Discussion Forum—A Way with Words, a fun radio show and podcast about language

Reminds me of a section in Gödel, Escher, Bach by Douglas Hofstadter. He suggested a class of words that describe themselves, words like "brief", "ugly", "written", "polysyllabic", "mispelled", "verbal" and "français". Let's call this class of words (since you said so) "aptronymic".

Now let's consider the opposite class, words that are nothing like their referents, for example "spoken", "monosyllabic", "onomatopoeic", "anglais", "misspelled" and "oral". Let's call these words "anaptronymic".

Now (asks Hofstadter), let's consider: Is "anaptronymic" anaptronymic?

If it is—if "anaptronymic" is in some way the opposite of the words that are members of its class—then it is indeed anaptronymic...and it is therefore descriptive of itself ("anaptronymic" is anaptronymic) and thus aptronymic. Wait, if it's anaptronymic then it's aptronymic. But if it's aptronymic—if "anaptronymic" is aptronymic—then it's anaptronymic.

Hofstadter was fascinated by such contradictions, and he managed to make them interesting to me too.

Is that good or bad? 🙂

Wow ... neither did I notice the GEB EGB wordplay. And I've read Hofstadter's book twice. Mostly it made my brain hurt, but there were so many great insights, especially into self-reference, that it made it worth the "pain."

Like, for example: Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.

Under this scenario, we can ask the following question: Does the barber shave himself?

Asking this, however, we discover that the situation presented is in fact impossible:

* If the barber does not shave himself, he must abide by the rule and shave himself.
* If he does shave himself, according to the rule he will not shave himself.
* He doesn't shave himself at all.

For more on this paradox, see: http://en.wikipedia.org/wiki/Barber_paradox

The paradox has been around for some time. Early 1900s. It's usually attributed to Bertrand Russell and occasionally called Russell's Paradox. I've often heard "Who shaved the barber?" used by geeks as a way to say "There is no answer to that question." I haven't read any Heinlein in ages, but I'm guessing that's how it was being used in the story.

That reference would be lost on 90% of the population. And maybe half of the people who read science fiction. Probably cracks up mathematicians though.

I have to wonder if that story predates the 1931 publication of Kurt Godel's "incompleteness theorem." Not like knowledge of it's implications immediately filtered down to the popular level ... many mathematicians didn't get it at first read, since it is very difficult to follow Godel's reasoning. I only came to understand it by reading a simplified explanation published by others much later.

Anyway, the language described as being "so exact that it was impossible to phrase a paradox" would be impossible according to Godel. His proof showed that in ANY formal system (math, logic, language, even science) there will always be statements that are "undecidable." They might be either true or false statements, but using the system in which they are expressed, it will be impossible to prove they are true or false.

That's why there are statements like "This statement is false."

Of course, science fiction is just that ... fiction, so I guess you're free to break whatever rule you want. Even faster-than-light travel.

Godel allows for systems that are perfectly unabiguous, but these would necessarily be "incomplete" in that they are restricted and do not address all possibilities. Godel asserts that all logical systems are necessarily either incomplete (silent on some valid situation), inconsistent (self contradictory), or ambiguous (indeterminate), or combinations of the three.

Bob Bridges said:

I suspect we're using "ambiguous" in two different ways. (Ironic, isn't it?) A statement that is ambiguous in my sense can be understood in more than one way; Glenn, I suspect you're using in Godel's sense, where an ambiguous statement is one that is "undecidable", ie that cannot be proven true or false.

Yeah, that's the way I read it too. As I said, I never attempted to read Godel's original paper ... way too deep for me. What I read was the "simplified" synopsis by Ernest Nagel and James Newman (both highly respected mathematicians). In their interpretation, "ambiguous" is a subset of "undecidable," as Bob points out.

I also have to agree with Bob's observation that the ambiguity of "ambiguous" is itself a bit ironic. I guess that would make it (as Draney pointed out) an aptronym?

CheddarMelt said:

There are a couple obvious solutions, rendering the setup non-paradoxical after all. Since the barber in the story is listed as the only male barber in town, one may easily imagine a barber in the next town could shave him, as could a female barber. Where is the paradox?

Er, not quite. If someone else shaves the barber, regardless of who or where they are, this violates the condition that the barber "shaves all and only those men in town who do not shave themselves." However, if you follow the link Heimhenge posted on Nov-24-10, it lists a few ways to cut this adoring knot.

My favorite answer would be to offer the logician the position of collecting only the paychecks of all the logicians who don't collect their own paycheck. Guaranteed to yield a solution before payday.