The barber paradox as bad technical writing

[kpreid]

In the unlikely event that you haven't heard of it already, the barber paradox is:

The barber [who is the only barber in town] shaves every man who does not shave himself. Does the barber shave himself?

Now, this can be considered just logically contradictory, or a gotcha (“the barber is a woman”). But how about considering it as a poorly-written specification? Under this principle I propose a correction:

The barber shaves every man who would not otherwise be shaved.

Comments

Interpreting specifications

[ccshan.myopenid.com (from livejournal.com)]

A natural way to interpret these specifications is as some kind of logical formula that can be satisfied or unsatisfied by each given model. Under such an interpretation, the original paradox is simply a specification that is satisfied by no model.

Your revision uses the words "would not otherwise". Perhaps we can treat these words as a modal operator and translate your revised specification to something like "forall x. man(x) & not(would_otherwise(exists y. shave(y,x))) -> shave(the_barber,x)", or some other formula. Still, there are two issues: First, what if Alice and Bob (neither the barber) shave each other? Your specification seems to allow the barber to shave neither then. Second, in an ordinary logic we would expect phi&phi to be equivalent to phi, but repeating your specification seems to change its meaning or even turn it into a paradox. And I don't see why a linear logic is called for.

Re: Interpreting specifications

[kpreid.livejournal.com]

Still, there are two issues: First, what if Alice and Bob (neither the barber) shave each other? Your specification seems to allow the barber to shave neither then.

And I'm sure Alice and Bob are perfectly happy with that.

Second, in an ordinary logic we would expect phi&phi to be equivalent to phi, but repeating your specification seems to change its meaning or even turn it into a paradox.

One of my prior versions was “The barber shaves everyone who is as yet unshaven”, which is essentially another form of that but timeful (and therefore explicitly transitioning from one world state to another). In this case, there is no paradox from doubling; the barber's second round (or the second barber; it doesn't matter) just shaves nobody. But some may be shaved that would prefer to have shaved themselves later.

BTW, when commenting, you're writing in HTML-plus-line-breaks; that's why your "phi&phi" turned into "phiφ".

[brokenhut.livejournal.com]

I am even more confused by the ambiguity in your restatement than in the original.

Does it mean, "the barber shaves every man who has hair growth" or "the barber shaves every man who does not have alternative arrangements"? The second one was my initial reaction, but seems tautological --- "the barber shaves who the barber shaves".

[kpreid.livejournal.com]

The latter. It is not tautological in that it implies that no one goes unshaven.

[https://www.google.com/accounts/o8/id?id=AItOawkLK5plVY4jv_uXVtUtNW_IDT3aL7xt9Do (from livejournal.com)]

This is essentially Russell's Paradox.

Your rephrasing is not the same as the original because there could be people who both do not shave themselves and the barber does not shave (e.g. two people shaving each other or just another barber in town).

Let's say you have two sets:
B, for the men who the barber shaves, and
S, for the men who shave themselves.

The original paradox says that
Forall x: if x ∉ S, then necessarily x ∈ B.

But your version allows that there could be some men who aren't in S, but could be shaved otherwise (by someone else). Therefore you lose the the dichotomy, the "necessarily."

[kpreid.livejournal.com]

I'm fixing the spec, not restating it.

https://www.google.com/accounts/o8/id

[(Anonymous)]

So you have shown that something else that is pretty close to the original barber paradox is not paradoxical?

[kpreid.livejournal.com]

An inconsistent specification should be corrected according to the best available judgement of the intent behind it. In this case, the intent I assume is that every man be shaved.

and if you want to translate that to database theory instead

[laurent-atl.livejournal.com]

congratulation, you just pretty much re-invented statelog

http://www.springerlink.com/content/fulcxc0dc2pxjlgq/