# Do we need 'such that' after qualifiers?

Alright this is driving me crazy. I'm trying to figure out when we actually need to use 'such that' in math/logical expressions. There seems to be quite a bit of inconsistency but I wanted to check to be sure.

I've seen 3 ways of doing it...

• My discrete mathematics professor IIRC always used a 'such that' after an existential qualifier but not after a universal qualifier... so he'd use ∃x ∈ N: x > 1, but then also ∀x ∈ N, x > 0 (I think he'd use a comma here)
• These guys and a couple others I've seen online use no punctuation unless indicating parentheses: What does a period in between quantifiers mean?
• But others still use 'such that' (:) before all qualifiers: Does order of qualifiers matter in FOL formula?

I believe my math professor did what he did because it translated cleanly to English. Since you'd say "There exists an x such that x > 3" but you could also say "For all x, x=x". But I'm trying to figure out what the 'such that' symbol actually means in math, because I don't think the way it works in English necessarily makes sense. Wolfram Alpha defines the 'such that' symbol as 'indicating a condition in the definition of a mathematical object', and this make sense but they introduce yet another convention of qualifiers after a such that since q∈Z ≡ ∀q∈Z. And of course this convention makes no sense when translated to English in the case when for example when we'd say "x > 3: ∃x ∈ N" which translates to "x is greater than three such that there exists an x in naturals".

So anyways my question is what are you actually supposed to do? It looks like there are multiple conventions so which is best and most commonly used?

10/9/18 EDIT:

The Root of the Problem:

I've realized that the source of all this inconsistency has to do with the way that we read an existential qualifier. We read it as "there exists a blank [in something] [such that...]", and still call it a qualifier while technically this 'qualifier' is actually an (English) statement. We could similarly distort a universal qualifier to be a statement that is post-qualified with a "such that", if we read it as "All blank [in something] have a property [such that...]". If we truly want to call an existential qualifier a qualifier we need to read it as "For at least one blank [in something], [something is true]".

When we do this the need for 'such that' disappears.

• It's similar to asking what "like" means in casual conversation. It's, like, simply a placeholder. It's, like, a verbal tic in the expression of an existential statement. I do it, many mathematicians do it. Sep 27, 2018 at 17:30
• Math isn't casual conversation though... Sep 27, 2018 at 17:33
• Math between human beings, whether spoken or written, is carried out in human language, though. The point is: clarity and communication. Placeholders can help (and they can, like, hinder). Sep 27, 2018 at 17:34
• Ok sure, so that would be "informal speach" which in English speaking cultures and many other cultures is grammatically incorrect. So why question is what is grammatically correct? Sep 27, 2018 at 17:35
• To answer your grammar question, putting the words "such that" in the correct position of an existentially quantified statement is correct mathematical grammar. Sep 27, 2018 at 17:37

Some people even write things like $$\exists z \text{ s.t. } \forall y, y\notin z$$ Personally I think this is a horrible practice. One can think what one wants about how it wastes space, but more importantly it reinforces the dangerous misconception that logical symbols are just shorthand for words in English sentences. They are not; they make up a separate language with its own syntax and semantics, and the way we usually pronounce it with English words can misrepresent that semantics. (Consider for example how many beginning students, and sometimes textbook authors, who get themselves into contortions trying to understand the truth table of $$\Rightarrow$$ as if it ought to be forced by the English words "if" and "then").

The "just shorthand for English words" mistake is also what makes people sometimes put quantifiers last, which leads to horrors such as $$\exists z \text{ s.t. }y\notin z,\; \forall y$$ where we have completely lost the information about whether $$z$$ is allowed to depend on $$y$$ or not -- is it $$\exists z \text{ s.t. }(y\notin z,\; \forall y) \qquad\text{ or }\qquad (\exists z \text{ s.t. }y\notin z),\; \forall y\;?$$ Succinct clarity about these matters is a big part of why we use symbolic quantifiers at all in the first place!

So always put quantifiers before the formula they range over. And eschew punctuation that pretends symbolic logic is a way to write down English.

