As a first guess, we might choose the linear transition, but many situations should be non-linear. For example, suppose we want the truth of the statement " our system is quite acidic." Our index is pH, but this is a logarithmic scale. There are ten times more hydrogen ions at pH 3 than at pH 4. Our truth decision should give more weight to lower pH values, and the transition line should be logarithmic.
We should stop and consider the abscissa for a truth graph. If this were a graph of the truth of the statement "It is too darn hot !", the abscissa might be temperature. Membership in the set of hot days could be 100 percent above some temperature and 0 percent below some temperature. Intermediate temperatures denote partial membership. For the testimony of a witness in a legal dispute, the abscissa might be credentials, how well the facts are remembered, speaking ability, and the like. In reality, all these may be lumped into some composite index that we call our faith in the testimony. It is very important in this context that the truth is some overall truth based on an accumulation of information. There might also be a truth graph for each question asked to a witness. Some sort of mental processing of all the fuzzy truth would result in a decision about innocent or guilty.
Truth graph for a warm day in March
Truth graph for "temperature is about 60 degrees"
Truth graph for "temperature is about 60 degrees (another opinion)"
Truth graph simply with straight lines
