If we're naming fallacies, then I would say that this post commits the following:
The Linguistic Consistency Fallacy: claiming, implicitly or otherwise, that a word must be used in the same way in all instances.
A word doesn't always mean the same thing even if it looks the same. People who worry about the purpose of life aren't going to be immediately reassured once you point out that they're just missing one of the relata. "Oh, silly me, of course, it's a three-place relation everywhere else, so of course I was just confused when I was using it here". If you ask people who are worrying about the purpose or meaning of life, "Purpose for whom?", in my experience they tend to say something like "Not for anyone in particular, just sort of "ultimate" purpose". Now, "ultimate purpose" may well be a vague concept, or one that we get somehow tricked into caring about, but it's not simply an example of people making a trivial mistake like leaving off one of the relata. People genuinely use the word "purpose" in different (but related) ways.
That said, the fact that everywhere else we use the word "purpose" it is three-place is certainly a useful observation. It might make us think that perhaps the three-place usage is the original, well-supported version, and the other one is a degenerate one that we are only using because we're confused. But the nature of that mistake is quite different.
If you think I'm splitting hairs here, think about whether this post feels like a satisfying resolution to the problem. Insofar as I still feel the pull of the concept of "ultimate purpose", this post feels like it's missing the point. It's not that "ultimate purpose" is just a misuse of the word "purpose", which, by the Linguistic Consistency Fallacy, must be used in the same way everywhere, it's that it's a different concept which is, for various reasons, a confused one.
FWIW I think "2-Place and 1-Place Words" is a bit dubious for similar reasons. Both this post and that make the crucial observation that we have this confusing concept that looks like it's a good concept "partially applied", but use this to diagnose the problem as incorrect usage of a concept, rather than viewing it as a perhaps historical account of how that confused concept came about.
Like I said, sort of splitting hairs, but it makes all the difference if you're trying to un-confuse people.
FWIW I think "2-Place and 1-Place Words" is a bit dubious for similar reasons. Both this post and that make the crucial observation that we have this confusing concept that looks like it's a good concept "partially applied", but use this to diagnose the problem as incorrect usage of a concept, rather than viewing it as a perhaps historical account of how that confused concept came about.
What do you mean by an incorrect use of a concept? If you curry a function, you get a new function, in this case a new concept that happens to be con...
I bet most people here have realized this explicitly or implicitly, but this comment has inspired me to write a short, linkable summary of this error pattern, with a name:
The Relation Projection Fallacy: a denotational error whereby one confuses an n-ary relation for an m-ary relation, where usually m<n.
Example instance: "Life has no purpose."
This is a troublesome phrase. Why? If you look at unobjectionable uses of the concept <purpose> --- also referenced by synonyms like "having a point" --- it is in fact a ternary relation.
Example non-instance: "The purpose of a doorstop is to stop doors."
Here, one can query "to whom?" and be returned the context "to the person who made it" or "to the person who's using it", etc. That is, the full denotation of "purpose" is always of the form "The purpose of X to Y is Z," where Y is often implicit or can take a wide range of values.
This has nothing to do with connotation... it's just how the concept <purpose> typically works as people use it. But to flog a dead horse, the purpose of a doorstop to a cat may be to make an amusing sound as it glides across the floor after the cat hits it. The value of Y always matters. There is no "true purpose" stored anywhere inside the doorstop, or even in the combination of the doorstop and the door it is stopping. To think otherwise is literally projecting, in the mathematical sense, a ternary relation, i.e., a subset of a product of three sets (objects)x(agents)x(verbs), into a product of two sets, (objects)x(verbs). But people often do this projection incorrectly, by either searching for a purpose that is intrinsic to the Doorstop or to Life, or by searching for a canonical value of "Y" like "The Great Arbiter of Purpose", both of which are not to be found, at least to their satisfaction when they utter the phrase "Life has no purpose."
Likewise, the relation "has a purpose" is typically a binary relation, because again, we can always ask "to whom?". "<That doorstop> has a purpose to <me>."
In some form, this realization is of course the cause of many schools of thought taking the name "relativist" on many different issues. But I find that people over-use the phrase "It's all relative" to connote "It's all meaningless" or "there is no answer". Which is ironic, because meaning itself is a ternary relation! Its typical denotation is of the form "The meaning of X to Y is Z", like in
Realizing this should NOT result in a cascade of bottomless relativism where nothing means anything! In fact, the first time I had this thought as a kid, I arrived at the connotationally pleasing conclusion "My life can have as many purposes as there are agents for it to have a purpose to."
Indeed, the meaning of <"purpose"> to <humans> is <a certain ternary functional relationship between objects, agents, and verbs>, and the meaning of <"meaning"> to <humans> is <a certain ternary relationship between syntactic elements, people generating or perceiving them, and referents>.
When I found LessWrong, I was happy to find that Eliezer wrote on almost exactly this realization in 2-Place and 1-Place Words, but sad that the post had few upvotes -- only 14 right now. So in case it was too long, or didn't have a snappy enough name, I thought I'd try giving the idea another shot.
ETA: In the special case of talking to someone wondering about the purpose of life, here is how I would use this observation in the form of an argument:
That being said, if you believe this argument, the best thing to do for someone lacking a sense of purpose is probably not to just say the argument, but to help them start satisfying their emotional needs, and have this argument mainly to satisfy their sense of curiosity or nagging intellectual doubts about the issue.