style parentheses to demonstrate that “[...] if and only if a and b” could mean either “[...] if and only if (a and b)” or “b and ([...] if and only if a)”. For my purposes, in almost all cases I want to specify that a and b are grouped. If there are multiple statements instead of just “a and b” (e.g. “a, b and c”), I almost always want to group these as well.

Limitations to known approaches in trickier scenarios

• In a lot of these scenarios one could replace “a and b” with “it is true that a and that b”. For example, “[...] if and only if a and b” can be changed to “[...] if and only if it is true that a and that b”. This usually makes it clear that a and b are grouped. But if statement b is something like "x is in set X" then the ambiguity remains. E.g. “[...] if and only if it is true that a and that x is in set X” could be interpreted as meaning "(that x is in set X) and ([...] if and only if it is true that a)" where "that x" refers to some other x.
• In a lot of these cases one could replace “a and b” with “both a and b hold”. For example, “[...] if and only if both a and b hold”. However, this only really seems to make sense if a and b are equalities. It does not grammatically make sense if statement b is of the form "no x exists in set X with…" for example.
• Changing “[...] if a and b” to “if a and b then [...]” is a solution that only works for “[...] if a and b”.
• In a lot of these cases one could put “a and b” in its own sentence. For example, “[...] if and only if a and b” can be replaced with "[...] if and only if the following is true. a and b.". But I rarely see this approach in my field and it can seem unnecessary to start a whole new sentence in cases where a and b are relatively short.

I try my best to avoid writing statements which require any groupings to be specified in the first place. But I always encounter scenarios where it is necessary. Any approaches for grammatically specifying groupings that apply to these trickier scenarios would help. I have a preference for approaches that generalize to as many scenarios as possible.

By the way, I already asked questions about closely related topics on the math StackExchange (some of the approaches I’ve listed here come from answers there) but the topics were deemed to be primarily about writing which led to some questions being closed.

Answer 1

Your question makes it clear that you already know that there are various ways of dealing with this matter, which kind of, sort of work, but none of which is guaranteed to work perfectly in every context. If one is looking for a solution that will work as well as parentheses work in mathematical formulae, and also sound elegant to average English speakers, one is bound to be disappointed. The reason why in mathematics, logic, etc. people resort to special notation is precisely that ordinary English (or French, etc.) does not work very well for the purposes of these fields.

It should be noted that this problem is only occasionally a real, practical problem, because in everyday communication, the context, and the nature of the subject matter, will often make what one is saying unambiguous for practical purposes, even if it is ambiguous so far as its syntax is concerned.

One way of dealing with the matter, in the cases in which it is necessary to completely eliminate the ambiguities (which is a variation of the last one you outline) is to use nested numbered clauses, as in:

If all of the following is true:

(1) I have run out of food,

(2) I am either

 (a) in the town centre, or
 (b) near the farmers' market, and

(3) I have not run out of money,

I shall buy the food in the amount that is adequate for

 (1) four days, if I plan to eat out on Friday,

 (2) otherwise, for five days.

This method can be extended to the ideas of any degree of complexity, unlike trying to do it within running text, which is bound to crash once one gets past a couple of levels of nesting. While this way of writing something seems odd when applied to mundane matters, variations of it are often used in legal drafting - law is a field that requires utmost attention to avoiding ambiguities and does not have the kind of notation that is used in mathematics. The method, admittedly, works only in writing, as there is no way of replicating the numbering and nesting in speaking.

Answer 2

For many simple situations where you need to change the operation order from that of the order of written or spoken operations, you can invoke parentheses with the word quantity. But you need to be maniacally consistent if you do this. For instance X divided by quantity A plus B.

Less commonly, and best used for familiar things, you can use term the same way. When people hear a recognizable element, they end the parse and hopefully, the next thing they here is term again.

Term the material derivative of density with respect to t plus term density times quantity gradient dot u equals term the partial of density with respect to t plus term the gradient of density dot u plus term density times the divergence of u. This sort of thing improves accessibility if you were displaying a slide and reading along.