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There's explanation in Tarski's book about belonging of the null class to every other class.

It goes like this:

“If x belongs to the null class, then x belongs to K. Whatever object we substitute here for "x", and whatever class for "K", the antecedent of the implication will be a false sentence, and hence the whole implication a true sentence (the implication--as mathematicians sometimes say--is satisfied "vacuously")."

But by this way I can prove otherwise just by replacing the consequent with “x doesn't belong to K”.

So this argument can't be valid, because it doesn't work exclusively for the first statement.

Am I wrong?

Егор Галыкин
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    Can you write down the full argument instead of saying "replacing this with that" and "prove otherwise"? This may help you sort out the problem. It is also true that if $x$ belongs to the null class, then $x$ doesn't belong to $K$. – Trebor Nov 25 '23 at 12:20
  • Your use of “belonging” is confusing, probably due to translation from another language. Since every element of the empty set (null class) is an element of $K$, the empty set is a subset of $K$. – Carsten S Nov 25 '23 at 12:26

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What the argument shows is that the empty set is a subset of any set $K$…. which I assume you agree to.

By changing the argument the way you do, you end up showing that the empty set is also a subset of the complement of any set $K$, because any object $x$ is an element of the complement of $K$ if and only if $x$ is not an element of $K$.

But that is of course also true, since the complement of any set $K$ is itself a set, and we already established that the empty set is a subset of any set.

So there is no problem here.

Bram28
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