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?
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.