• I don't think a colon necessarily means "such that", though (written English doesn't use a colon for that function at all), but a comma after $\forall y$ instead of a dot or colon, that drives me up a wall. Sep 27, 2018 at 20:11
• Re: "Contortions trying to understand the truth table of ⇒ as if it ought to be forced by the English words 'if' and 'then.'" It seems to me that, if we are dealing with logical propositions that are unambiguously either true or false, then material implication quite adequately models the if-then constructs of natural language. Sep 27, 2018 at 20:39
• @DanChristensen: the problem Henning is pointing out already exists if you write the quantifiers in natural language: "there exists a natural number $n$ such that $n > m$ for any natural number $m$" is ambiguous whereas "there exists a natural number $n$ such that for any natural number $m$ $n > m$" and "for any natural number $m$ there exists a natural number $n$ such that $n > m$" are not (the former being false and the latter being true). Sep 27, 2018 at 21:45
• @RobArthan I myself was confused by the quantifier at the end of the statement and deleted my comment once I realized what it was. I have never seen that sort of thing. I didn't see your comment at the time. Sep 27, 2018 at 22:23
• @profPlum: I would view most of the formulas you see as "pseudocode" in comparison with an actual formal language for logic, which would be the corresponding "code". Like in pseudocode, people are just trying to convey an idea, rather than to stick to any particular formalism. This is why they seem to sprinkle colons, commas, and other symbols into formulas, even if the formal definition of logical formulas doesn't include those symbols. Oct 9, 2018 at 17:47

And in Enderton's Elements of Set Theory, he presents the axioms of ZFC in first-order logic as follows:

There's neither ':' nor ','. Also, if you look at how first-order formulas are defined, you'll see no mention of ':' or ','.

So I don't think there's a standard think or rule to follow about these kind of things when you're doing mathematics in general. See what others use in general, and use it similarly, in a way that would be clearly understood.

Edit: This kind of questions may be relevant when you're doing things related to mathematical logic or computer science.

• Ok interesting, I'm starting to think this no punctuation notation is most common and "formal". Thanks for your answer Sep 27, 2018 at 17:44
• Also yes it is, I ask mostly for personal note-keeping but also in case I need to include it in any research papers. Sep 27, 2018 at 17:45
• You're welcome :) Glad you found the answer useful. Oh don't worry about that. People here answer for the pleasure of mathematics ;) Sep 27, 2018 at 18:17
• The notation shown in this answer is, I believe, the standard in mathematical logic and very common in the rest of mathematics. May 23, 2021 at 1:09

There are no universally accepted standards in this case. You really must get used to slight variations in notation from one author to the next.

To add to the confusion, I often write $$\forall x: [x \in N \implies P(x)]$$ and $$\exists x: [x \in N \land P(x)]$$. It uses more symbols but it is often easier to work with than other notations. Tricky proofs in logic for the beginner, can often be made much simpler using this notation.

Warning: Avoid using $$\exists x: [ x\in N \implies P(x)]$$. Very weird things can happen. Using ordinary set theory here, this statement will be true for any set $$N$$ and for even the most nonsensical proposition $$P(x).$$

In my experience, mathematical logicians never use the form with ":", but split into two camps as regards the use of a ".": many people don't use a "." after quantifiers and take the quantifiers to have high precedence, so that $$\forall x\forall y(x > y \implies x \ge y + 1)$$ requires the brackets to make $$x \ge y + 1$$ fall in the scope of the universal quantifiers; others use a "." or a "$$\bullet$$" and take it as indicating that the quantifiers have low precedence, so they would write $$\forall x.\forall y.x > y \implies x \ge y + 1$$ (with the scope of the quantifiers extending as far to the right as possible). The former usage is fairly standard in the traditional logic literature, while the latter is perhaps more common among computer scientists and is often adopted in the syntax for proof assistants like HOL (and saves a lot of brackets in my experience of using such systems).

• It may help readers to know that the . and : are, in many cases, holdovers from the olden days, when they were used instead of parentheses, with the special rule that closing punctuation at the end of a formula could be dropped. So ∀x.x>y would be ∀x(x>y) in parentheses notation. The article The Notation in Principia Mathematica on the SEP has a useful introduction to the complexities of the dot notation that was adopted by PM. Oct 9, 2018 at 17:45
• @CarlMummert: thanks for the reminder that something that looked like a colon (":") was used in times gone by, thought of as a heap of 2 dots in Russell & Whitehead's notation. Maybe we should think of the modern "$\bullet$" used after a quantifier (by some) as a superdot that outweighs anything else $\ddot{\smile}$. Oct 9, 2018 at 18:55