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For nonnegative functions on a finite Boolean lattice, the condition $\alpha (A)\beta (B)\le \gamma (A\cup B)\delta (A\cap B)$ for all pairs implies $(\sum \alpha )(\sum \beta )\le (\sum \gamma )(\sum \delta )$. Coordinate elimination and the two-point four-functions inequality prove it by induction. Indicators give inequalities for the union and intersection of set families. The Boolean-lattice statement appears in Theorem 3.1 of Nicholas Ruozzi's paper.

In the four functions theorem, take $\alpha =1_{\mathcal{A}}$, $\beta =1_{\mathcal{B}}$, $\gamma =1_{\mathcal{A}\vee \mathcal{B}}$ and $\delta =1_{\mathcal{A}\wedge \mathcal{B}}$, where the union and intersection of set families are the collections of all pairwise unions and intersections.

If $\alpha (A)\beta (B)=1$, then $A\cup B\in \mathcal{A}\vee \mathcal{B}$ and $A\cap B\in \mathcal{A}\wedge \mathcal{B}$, so the pointwise hypothesis holds. Otherwise its left side is zero. Summing these indicator functions gives