Translating English to Symbolic Logic - Multiple Quantifiers

Any hints on translating this English sentence into symbolic logic:

Something is between everything.

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Is there any more context? "Between everything" is not really well-defined; "between" is usually between two elements. Moreover, if an element X is between everything, and X is part of everything, is X between itself? – Gadi A Jun 4 '11 at 7:37
@Gadi, when doing these translations from English to symbolic logic, I use Between(x, y, z) to say "x is between y and z." Also, we use x != y && x !=z && y != z. – KerxPhilo Jun 4 '11 at 7:44
That doesn't answer Gadi's question. "everything" is not two elements, so it's unclear what $y$ and $z$ might be in your predicate. – joriki Jun 4 '11 at 7:56
The closest thing to this sentence that makes sense might be "There is something between any two things". That would be $\forall y\forall z\exists x\,\text{Between} (x,y,z)$. – joriki Jun 4 '11 at 7:58
This question is ill-formed and does not really parse. I am having a Russell-paradox headache here. – ncmathsadist Jun 6 '11 at 2:06

1 Answer

Here is a quick list of rule of thumbs for translation:

• "Some" means $\exists x$
• "Every" and "All" mean $\forall x$
• "If x then y" or "x implies y" or "y only if x" mean $x\rightarrow y$
• "Either x, or y" means $x\lor y$
• "Both x and y" means $x\land y$
• "Not x" means $\lnot x$

The actual predicates you would have to fill in on your own.

As joriki and Gadi commented, "Between" is a trinary predicate (i.e. takes three variables, "x is between y and z") and the closest interpretation of the current sentence would be $\exists x\forall y\forall z\operatorname{Between}(x,y,z)$.

(Note the difference in the order I put the quantifiers and joriki put them, in his version between every two elements there is a third; in my version one specific element is between every two others, including itself.)

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The other difference between you and joriki is that your statement is less likely to be true (though it does depend on what "Between" means). – Henry Jun 4 '11 at 13:53
@Henry, assume $x$ is God in the Judeo-Christian tradition, then god is between everything including itself and itself. :-) – Asaf Karagila Jun 4 '11 at 14:58