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I have some problems identifying the essential difference between using "and" and "such that" in statements. Consider the property of holding almost everywhere i.e

$$ \exists N \in \mathcal{F} \,\,\text{s.t} \,\,\mu(N) = 0 \,\,\text{s.t} \,\,\ \forall x \in \Omega \setminus N , P(x) \ \text{holds}$$

$$ \exists N \in \mathcal{F} \,\,\text{s.t} \,\,(\mu(N) = 0 \wedge \forall x \in \Omega \setminus N , P(x) \ \text{holds})$$

somone know how to think about this?

user123124
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    I can't give meaning to the first sentence. –  Apr 17 '20 at 07:06
  • @YvesDaoust I sort of can..but I know I should not be..but why..why cant we say it in that way? – user123124 Apr 17 '20 at 07:08
  • The first statement just does not make sense grammatically. I guess it depends if you are asking from the perspective of English or formal logic. – pancini Apr 17 '20 at 07:11
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    You should not be what ? –  Apr 17 '20 at 07:12
  • @YvesDaoust being able to give meaning to the first. As I wrote in a comment to an answer. Why cant we use "such that" "recursively" to assign properties like in the first? – user123124 Apr 17 '20 at 07:14
  • Because "such that" describes $N$ and the second clause is a property of $N$. Grammatically, you could say, "Such that $\mu(N) = 0$ and such that $\forall x \in \Omega \setminus N , P(x)$ holds." Substitute "having the property that" for "such that" if you don't see what I mean. – saulspatz Apr 17 '20 at 07:43
  • @saulspatz wait! a property cant have a property? – user123124 Apr 17 '20 at 07:45
  • @YvesDaoust I think the answer to my problem is that a property cant have a property. Does that seam reasonable to you? – user123124 Apr 17 '20 at 07:53
  • @user1 - In second-order logic (which is not an esoteric logic) it is perfectly legitimate that a property has a property, see for instance here. – Taroccoesbrocco Apr 17 '20 at 08:08
  • @Taroccoesbrocco ok nice, well in my example the property cant have the other i think..? – user123124 Apr 17 '20 at 08:15
  • So, your question is only grammatical, not about the logical form. – Taroccoesbrocco Apr 17 '20 at 10:40
  • @Taroccoesbrocco I dont think so, if a property can have a proeprty as I written above then there is nothing wrong with either way of expressing it riight? my thinking is that the property of a property has to make sense in order to be meningful and the above dosnt – user123124 Apr 17 '20 at 10:46
  • @Taroccoesbrocco I think you might be right, in the end this might have been a grammar question. The way I written that my property should have a property is not GRAMMATICALY correct. It hasnt anything to do with the relation of the properties per se – user123124 Apr 18 '20 at 07:05

2 Answers2

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"Such that" is technically not a part of a structured logical sentence - it is simply a phrase we insert to make the sentence more English-like. This is used after an existential quantifier $\exists$. E.g. the sentence $\exists n (n+n=n)$ could be rendered in natural language as "there exists $n$ such that $n+n=n$" (note how we inserted this phrase which was not present in the logical sentence).

In contrast, and ($\land$) is a logical symbol and a necessary part of the sentence. Your second example is correct, not the first.

Jordan Mitchell Barrett
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    right "such that" and $,$ are "common language" things. "such that" describes properties of something mentioned..well why cant we assign propertites after one another like this? – user123124 Apr 17 '20 at 07:11
  • It doesn't really make sense in English. "Such that" should refer to an object which was just defined, but in your second example, nothing was defined immediately before the second "s.t.". Again, this is only an issue with common language - "such that" doesn't exist in the logical world. – Jordan Mitchell Barrett Apr 17 '20 at 07:25
  • thanks for that I still I am getting it soon... Taken as you suggested, whats wrong with $$ \exists N ,,( ,,\mu(N) = 0 ,,(,,\ \forall x \in \Omega \setminus N , P(x) \ \text{holds}))$$ because this is what the first one like right? – user123124 Apr 17 '20 at 07:32
  • If you want a formal logical sentence (a "well-formed formula"): $$\exists N ( \mu(N)=0 \land \forall x \in (\Omega \setminus N)\ P(x) )$$ To render it more English like: $$\exists N \text{ s.t. } \mu(N)=0 \text{ and } \forall x \in (\Omega \setminus N), P(x) \text{ holds}$$ – Jordan Mitchell Barrett Apr 17 '20 at 07:40
  • yes, but why or how does mine pre your last comment falter? – user123124 Apr 17 '20 at 07:41
  • For formal logic, "holds" is not allowed - formulae can only be formed according to the rules here: https://en.wikipedia.org/wiki/First-order_logic#Formation_rules. Your sentence is fine in informal logic, except you still need the "and". – Jordan Mitchell Barrett Apr 17 '20 at 07:44
  • I think I see it now..a property cant have a property, right? I think thats what I am saying..which messes it up. unless I use the "and" – user123124 Apr 17 '20 at 07:46
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    Yup, think you got it. – Jordan Mitchell Barrett Apr 17 '20 at 07:53
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When in the scope of an existential quantifier $\exists$, "such that" usually stands for "and", from a logical point of view, when it connects more elementary statements.

In particular, your two statements are logically equivalent, even though the second one is slightly more elegant from a grammatical point of view. Their rigorous logical form is the following: \begin{align} \exists N \, (N \in \mathcal{F} \land \mu(N) =0 \land \forall x (x \in \Omega \setminus N \to P(x))) \end{align}

Taroccoesbrocco
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  • @YvesDaoust disagree to your last part. He cant make sense of the first way to write it. – user123124 Apr 17 '20 at 07:18
  • Your logical form is very helpful. Still dont know what to think tho..a lot of different answers here..Removing common language should help I suppose. However defintions often include it. – user123124 Apr 17 '20 at 07:24
  • @user1 - If the question is about English grammar, I'm not entitled to answer because I'm not a native English speaker. If the question is about the logical meaning of a sentence, trust me, "such that" in the scope of an existential quantifier is the same as "and". The only meaningful definitions have a logical form. Often they are written using common language because in this way it is less heavy and more accessible for readers. But there is always a logical form behind them. – Taroccoesbrocco Apr 17 '20 at 07:54
  • ill keep that in mind, thx – user123124 Apr 17 '20 at 07:55