The requirement that triangle equality for the inner product is satisfied, does not allow linear superposition and one must choose some orthogonal basis for the space. The absence of quantum superposition means that theory is completely classical. Thus it seems that Boolean QFT is completely classical and the transition from classical mechanics to quantum theory could be regarded as a transition from binary QFT to complex QFT or from a binary logic to complex logic. Hence a direct connection with earlier ideas about genetic code emerges. Multiplication corresponds to the intersection of the sets.
Entire set represents unit element and empty set zero.
The states of this QFT are constructed in the usual manner. Thus Boolean particles are fermions always. The proposition and its negation correspond to the states created by, say I oscillator operators and the dual of this state created by the remaining N-I oscillators operators. One can change state basis by unitary transformations. In particular, a nonlinear transformation of the oscillator operators in general occurs.