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

"The meaning of <the sound 'owe'> to <French people> is <liquid water>" or
"The meaning of <that pendant> to <your mother> is <a certain undescribed experience of sentimentality>".

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:

First of all, you may be lacking satisfaction in your life for some reason, and framing this to yourself in philosophical terms like "Life has no purpose, because <argument>." If that's true, it's quite likely that you'd feel differently if your emotional needs as a social primate were being met, and in that sense the solution is not an "answer" but rather some actions that will result in these needs being met.

Still, that does not address the <argument>. So because "What is s the purpose of life?" may be a hard question, let's look at easier examples of purpose and see how they work. Notice how they all have someone the purpose is to? And how that's missing in your "purpose of life" question? Because of that, you could end up feeling one of two ways:

(1) Satisfied, because now you can just ask "What could be the purpose of my life to <my friends, my family, myself, the world at large, etc>", and come up with answers, or

(2) Unsatisfied, because there is no agent to ask about such that the answer would seem important enough to you.

And I claim that whether you end up at (1) or (2) is probably more a function of whether your social primate emotional needs are being met than any particular philosophical 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.