Translating “there exists an element that comes after all others”

logiclogic-translationquantifiers

Consider the predicate language with predicate symbol $<,$ where $x < y$ means that "$x$ is an object different from $y$ that comes before $y$", and $=$ is the usual equality symbol.

Choose the first-order logic formulas that correctly translate the sentence:

"There exists an element that comes after all others".

A. $\exists x\, \forall y\, (y<x)$

B. $\exists x\, \forall y\, (\neg(y<x) \to (y=x))$

C. $\forall x\, \exists y\, (x<y)$

D. $\exists x\, \forall y\, ((y<x) \vee (y=x))$

A to me says: "There exists an element $x$, for all $y$, $y$ comes before $x$" so an element comes after all others.

C to me says: "For all $x$, there exists a $y$, such that $x$ comes before $y$" so an element comes after all others.

So, surely both A and C should be correct but I am unsure if I am translating them into English correctly. I am stuck between A and C.

Best Answer

A. $\exists x\, \forall y\, (y<x)$

A to me says: "There exists an element $x$, for all $y$, $y$ comes before $x$" so an element comes after all others.

Try to at least translate grammatically, so that the result is easier to parse and make sense of; here, the first comma ought to be replaced with "such that".

Or, simply, "some element is larger than all elements."

Some other natural translations:

  • for some element $x,$ for each element $y,$ $y<x$
  • there is some element $x$ such that each element $y$ is smaller than $x$
  • for some element $x,$ each element $y$ is smaller than $x$

Notice from my first suggested translation that the quantified (bound) variables $x$ and $y$ do not actually need to feature in the sentence's semantics and are merely placeholders. As such, it is perfectly legitimate for them to point to the same object.

C. $\forall x\, \exists y\, (x<y)$

C to me says: "For all $x$, there exists a $y$, such that $x$ comes before $y$" so an element comes after all others.

Nitpicking for the same reason as above: omit the second comma.

Or, simply, "every element is smaller than some element."

Does any of the above translations match the required "there exists an element that comes after all others"?

B. $\exists x\, \forall y\, (\neg(y<x) \to (y=x))$

D. $\exists x\, \forall y\, ((y<x) \vee (y=x))$

The correct answer(s) is actually from options B and D.

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