I was wondering if you could explain Dialetheism for me. That shit is confusing. How can a contradiction be true?
Right, so, for those of you playing along at home, dialetheism is (roughly) the belief that there are true contradictions, or dialetheias - for some p, both p and not-p are true. Which means that, whatever p is, you could ask, "Hey, bro, uh... p?" and I could legitimately respond "Yes! Also, no."
And of course, at first blush that sounds fucking crazy. How can a thing be true and false at the same time? If you're a Randian, you are perhaps stomping your foot here and just insisting that contradictions can't be true, and that's all there is to it. (Also, if you're a Randian you should know that I'm going to be presenting two sides to an issue here, so get out while you can.)
It turns out, some sentences just don't fit neatly into the true/false paradigm. Sentences like the Liar's Paradox: "This sentence is false." What do you do with something like that? Is it false? Well then it's true, so suck it. Maybe instead you want to say it's true? Well if it's true, then it's false, so also suck it.
And now maybe you're like, "Well gosh, this is embarrassing. Okay, well, you've put me on the spot here, Bro, but can we just say it's neither?" I mean, we totally can say that, sure, and in fact for millenia we did just say that. No one really knew what to do about those paradoxes and they tried to sweep them under the rug like, "Huh! Aren't you an amusing little trick of language? Well, scurry off, we're trying to say true or false things here!" But they just won't go away. And then you get something like this sentence:
This sentence is either false or neither true nor false.
If that sentence is true, then you get either "true and false" or "true and not true or false"; and if it's false then you get "true and false", and you're right back where you started.
If that last bit confused you, don't worry about it. The important thing is, you can't just add new possibilities and get out of the paradox. No matter how many levels up you try to go, someone's going to find a way to stick you with a sentence that is both true and not true. Now you could try to just keep saying, "Oh, well, if we go one level up in language we see that that sentence doesn't make sense!" But really, you're building these arbitrary metalanguages and they're not doing anything except getting you out of this one problem. And you can never build enough. A dialetheist is going to roll his eyes at you and say, "Just... will you just let it be true and false, for fuckssake?"
But of course, allowing there to be contradictions opens up a whole new goddamn can of worms, which is my least favorite kind of can to open, and also my most favorite (see what I did there? Right, I know, fuck me.) For starters, it blows classical logic all to hell. It turns out that in classical logic, if you can prove a contradiction, you can prove literally whatever you want. Which, spoiler alert, is bad. We don't need people running around like, "The Liar's Paradox! Therefore, it follows that NASCAR is a sport!" See how that could get out of hand quickly? So if you accept dialetheism, you have to develop some new logical systems to make it work out. And that work is being done! Dialetheism is actually a pretty recent development, and it's some of the coolest stuff being done right now in philosophy.
One problem dialetheism faces is deciding which sentences get to be true and false. Obviously no one wants to make every contradiction true. But now, when someone says, "Well, I think p" you can't be sure they don't also think not-p. What do you do with that? WHERE DOES THE MADNESS STOP?! The obvious candidates are the Liar's Paradox and related statements - the tricky, self-referential bullshit, really. Those are statements that more or less force you to admit they're true and false, because they can't be one without the other and there's no neat solution in 'neither.' Graham Priest, maybe the most influential dialetheist still working in the field, has suggested - reminded us, really - that you should accept those statements that you have good evidence for, which he takes to be pretty limiting - you have good evidence that the Liar's Paradox is both true and false; on the other hand, you have lots of evidence that Elvis is dead, almost no evidence that he is not dead, and definitely no evidence that he both is and is not dead. So you don't have good reason to believe that "Elvis is dead" is both true and false. It looks like very few statements will turn out to be dialetheias on an evidence based view. Priest also suggests early on that transition states, like when you pass from one room to another, might give us dialetheias - it is true that I am in the room and also that I am not in the room. I'm halfway! These seem less obviously dialetheias to me - "You're inside and not inside so contradictions can be true!" seems like the kind of bullshit that would make Wittgenstein want to put his head through a goddamn wall.
So we have this hilarious image, an impetuous child angrily stomping her foot and screeching, "Contradictions CAN'T be true! They just can't!" But really, that's more or less all we have. Aristotle called the Law of Non-Contradiction the most basic law of reason, period, but that was just his way of gesturing and saying, "Well, I mean, obviously." He actually gave some reasons, but none of them were super-convincing. It was more like everyone wanted it to be true, so when Aristotle gave some bullshit argument they were like, "Oh, yeah, good enough. Okay. NEXT!" (If there's enough demand for it, I can summarize that text and/or Priest's two-millenia-late response.) And can you blame anyone for wanting that to be true? That seems pretty reasonable to me - you'd want some things to be true, and others to be false, and never the 'twain shall meet. When someone says "Hey dude, there are no velociraptors coming," you don't want to have to worry, "...but what if there also are velociraptors coming?" So it's no surprise that Aristotle got away with just sort ofs claiming contradictions are false for so long - we all think like that, which is why shit like the Liar's Paradox is so goddamn irritating. I mean, Thomas Aquinas thought that even God Almighty Himself couldn't make contradictions happen. When you think about how many annoyingly obvious things philosophers do question, it's a testament to how entrenched in our thought non-contradiction really is, that it took two millenia for someone to go, "Uh, guys? Are we sure about this?"
But eventually, someone did. So, here we are. So the short answer to your question, Matthew, might just be "Well, it turns out there just are true contradictions. Sorry for partying."
For more on dialetheism, the SEP page is really good if you've got the time - it is a technical introduction, but it is minimally technical.