Which of these translates the universal and existential quantifiers better?

Question

Consider this proposition (which I know is false):

$$(\exists y{\in}\mathbb Z)\,(\forall x{\in}\mathbb Z)\,(y > x).$$

I am wondering whether the analogy of picking a variable value according to the order in which their quantifiers appear in the statement is correct: in the above, we pick an integer due to the existential quantifier then afterwards pick an integer according to the universal quantifier.

I am also wondering whether these informal translations (in which there are two distinct pickings, the first before the second) are accurate:

• We can pick an integer which is greater than any integer we could pick.
• We can pick an integer which is greater than every integer we could pick.

Is ‘every’ or ‘any’ better here for the universal quantifier?

Answer 1

Your interpretation of picking is correct, there are two distinct "pickings" that happen on after another. The $\exists$ quantifier means we have control of what to pick, and $\forall$ means that we don't have control of what is picked (i.e., it means "no matter what we pick").

As for the difference between "every" and "any", in terms of mathematical expression, the two expressions are more or less interchangeable.

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That said, be aware that any translation of rigorous mathematical symbolical statements into natural language is inherently a practice of not only mathematics, but also linguistics. Sure, you could rigorously translate your statement into

We can pick an integer $y$ such that no matter how we pick the integer $x$, the statement $y>x$ is true.

but any further simplification of that statement will heavily depend on the particularities of the language you are using. In English, the sentence above can simplify into

We can pick an integer $y$ that is bigger than any integer $x$ we can pick.

but in my native language, the same simplification would sound ambiguous...

Answer 2

$$(\exists y{\in}\mathbb Z)\,(\forall x{\in}\mathbb Z)\,(y > x)$$

Your proposition means $\Big(\exists y\;\big(\forall x \; y > x\big)\Big);$ here we want to find an integer, $y,$ that is greater than every integer including $y$ itself.

In contrast, in the proposition $\Big(\forall x \;\big(\exists y \; y > x\big)\Big),$ for each integer, $x,$ we want to pick an integer that is greater than $x.$

I am wondering whether the analogy of picking a variable value according to the order in which their quantifiers appear in the statement is correct

No: observe that in the second proposition above, we are iteratively checking that $y>x,$ going back and forth between—rather than chronologically handling—the variables $x$ and $y.$

• We can pick an integer which is greater than any integer we could pick.
• We can pick an integer which is greater than every integer we could pick.

Both suggestions are clear and good. Or how about ‘we can pick an integer that is greater than any/every integer we could pick’? Use ‘that’ for a restrictive clause, which adds specification or narrows down a class of objects; use ‘which’ for a nonrestrictive clause, which adds supplementary, non-essential information. Though this distinction isn't universally observed, precision never hurts when translating between formal logic and English.

Is ‘every’ or ‘any’ better here for the universal quantifier?

In this case, ‘every’ and ‘any’ interchangeably correspond to universal quantification. However, sometimes, the word ‘any’ corresponds to ‘∃’ instead of ‘∀’, so it should be used judiciously! To wit:

1. The phrase ‘if any’ frequently means ‘if some’:

   • $\color\red{\textbf{If any}}$ intruder enters, the alarm will go off.

     $(\color\red{\boldsymbol{\exists}} x\, Ex)\color\red{\boldsymbol{\implies}} A$

   • $\color\red{\textbf{If any}}$ command is understood by the dog, then it is a genius.

     $(\color\red{\boldsymbol{\forall}} x\, Ux)\color\red{\boldsymbol{\implies }}G$

2. The phrase ‘...not...any...’ actually means ‘...not...some...’:

   • She has $\color\red{\textbf{no}}$ disease.

     She does $\color\red{\textbf{not}}$ have $\color\red{\textbf{any}}$ disease.

     $\color\red{\boldsymbol{\lnot \exists}} x\, Hx$

   • She does not have every disease.

     $\lnot \forall x\, Hx$