Translate this logical statement into natural language.

Question

I am trying to translate this logical statement into English: $\forall x(x<10 \implies \forall y(y < x \implies y < 9))$.

Note that the statement is true when the universe of discourse is $\mathbb{N}$.The best sentence I could come up with was "all numbers less than any number that is less than 10 are also less than 9." Is there a more clear way to translate this statement?

EDIT: The English statement should not include variables or statements such as "for all".

Answer 1

Here is a somewhat more idiomatic restatement, I think: “If $x$ is less than $10$, then every $y$ less than $x$ is less than 9. (It’s common, whether it’s a good idea or not, to leave out the universal $\forall$ quantifier when expressing universally quantified implications, either notationally or in English.)

David appended a similar restatement to his answer in his “Simplified:” sentence.

See https://academic.oup.com/teamat/article-abstract/35/1/41/2461443?redirectedFrom=fulltext

Answer 2

For all $x$ if $x$ is less than $10$ then it must be the case that for all $y$ such that $y$ is less than $x$ it is the case that $y$ is less than $9$.

Simplified: If an integer $y$ is less than an integer that is less than 10, then $y$ is less than $9$.

(A true statement, by the way, so long as we're dealing with integers... but that wasn't part of your question.)

Answer 3

I would say that the formula holds for $x,y \in \mathbb{Z}$, then it would mean that for every integer which is smaller that $10$, there exists (or you can pick) infinitely many integers which are smaller than $9$.

Answer 4

tl;dr– If $y<x<10 ,$ then $y < 9 .$

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$$ \underbrace{\forall x}_{\text{For any}~x,} ( \underbrace{x<10}_{\begin{array}{c}x~\text{being less}\\[-25px]\text{than 10}\end{array}} \underbrace{\implies}_{\begin{array}{c}\text{implies}\\[-25px] \text{that},\end{array}} \underbrace{\forall y}_{{\begin{array}{c}\text{for}\\[-25px]\text{any}~y,\end{array}}} ( \underbrace{y < x}_{{\begin{array}{c}y~\text{being less}\\[-25px]\text{than}~x,\end{array}}} \underbrace{\implies}_{\begin{array}{c}\text{implies}\\[-25px] \text{that}\end{array}} \underbrace{y < 9}_{\begin{array}{c}y~\text{is less}\\[-25px]\text{than}~9.\end{array}} ) ) $$

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$$ \underbrace{\forall x}_{\text{For any}~x,} ( \underbrace{x<10}_{\begin{array}{c}\text{if}~x~\text{is}\\[-25px] \text{less} \\[-25px]\text{than 10}\end{array}} \underbrace{\implies}_{\text{then}} \underbrace{\forall y}_{{\begin{array}{c}\text{for}\\[-25px]\text{any}~y,\end{array}}} ( \underbrace{y < x}_{{\begin{array}{c}\text{if}~y~\text{is}\\[-25px]\text{less}\\[-25px]\text{than}~x,\end{array}}} \underbrace{\implies}_{\text{then}} \underbrace{y < 9}_{\begin{array}{c}y~\text{is less}\\[-25px]\text{than}~9.\end{array}} ) ) $$

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$$ \underbrace{x<10}_{\begin{array}{c}\text{If}~x~\text{is}\\[-25px] \text{less} \\[-25px]\text{than 10}\end{array}} \underbrace{\implies}_{\text{then}} \underbrace{y < x}_{{\begin{array}{c}\text{if}~y~\text{is}\\[-25px]\text{less}\\[-25px]\text{than}~x,\end{array}}} \underbrace{\implies}_{\text{then}} \underbrace{y < 9}_{\begin{array}{c}y~\text{is less}\\[-25px]\text{than}~9.\end{array}} $$

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$$ \underbrace{x<10 \implies y < x}_{\begin{array}{c}\text{If}~x~\text{is less than 10} \\[-25px] \text{and} \\[-25px] \text{if}~y~\text{is less than}~x,\end{array}} \underbrace{\implies}_{\text{then}} \underbrace{y < 9}_{\begin{array}{c}y~\text{is less}\\[-25px]\text{than}~9.\end{array}} $$