## October 30th, 2006

### Dr. Zhang's Left Foot Implies tugrik

The math in this post will only display correctly if your computer and browser have the Symbol font installed. Otherwise, " means "for all", \$ means "there exists", Þ means "implies", Ù means "and", and ' means "element of".

I got my Artificial Intelligence exam back, the one I took last Wednesday. I am not happy with my score, but I'm not pissed at myself so much as the professor. My 71% (C-) is a bit above average for the class, which is a sign of the class going wrong more than an error on my part. I think this question on the exam and the professor's answer pretty much explains what's going on.

I need some explanation first. For those of you familiar with first-order symbolic logic (all three of you), you probably know that whenever you use universal or existential qualifiers, the variable is supposed to be over a particular set, otherwise the statement has no meaning. (It works for all X, but what sort of X?) Dr. Zhang gave me massive points off for using the correct syntax, as taught by six other classes I've had- "(x ' S), for example. Dr. Zhang required, in this case, "x S(x) Þ (rest of expression), which is like "for everything in the world, those that are in set S imply...", which works just fine in "for all". But in case of an existential qualifier ("there exists"), you'd need an "and", otherwise the existence of anything not in the set makes the expression true. Dr. Zhang fails to grasp this.

Exam problem 2c. "Express the following sentence in first-order symbolic logic: Some cats are big."
My answer (considered wrong): \$x cat(x) Ù big(x)
His answer: \$x cat(x) Þ big(x)

Unfortunately, this doesn't hold up. Let's say our entire universe consists of three things:

{kitten, puppy, Dr. Zhang's left foot}
We will define this such that:
cat(kitten) = true; big(kitten) = false
cat(puppy) = false; big(puppy) = false
cat(Dr. Zhang's left foot) = false; big(Dr. Zhang's left foot) = true

So in our universe, there are no big cats, and the sentence is false. That works correctly in my version of the sentence: it is never true that cat(x) Ù big(x) for any value of x.

For an existential quantifier (there exists), there only needs to be one to prove it true- that is to say, if I can construct a single value of x for which it holds, the statement holds true. So let's try, for example, Dr. Zhang's left foot in his version of the expression.
\$x cat(x) Þ big(x)
cat(Dr. Zhang's left foot) Þ big(Dr. Zhang's left foot)
F Þ T
T (F implies everything; a F on the left side of an if-then is always T, so "puppy" would also work here)

So if Dr. Zhang's version of the sentence really means that "some cats are big", then the mere existence of Dr. Zhang's left foot has proven the existance of . Sort of.

This is why I am going to be filing a regrade request in the near future.
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