A couple months ago, I wrote a dialogue for my satirical series, Somewhere Atop the Ivory Tower, on “The Present King of France is bald” proposition. That piece was inspired by one of the most peculiar puzzles in the Philosophy of Language, and if it went over your head, that’s fine. So to help decipher the madness of this famous proposition, and because today is Bastille Day, the mark for the beginning of the end of the reign of kings in France, I have decided to write a special philosophical editorial for it here.
The proposition is an example of definite description. What logicians and philosophers mean by that is this:
∃x((Fx & ∀y(Fy → y = x))
And if you’re like the average human being who doesn’t speak in logical symbols, this effectively means that “there exists one F, and it is the only F,” or “there is only one F” (F being a variable for anything you want to put there). It’s a useful description if you’re trying to pinpoint exactly one object in a set of whatever you’re trying to categorize and postulate other descriptions onto it, hence why it is called a definite description.
So what’s the problem then? Well this British philosopher named Bertrand Russell (whom I kiddingly and affectionately call “Berty”) was very interested in the semantics of definite description in his early philosophical career, and what he discovered was something that he believed would be a threat to classical logic as we know it!
Russell and the Law of Excluded Middle
Now before my fellow logicians, mathematicians, and philosophers start to panic, Berty has since been debunked many times over for this and many other metaphysical faux pas that he has made; and while he will be remembered in the field for these puzzling propositions, his legacy also includes very important commentaries on social philosophy, and oftentimes going to the front lines on radical issues of his day. Apparently he also had a taste for really good porn too, so you know. He’s still human.
As he was playing around with definite descriptions, Russell considered the most obvious things to add to the sentence: an additional feature of the definite description. So he did:
∃x((Fx & ∀y(Fy → y = x)) & Gx
Again, if you don’t speak in logical symbols, this reads as “there is only one F, and it is also G.” Now in this case, G is not definite, so other things can be G. What the sentence is saying though is that the only F in the universe is also G. Still no problem with that sentence? Probably so. But what if we were to make a case where this sentence and its opposite have the same truth value?
This is where Russell’s famous proposition comes in. One interpretation of this logical sentence could be “The present King of France is bald.” After all, there is only one present King of France, thus fitting the definite description, and we’re making a claim that he is also bald. That’s great and all, but last I checked, there is no present king of France. They and their progeny were all violently executed in the wake of the first French Revolution in 1789, well before Berty started pondering this proposition, and obviously long before ours as well.
On these grounds, “The present king of France is bald” would be false, since there wouldn’t exist a present king of France in the first place. And since the first half of that sentence before the ampersand is false, we have reached our conclusion.
The same goes for the sentence’s opposite, “The present king of France is not bald.” This sentence would also be false for the same reason: since there is no present king of France, it stands to reason that the entire sentence is false.
This is the problem with the law of excluded middle, a logical conundrum where two complementary sentences (i.e. a sentence and its opposite) result in contradictory truth values. In most cases, if one sentence is true, its opposite would be false. This is how classical logic dating as far back as Aristotle would have worked! And yet by playing with the semantics of just one sentence using this logical form of definite descriptions, Russell would have had the logical world stumped!
That is, had it not been for the fact that another guy was working on this very problem an entire generation before him and solved it with a much more obvious logical feature.
Frege and the Principle of Indifference
So right around the time young Berty was still but a toddler, a relatively obscure German philosopher named Gottlob Frege was also positing the semantics of logical sentences. Sadly I don’t have an elaborate backstory on Frege, but he is considered to be the pioneer of the Philosophy of Language in the Western canon. So yeah. Kind of a big deal.
While Frege didn’t explicitly use the example of “The present king of France is bald” as a proposition (although he easily could have given the timeline), he was also playing around with definite descriptions. Yes, he got around to playing with this idea that there might be a sentence and its opposite would hold the same truth value or whatever, except that he stopped short of one thing: who ever said that we were certain the definite describer does not exist?
This was the feature that Russell’s contemporary Peter Frederick Strawson had noticed in response to the controversial proposition, and piggybacked on Frege’s work to resolve the problem. By adding an epistemological take on the sentence, we are simply not sure whether or not the present king of France is bald. At a platonic level, we can understand that a present king of France exists as a concept, even as we speak right now in 2018. However, since no present king of France exists in our real, tangible universe, there is no way to check if the present king of France is in fact bald (or not bald). Therefore, the sentence “The present king of France is bald” and its opposite “The present king of France is not bald” still have the same truth value, but they’re a lot easier for most of us to resolve without it not being a metaphysical conundrum: they’re both undefined, or inconclusive.
So there you have it. If you ever come across a problem that threatens the fabric of classical logic, you can always use the next order of logic (I guess modern?) and say that we just can’t tell. Next time, Berty. Next time.
But I’m not done with this proposition just yet, and if Philosophy of Language classes have anything to say about it, neither are they. Because at the end of the day, some of us are probably not very comfortable with a cop-out answer like “they’re undefinable because we just don’t know.” And just when you thought that Berty was alone in this seemingly one-sided fight, he’s got a rather influential ally.
Tarski and the Set of all Things Bald
So unlike these British and German philosophers that I have named thus far, Alfred Tarski is the American of the bunch! And while he’s not considered a philosopher by title, his set theories have become a staple of logic that philosophers critique. So there you have it. I’m considering him a philosopher.
Again, Tarski doesn’t exactly respond to the direct problem of “The present king of France is bald” proposition, but his theory certainly does. For the sake of this argument, Tarski’s set theory includes a very quick and easy way to check if a sentence is true or not.
A (definite) description such that F is G is true if and only if F belongs in the set, or an element of the set G.
So in other words, “The present king of France is bald” is true iff in the set of all things bald, there is the present king of France. And since the present king of France does not belong in that set (because as we have established, a present king of France does not exist), there we have it. The sentence is false. Berty was right all along!
Now of course, Tarski’s addendum to the problem still doesn’t really solve the problem that the present king of France would technically also be in the set of all things that are not bald, thus starting the problem over again, but somehow I don’t think that Tarski was ever really concerned about the set of opposites. Countless other logicians and philosophers have theorized their own semantics to solve Russell’s problem, whom I have included somewhat in my original dialogue on the issue (see link at the very top of this post), but I don’t think that we as philosophers will ever be satisfied with the problem one way or another. Such is the nature of our love of wisdom: we relish in the process of finding out the truth more than actually knowing the absolute truth… or something like that.
So I leave you all with this final question on the matter: how would you resolve the proposition? Is the present king of France bald absolutely false? How would we know? I suppose as long as liberty prevails and France doesn’t decide to establish a king any time soon, we will still have much to ponder, and argue about.
Happy Bastille Day!