S(☹)⇔(∀x∈☺)(¬S(x))

That is to say, a sentence is true of nothing (☹) if and only if, for every x that is a member of the set of all things (☺), the sentence is not true of x.

It’s complete nonsense, and only fit for the purpose of adding a little humour to a mathematical conversation. For a start, if you plug in S(x) = ∃x, you get out the result that nothing *doesn’t* exist, so why are we talking about it? As I wrote at the time, “Once you’ve established that something doesn’t exist, you don’t say, ‘OK, so it doesn’t exist, but what OTHER properties does it have?’ Mathematical dragons aren’t allowed to be big, scaly, AND nonexistent.”

Then again, if you plug in S(x) = (∃x : 1+1=3), you’ll find that indeed no x exists with the magical power to make one plus one equal three, and therefore clearly “nothing” DOES have that power, i.e. ∃☹ : 1+1=3. But this simplifies to simply ∃☹, contradicting the earlier conclusion.

Moreover, if you plug in S(x) = (x=☹) you get ☹=☹, whereas if you plug in S(x) = (x=x) you get ☹≠☹, just as in English, “nothing equals nothing” is true but “nothing equals itself” is false.

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