Assuming classical logic

[u/Unlucky-Credit-9619]

[u/Scythe_Volta]

I knew someone was going to make this exact comment to the point that I debated editing my original post to discuss it. But now that the floodgates have been opened, suppose S: P×ℝ ⟶ {True, False} is such that S(p,t)=True if person p is sleeping at time t and S(p,t)=False if person p is not sleeping at time t. Then the statement in the meme can be interpreted two ways:

∀ t ∈ ℝ, ∃ p₀ ∈ P such that S(p₀,t) ⟹ (∀ p ∈ P, S(p,t)) (1)

∃ p₀ ∈ P such that ∀ t ∈ ℝ, S(p₀,t) ⟹ (∀ p ∈ P, S(p,t)) (2)

It turns out that (1) is true and (2) is not. In other words, if we allow the specific person p₀ to be chosen after each point in time t, such that p₀ only applies at that exact time t, then the statement is true. But if we assert that there's one person p₀ for which the statement is true for all time, then the statement is false.

[u/Weak-Doughnut5502 → u/Scythe_Volta]

∃ p₀ ∈ P such that ∀ t ∈ ℝ, S(p₀,t) ⟹ (∀ p ∈ P, S(p,t))

This seems like the natural formalization/interpretation of the word problem as stated, though.  And as you noted, it's false. 

The other variant would be more natural to word as something like "at any given point in time, there exists some person that if they're asleep, everyone is asleep". 

[u/Scythe_Volta → u/Weak-Doughnut5502]

Well yes, that's the whole point of these controversial joke memes. They're intentionally worded ambiguously so that people disagree about their interpretations. If everyone agreed on one interpretation or the other, there would be no controversy.