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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
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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
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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
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1 Answer 1

up vote 6 down vote accepted

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
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@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
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