

TruthFunctors
Consider "j and y". If we had a sentence consisting of this together with two constituent sentences, we would be able to tell the truthvalue of the sentence as a whole given simply the truthvalues of the constituent sentences. We would know that the whole sentence would be true if both the constituent sentences were true, and otherwise would be false. We could represent this fact about "j and y" by the following diagram:
A diagram of this sort is called a truthtable. So, a truthtable for a sentencefunctor is a diagram which shows (in the case of a sentence consisting of it together with constituent sentences) what value the sentence as a whole will take for every possible combination of truthvalues in the constituent sentences. We can define a truthfunctor thus: A truthfunctor is a sentencefunctor such that the truthvalue of any sentence consisting of it together with constituent sentences is determined solely by the truthvalues of the constituent sentences (together, of course, with the meaning of the sentencefunctor).Alternatively, we could define a truthfunctor as a sentencefunctor with a truthtable  but the usefulness of this depends, of course, on the fact that we have already defined truthtable. Some sentencefunctors are not truthfunctors: "j because y", for example. This is because, although the falsehood of either of the constituents is enough to determine the truthvalue of the sentence as a whole (it must be false), the truth of both constituents is not enough to determine the truthvalue as a whole. For instance "Bill Clinton was president of the USA because he secured most votes in the electoral college" is true. But "Bill Clinton was president of the USA because the battle of Hastings took place in 1066" is not. So we can draw at most a partial truthtable for "j because y".
The following is also not a truthfunctor: Bill Clinton was president of the USA and j. This is because, although the falsity of j is enough to determine the truthvalue of the whole, its truth is not. You, of course, will know that the whole is true if the constituent is true. But that is because you know some history. The fact that it is true is not determined solely by the truth of the constituent and the meaning of the sentencefunctor. 

