Steven D'Aprano steven at REMOVE.THIS.cybersource.com.au
Mon Jun 15 06:50:27 CEST 2009

```On Sun, 14 Jun 2009 19:14:10 -0400, Terry Reedy wrote:

> Steven D'Aprano wrote:
>>
>> So-called "vacuous truth". It's often useful to have all([]) return
>> true, but it's not *always* useful -- there are reasonable cases where
>> the opposite behaviour would be useful:
[...]
> It seems to me that the absurd conclusion implied by the theorem
> invalidates the theorem rather than supporting your point.

I wouldn't go so far as to say the vacuous truth theorem ("empty
statements are true") is invalidated. The Wikipedia article discusses
various reasons why it's more correct (or at least more useful) to treat
vacuous statements as true:

http://en.wikipedia.org/wiki/Vacuous_truth

But it's not without difficulties -- however those difficulties are
smaller than those if you take vacuous statements as false in general.

[...]
> Try finding another 'reasonable case'.

Any time you do something like:

if not x and all(x):
process(x)

if all(x):
process(x)

I can't think of a real world example off the top of my head, but here's
a contrived example demonstrating that vacuous truths imply both a fact
and it's opposite:

def startswith(s, chars):
"""Return True if s starts with any of chars."""
for c in chars:
if s.startswith(c): return True
return False

if all([startswith(w, "aeiou") for w in words]):

if all([startswith(w, "bcdfghjklmnpqrstvwxyz") for w in words]):

If words is empty, this code claims that all of the words start with
vowels as well as starting with consonants.

There are, of course, ways to work around that other than rejecting
vacuous truths. One is to simply use an "elif" for the second test.

--
Steven

```