# What is the logical interpretation of this set?

$\{f \in C : f(x)>d$ for each $x$ for some $d\}$

Do you read the above set as "the set of functions in $C$ such that there exists $d$ such that for each $x, f(x)>d$" or do you read the $d$ as depending on $x$ i.e. "the set of functions in $C$ such that for each $x$, there exists $d$ such that $f(x)>d$" ?

I always read the quantifiers in reverse order if they are written behind the proposition (like in the set above), but I've been wondering if this is universal or merely my personal quirk. Thanks.

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I don't think there is a universally accepted meaning for things that look like formulas but with the quantifiers in the wrong place. I think you should avoid such things. –  Trevor Wilson Sep 30 at 17:46
I will assume you are aware that the two readings you have mentioned produce different restrictions on the set, even if (in some cases) the sets produced may be the same. –  abiessu Sep 30 at 17:49
Given a set description like that, I would make every effort to get clarification from the source as to the meaning, as I would treat that set description as "unclear". If I were forced to choose a set description, I would work through whatever problem used it and choose the set that produces the answer I like the best. –  abiessu Sep 30 at 17:53
@abiessu Ah, thanks. –  Ryan Sep 30 at 17:54
@TrevorWilson Do you mean that all the quantifiers are best written in front of the proposition in left-to-right order? I frequently see things like "the set of functions in C such that there exists d such that f(x)>d for each x " where the quantifiers are both in front of and behind the proposition-- this I do studiously avoid. By the way, if you don't mind, please post your comment as an answer so that I can accept it. –  Ryan Sep 30 at 17:55

I don't think there is a universally accepted meaning for things that look like formulas but with the quantifiers in the wrong place (that is, not in front of the part they apply to.) I think you should usually avoid such things.

In natural language I think it is fine to put a quantifier at the end if there is only one quantifier, as in "let $E$ be the set of natural numbers $n$ such that $n = 2m$ for some natural number $m$." Even with more quantifiers there may be some cases where it is okay to put them in at the end or in the middle but I think the best way to avoid ambiguity is usually to put them at the beginning.

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It's a bit forced, but in some contexts, even this description can be a bit ambiguous. For instance, I could say that "$\{6\}$ is the set of natural numbers $n$ such that $n = 2m$ for some natural number $m$." That's a stretch, no doubt about it, but… –  Joshua Taylor Oct 11 at 17:57
@Joshua You could say that, but I would suggest that you don't :-) –  Trevor Wilson Oct 11 at 19:02
We give students exercises in translation between English and first-order logic, and you'd be surprised as some of the phrasing that appears now and then. When trying to be very explicit, I tend to liberally apply "such that [there exists|for all/every]". The "sub" of subordinate clauses often helps express nesting and scope. –  Joshua Taylor Oct 11 at 20:31
I think the reason that the meaning of "let $E$ be the set of natural numbers $n$ such that $n = 2m$ for some natural number $m$" is more or less clear is that it translates to the definition "$E = \{n \in \mathbb{N} : n = 2m \text{ for some natural number$m$}\}$", so the "such that" here is a colon in set-builder notation and the "for some..." can't cross it. But you're certainly right that more parentheses can only help. –  Trevor Wilson Oct 11 at 21:16
Yes, I don't mean to seriously imply that this particular example is likely to be misread, but do notice how much is determined by the "the" in "Let E be the". Using the definite article indicates to a reader that the rest of the phrase should denote a unique entity. Saying "Let E be a set of natural numbers such that n = 2m for some natural number m" makes the scope of m much less clear. That's OK, but it means that there's something meaningful in the whole phrase that's not just part of the description of the elements of the set. –  Joshua Taylor Oct 11 at 21:25
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