I am curious how one accounts for language and logic in this approach?

After all, a logical statement involves something like X iff Y and Z and Q, so the truth of X might varying discontinuously with average truth values of Y, Z and Q.

You may have one population where 50% think Y, Z and Q are truth and 50% don't and another population where 50% think only Y and Z are truth and 50% think only Z and Q are true. In the first population, 50% think X is true and in the other population 0% think X. But the average truth values of Y, Z and Q would higher in the second population.