“The King of France is Bald” is Tricky

The king of France is bald. What does this mean? Is it this?

∃x [KingOfFrance(x) ∧ ∀y [KingOfFrance(y) → x=y] ∧ Bald(x)

Or maybe this?

How about this?

All of these depictions are useful. The translation of natural language into predicate calculus—pioneered by Bertrand Russell and later taken up and run with hard by Richard Montague—remains an invaluable tool for formal semanticists studying the interaction between a sentence’s grammatical structure and literal meaning. Lexicalized grammars like HPSG are formal systems that enable us to describe our syntactic and semantic intuitions in a manner that is precise, repeatable, and applicable to a broad range of expressions, bringing syntax out of the armchair and into the streets. Word embeddings reverse the discretizing movement of language–the conversion of the smooth flow of human experience into discrete symbols–by mapping those symbols back into a high-dimensional continuous vector space. This may seem like a perversely obscure thing to do, but actually proves quite helpful in software applications.

So these representations are all useful for performing various tasks related to a sentence’s meaning, but you can’t say that any are what the sentence actually means. This is most obvious in the formal logic case because that still looks like a sentence, except with the words out of order and some math symbols tossed in willy-nilly. It remains distractingly legible because it doesn’t manage to expunge all the natural language elements—there are still “functions” with names like Bald(x). This is less apparent with the HPSG representation because it’s harder to read—a visually busy grid, with non-standard mathematical notation. Still, look closely and you’ll see that tokens like “the”, “king”, “named”, and “bald” have again managed to insinuate themselves into the formalism. The theory may call them unanalyzed predicates, but they sure look like words to me.

The vector space representation does not have this problem. No one would ever accuse that list of three hundred numbers in the vicinity of zero of making any kind of sense. But unless you’re a computer, it’s not clear what to do with it. If I wanted to convey to another person that the king of France is bald, I wouldn’t clear my throat and then say, “Ok here goes…Point zero zero six. Negative zero point five three. Negative point one one six…”

All of these representations are helpful in one way or another, and it is their very helpfulness that serves as a clue as to why they must ultimately fail at representing an utterance’s essential meaning. In order to be helpful, a thing must be helpful for some task, and while the tasks described above may be compelling to linguists, programmers, and philosophers, the one most compelling to us all is the task of speaking itself. There simply is no better tool for the job of joking, flirting, arguing, and bonding (over one’s deepest passions, or just over whether it’s hot enough for you today) than the words we already use. There is no essence behind the curtain. It’s curtains, curtains, curtains all the way down.

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