Re: Knowledge in Action (Reiter) - example 2.1.1

On 11 Mar 2007 12:56:59 -0700, Daryl McCullough
<stevendaryl3016@xxxxxxxxx> said:
In article <slrnev8k89.103d.cmenzel@xxxxxxxxxxxxxxxxx>, Chris Menzel says...

On Sun, 11 Mar 2007 17:40:38 GMT, Neil Madden <nem@xxxxxxxxxxxxx> said:

T(x, y) <=> G(x, y) \/ (Ez).(G(x, z) /\ T(z, y)).

(Hopefully that is readable: E is existential quantification here).

The author then describes a counter-example (example 2.1.1):

"Consider the directed graph with two vertices a and b, and with a
single directed edge G(b, b). Consider a structure with universe {a,
b} that interprets G as {(b, b)} and T as {(a, b), (b, b)}.

Actually, it seems to me that the textbook is wrong. The definition
of T(x,y) has the consequence

T(x,y) -> Ez G(x,z)

So in particular

T(a,b) -> Ez G(a,z)

Since the right-hand side is false, then so is the left-hand side.

Of course you're right. I was seeing the example, dyslexically, in
exactly the way you suggest in your followup to the OP. I was focused
on the lesson about the limits of first-order expressiveness that Reiter
was intending his example to show, as I thought from the git-go that
that was what the OP was missing, but he was obviously just puzzled by
the fact that the example as stated simply doesn't seem to work.

.

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