When I, attempted, to read this paper I felt as though I was Remy's brother in Ratatouille where Remy is trying to show him the power of taste variance but something internally doesn't compute. It's as if sight is a clear understanding and I'm looking at the subject of the paper through frosted glass.
I also find much of the article dense, but here's my attempt at a summary.
Logicians like the idea that words and phrases have "meanings" or "references."
Like, "my computer" should be interpreted, when I say it, as referencing a particular gadget that I own.
If you sit around and speculate about such references, you might wonder, for example, what the reference of "the unit sphere" is. Your Platonist friends will tell you that the reference is to an actually existing abstract object, or something like that.
The question of truth values seems to revolve around the question of the reference of a claim such as "Socrates is mortal." And the proposed answer is that this claim is a kind of name referring either to "the Truth" or "the False."
Note that this is how you tend to think of such claims when you work in predicate logic: the term Mortal(Socrates) is assumed to have the value True or False.
A quote cited by the article, by Jan Łukasiewicz (incidentally the inventor of "Polish notation"):
"All true propositions denote one and the same object, namely truth, and all false propositions denote one and the same object, namely falsehood. I consider truth and falsehood to be singular objects in the same sense as the number 2 or 4 is. … Ontologically, truth has its analogue in being, and falsehood, in non-being. The objects denoted by propositions are called logical values. Truth is the positive, and falsehood is the negative logical value. … Logic is the science of objects of a special kind, namely a science of logical values."
These thoughts about meaning were fundamental to the development of mathematical logic. Frege's notion of truth values was a prerequisite for his development of formal first-order logic and second-order logic, and algebraic logic in general (Boole's algebraic logic is an early version).
I am in the exact same position with respect to type theory, it's maddening and depressing in equal measure.
Not claiming to be any kind of expert here but maybe what is making the glass frosted for you is not quite grasping the following (not saying that I fully grasp it):
> This new and revolutionary idea has had a far reaching and manifold impact on the development of modern logic. It provides the means to uniformly complete the formal apparatus of a functional analysis of language by generalizing the concept of a function and introducing a special kind of functions, namely propositional functions, or truth value functions, whose range of values consists of the set of truth values. Among the most typical representatives of propositional functions one finds predicate expressions and logical connectives.
Predicate expressions[0] are: is tall? is green? is a woman? (adjectives and nominal and so on.)
Logical connectives[1] are: and, or, not, …
Imagine thinking about these concepts before the notion of using functions[2] arrived on the scene in mathematical logic and thinking about these concepts after.
A part of me would love to believe that the problem (such as it is) resides with the communicative powers of the entire community of mathematical logicians and not with the learning power of my cerebral cortex but alas I fear the failure is mine.
It saddens me that my mind can't quite grasp it.