I can appreciate the depth of research behind the article. The many-valued notions of logic are quite helpful. Here's are two ways that I would distill the message for different audiences.
1. A computer science audience: Not all sentences have a computable boolean truth value. Here's why. To find the truth value of the sentence "This sentence is false.", you have to figure out the truth value of the following statement (call it X1): "Whether or not this sentence is false depends on if X1 is true or false." Put that way, you can immediately see the circularity in computation. From a computer science perspective, there is no terminating condition. The function will never complete, so there is no computable boolean truth value.
2. To an educated, but non-technical, audience: Just because a sentence asserts something doesn't mean there is any guarantee of the result being "truth" or "falsehood". The truth value of this "This sentence is false." is simply "undecidable" or "ineffable" -- pick a word. So, just live with the fact that not everything is true or false. Note: I'm not saying that the statement is really one or the other but we just can't figure out which (as in Schrödinger's cat); I'm saying that neither "true" or "false" makes any sense for that kind of self-referential sentence.
This really isn't mind-blowing. Many articles write up these "paradoxes" as if they are insurmountable. They aren't.
All of this said, I think many philosophical writings, especially Buddhist writings, use contradiction as way of promoting thinking and careful decomposition of the essence of things. In short, complex things consist of parts that vary over time. So their components or transient values can seem to change or stand in contradiction.
There is no contradiction in Dickens writing "It was the best of times and it was the worst of times." in my opinion. This is just a literary device to show contrast eloquently, because saying "In some regards, it was the best of times. In other regards, it was the worst of times." is not as memorable or striking.
1. A computer science audience: Not all sentences have a computable boolean truth value. Here's why. To find the truth value of the sentence "This sentence is false.", you have to figure out the truth value of the following statement (call it X1): "Whether or not this sentence is false depends on if X1 is true or false." Put that way, you can immediately see the circularity in computation. From a computer science perspective, there is no terminating condition. The function will never complete, so there is no computable boolean truth value.
2. To an educated, but non-technical, audience: Just because a sentence asserts something doesn't mean there is any guarantee of the result being "truth" or "falsehood". The truth value of this "This sentence is false." is simply "undecidable" or "ineffable" -- pick a word. So, just live with the fact that not everything is true or false. Note: I'm not saying that the statement is really one or the other but we just can't figure out which (as in Schrödinger's cat); I'm saying that neither "true" or "false" makes any sense for that kind of self-referential sentence.
This really isn't mind-blowing. Many articles write up these "paradoxes" as if they are insurmountable. They aren't.
All of this said, I think many philosophical writings, especially Buddhist writings, use contradiction as way of promoting thinking and careful decomposition of the essence of things. In short, complex things consist of parts that vary over time. So their components or transient values can seem to change or stand in contradiction.
There is no contradiction in Dickens writing "It was the best of times and it was the worst of times." in my opinion. This is just a literary device to show contrast eloquently, because saying "In some regards, it was the best of times. In other regards, it was the worst of times." is not as memorable or striking.