One-preserving Boolean function ($f(1,\dots ,1)=1$)

A Boolean function satisfying the displayed condition. Composition of one-preserving Boolean functions is one-preserving, since each inner output is one on an all-one input. Consequently structural induction shows that a variable-built propositional formula using only one-preserving logical connectives, such as $\wedge ,\vee ,\to$, is true under the all-one Boolean valuation.