Use of "any" in predicate logic?

Question

How would you translate these two sentences into predicate logic?

"X does not know anything" vs. "x does not know everything?"

Both sentences seem quite similar. Hence the confusion.

Answer 1

The first sentence means “for every piece of knowledge, X is unware of it”, whereas the second one means “the is some piece of knowledge of which X is unaware”. Can you take it from here?

Answer 2

Both the sentences are not similar, at least by its meaning. And, secondly, a predicate is just a boolean-valued function, which indicates that either a statement is true or false. So from a mathematical point, both the statements are different and can be both true or false or one of them can be true.

Answer 3

$\newcommand{\know}{\mathop{\rm know}\nolimits}$ "Any" is a bit tricky.

In the context of negation, "not any" usually translates as "no(thing)":

X does not know anything
= X knows nothing

Without negation, "any" can mean "every(thing)":

Any child can do this with a bit of help.
= Every child can do this with a bit of help.

And sometimes it means "some(thing)":

Pick any of the cards.
= Pick some (no matter which) of the cards.

What any means in a particular context is more of a question of English linguistics than logic. You will have to rely on your intuition to find that out.

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In any case, "all" or "every(thing)" translates as $\forall x \phi$:

"I know everything"
$\forall x (\know(i,x))$

Correspondingly, a negated "every" translates as $\neg \forall x \phi$, which is equivalent to $\exists x \neg \phi$:

"I do not know everything"
$\neg \forall x (\know(i,x))$
"There is something that I do not know"
$\exists x \neg (\know(i,x))$

"No(thing)" translates as $\neg \exists x \phi$ or, equivalently, $\forall x \neg \phi$:

"I do not know anything"
$\neg \exists x (\know(i,x))$
"For all things it holds that I do not know them"
$\forall x \neg (\know(i,x))$

And finally, "some(thing)" translate as $\exists x \phi$:

"I know something"
$\exists x (\know(i,x))$

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So your two sentences are not that similar:

"I don't know anything"

is

"There doesn't exist a thing I know"
$\neg \exists x (\know(i,x))$
or equivalently
"For all things it holds that I do not know them"
$\forall x \neg (\know(i,x))$

whereas

"I don't know everything"

is

"Not for all things it holds that I know them"
$\neg \forall x (\know(i,x))$
which is equivalent to
"There exists a thing that I don't know"
$\exists x \neg (\know(i,x))$