# Translate this logical statement into natural language.

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".

• But is this really a mathematical question... Feb 2 '20 at 2:39
• What do you mean? Feb 2 '20 at 2:42
• You might want to say "integers" or "natural numbers" rather than just "numbers" Feb 2 '20 at 2:54
• For instant, $9.56$ is a number (and clearly less than $10$), but not every number less than $9.56$ is less than $9$. Feb 2 '20 at 5:05

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.

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.)

• For what it’s worth, here is a somewhat more idiomatic restatement, I think: “If $x<10$, 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.) [After posting, I see that David has done something similar in his “Simplified:” sentence.] See academic.oup.com/teamat/article-abstract/35/1/41/… Feb 2 '20 at 2:53
• @SteveKass That makes more sense to me. Could you make that an answer so I can accept? Feb 2 '20 at 2:55
• I mildly object to the use of "<" in a natural-language sentence... but that's easily fixed. Feb 2 '20 at 3:00

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 $$9$$.

• This isn't quite what it says (though this is true). What's said is “For every integer you choose that is smaller than 10, any integer smaller than the one you chose will be smaller than 9.” The original statement is true for the positive integers, but for those, there are not infinitely many choices for $y$. Feb 2 '20 at 2:59

tl;dr If $$y then $$y < 9 .$$

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

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

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

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