# Logic translation involving the existential quantifier and “such that”

A: "There exists an integer greater than 5 such that it is less than 10"

B: "There exists an integer such that it is greater than 5 and less than 10."

C: "There exists an integer less than 10 such that it is greater than 5."

D: "There exists an integer such that it is less than 10 and greater than 5."

I know that A can be translated to B (likewise C to D).

B and D are obviously equivalent, whereas I don't think A and C are.

Thus, I believe that A implies B but not vice-versa (likewise C implies D but not vice-versa). But is this correct??

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Why do you think any of them are logically unequivalent? – anon May 17 '12 at 1:03
@anon Since the part after "such that" is the clause, I don't think "...X such that Y" is equivalent to "...Y such that X". But am I confusing the linguistics of the statement with its logical interpretation? – Ryan May 17 '12 at 1:29
First off, all four statements are clearly true and implication is truth-functional... but let's ignore this. What do you think X such that Y means if $X$ and $Y$ are generic propositions? For example, "It is two o'clock such that apples are red." That's not the real form of these statements. Rather, let $\Bbb Z$ denote the integers, $P(x)$ the claim "$x>5$" and $Q(x)$ the claim "$x<10$," and finally the sets $A=\{x:P(x)\}$ and $B=\{x:Q(x)\}$. Now try to rephrase the four items with set inclusions, the two propositional functions and the existential quantifier and see how malleable they are. – anon May 17 '12 at 1:29

They're the same. "Such that" would be the colon in my interpretation of A: $$\exists n>8 : (n<15)$$

However, in this case "n>8" is itself a statement, so what we're really saying is that there exists some n that satisfies both the conditions: $$\exists n : (n>8) \wedge (n<15)$$

Because "and" is commutative, and "such that" simplifies to "and" in this case, all of the statements are equivalent.

Before you ask, "for all" quantifier is different. If you say the following: $$\forall n>8 : n<15$$ what you're really saying is that any $n$ greater than 8 implies that $n$ is less than 15, or: $$\forall n :(n>8) \rightarrow (n<15)$$

This is the reason why the statements

for all x such that x>0, x<10

and

for all x such that x<10, x>0

are not the same. However, the statement:

there exists an x greater than zero such that x is less than ten

is equivalent to

there exists an x less than ten such that x is greater than zero

There are lots of linguistic issues like this. "Such that" and "but" are the ones I've had the most trouble with.

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Yes, I'm aware of the if-then translation for the "for all" quantifier and that "but" is, amusingly enough, translated as "and". But the reason why I've been struggling with "such that" is because the set {2,3,4}={x: x=2,3,4} cannot be written as {x=2,3,4: x}. So my mind is fixated on the notion that you cannot switch around the preceding and succeeding parts around the colon/"such that" ! Thanks for clarifying that I'm indeed confusing the linguistics of the "such that" statements with their logical interpretation! – Ryan May 17 '12 at 1:41