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In natural language, we often use phrases like:

  1. "fix arbitrary"
  2. "such that"
  3. "we have"

In general, how to translate those phrases into formal logic?

One example condition is:

Fix arbitrary $x,y$. For all $f\in A$ such that $f(x)=0$, we always have $f(y)\geq0$, or we always have $f(y)\leq 0$.

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  1. "fix arbitrary"

for each

  • Fix an arbitrary $x$ (or:  arbitrarily fix $x$);  then $P(x)$ is true.
    In other words:  $P(a)$ is true; $P(b)$ is true; $P(c)$ is true; etc. $$\forall x\;P(x).$$ Literally:  For each $x,\:P(x)$ is true.
  1. "such that"
  • For each $x$ such that $P(x)$ is true, $Q(x)$ is true. $$\forall x\;\big(P(x)\implies Q(x)\big).$$ Literally:  For each $x,\,$ if $P(x)$ is true, then $Q(x)$ is true.
  • There is some $x$ such that $P(x)$ is true. $$\exists x\;P(x).$$ Literally:  For some $x,\,P(x)$ is true.
  1. "we have"
  • We have $P(x)$ being true.
    (or:  We have that $P(x)$ is true.) $$P(x).$$ More crisply:  $P(x)$ is true.

Fix arbitrary $x,y$. For all $f\in A$ such that $f(x)=0$, we always have $f(y)\geq0$, or we always have $f(y)\leq 0.$

$$∀x\;∀y\;∀f{\in}A\;\Big(f(x)=0\implies f(y)\ge0\;∨\;f(y)\le0 \Big);\tag1$$ more plainly:  for each $f,x$ and $y$ for which $f$ is in $A$ and $f(x)$ equals zero, $f(y)$ is either nonnegative or nonpositive.

Equivalently: $$∀f{\in}A\;\Big(∃x\;f(x)=0\implies ∀y\;f(y)\ge0\;∨\;∀z\;f(z)\le0 \Big),\tag2$$ i.e.,  every function in $A$ that has a root is either identically nonnegative or identically nonpositive,
i.e.,  every function in set $A$ that has a root never crosses (cuts through) the $x$-axis.

(Observe that formalisation (1) is the most succinct, while verbalisation (2) is the easiest to understand.)

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    $\begingroup$ This is very concise! In the first formal logic statement, is it possible to have: $g(0)=h(0)=0$ and $g(1)>1$, and $h(1)<1$ simultaneously hold? $\endgroup$
    – High GPA
    May 28 at 7:35
  • $\begingroup$ Re: the new plain statement. I guess when we say a function $f$ is nonnegative, it usually mean that $f(\mathbb R)\geq 0$? Will people confuse "$f(y)$ is nonnegative" with "$f$ is nonnegative"? $\endgroup$
    – High GPA
    May 28 at 7:37
  • $\begingroup$ 1. Yes. $\quad$ 2. Not technically: $f$ being identically nonnegative means that for each $x$ in $f$'s domain (in real analysis, this is some subset of $\mathbb R),\;f(x)≥0.\quad$ 3. "$f(y)$ is nonneg" is an incomplete sentence: the reader is left wondering whether you mean that $f$ is identically nonneg, or that $f$ is nonneg somewhere, and if the latter, whether that nonneg location is unknown or has been specified. $\endgroup$
    – ryang
    May 28 at 8:28
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    $\begingroup$ Very helpful! So it seems like "for all constant", "fix arbitrary constant" and "fix a constant" all have the same meaning? If so, why do people often use “fix a constant” instead of "for all"? $\endgroup$
    – High GPA
    May 28 at 14:20
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    $\begingroup$ Writing style! Alternatively, maybe think of “For all $x,$ blah blah.” as the theorem statement, and “Let us arbitrarily fix $x.$ blah blah. more blah blah.” as its proof. Personally, I might write “Let $x\in X$ (i.e., let $x$ be an arbitrary element of set $X)$, and suppose that $P(x).$ Then blah blah blah.” $\quad$ P.S. The comments at the top of Using “we have” in maths papers cracked me up; this response is particularly eloquent. $\endgroup$
    – ryang
    May 28 at 23:40

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