What does it mean, intuitively and then also precisely, when we say that a particular English word is not truth functional? Let me present some examples.
Example 1 As far as I can tell from a book I am reading about propositional logic, the truth value at a particular time of the sentence Max is home whenever Claire is at the library (call this S) is not determined by the truth values, at that same time, of the atomic sentences Max is home (call this A) and Claire is at the library (call this B).
S appears to be similar to the sentence B -> A. However, it is possible that Claire is in fact at the library (B is true), Max is in fact home (A is true), but the sentence as a whole is false. My interpretation is that there is the possibility of the sentence being false in this case because whenever does not refer only to one moment or instant in time, but instead refers to all instants in time in which Claire is at the library. The atomic sentences on the other hand refer only to one instant in time.
If we had the sentence D being Max is home if Claire is at the library, apparently if refers to one instant in time. Therefore, the truth values of the atomic sentences relate to one instant in time, and the sentence D with the word if also has a truth value that is related only to that same instant in time. So the truth value of D depends solely on the truth values of the atomic sentences, and it is therefore truth functional.
Example 2 Consider sentence S If Max had been home, then Carl would have been there too.
Looks like a sentence with a material conditional: B -> A, where B is Max had been home and A is Carl would have been there too.
From what I read in my book, it is possible that B is true and A is true, but the sentence as a whole is false. First of all, it is confusing to think of a truth value for Max had been home, because of the tense of the verb.
Intuitively, if it is possible that Max had been home is false, but the sentence as a whole is also false, then this means that the sentence does not follow the truth table of a material conditional, and hence does not depend solely on the truth values of the constituent atomic sentences.
For example, say Carl is a dog, and the person who says S thinks Max is taking Carl with him everywhere he goes. But what if instead of this circumstance being true, someone else altogether has Carl. Even if Max were home, Carl would not be with him, because Carl is with this other person.
Example 3 If the book is here then the mailman delivered it.
A is The book is here, B is the mailman delivered it, and S is the sentence A -> B.
If the book is here, and the mailman delivered it, then the sentence is true. If the book is here, and the mailman did not deliver it (say someone else brought it over), the sentence is false. If the book is not here, whether the mailman delivered it or not, the sentence is true.
It seems like the concept of whether a sentence is truth-functional or not is not super sharply defined. Intuitively, in example 3, it is truth-functional because when I assign truth values to the atomic sentences, my imagination cannot come up with a scenario wherein the sentence as a whole has a truth value not equal to the truth value we would expect from a truth table analysis. In examples 1 and 2, I did come up with such a scenario.
Is it correct to say that if I can come up with a scenario wherein the truth value of the sentence differs from the truth-table expected value, then the particular English connective is not truth-functional?
I would imagine there is no airtight criterion for determining whether a word in English is or is not truth-functional.