Re: [aima-talk] Digest Number 303
- I doubt if I can make the issues more clear, but I'll try.
I still think the main problem lies in the phrase "if it were false". If
a proposition in a set is "shown" to be false from observation rather
than through logical proof, the proposition does not make the set
inconsistent, because it *could* be true in some other possible world.
If I recall correctly, that is how logical possibility is defined. Your
ABC example is indeed inconsistent because it can be proved without
observing the world state that the set contains at least one false sentence.
So, according to the defintion of consistency, the authors mostly likely
do not mean "if it were provably false". Otherwise, the argument would
be employing a tautologus sentence: "if the sentence were provably
false, then he (or in fact, anyone) could not consistently assert it to
be true". One derives nothing from a tautology
But, without the word "provably", I don't see how the contingent truth
or falsity of a sentence have anything to do with consistency, for
reasons above. Furthermore, it seems to me that J. R. Lucas just cannot
consistently assert the sentence anyway, whether it is true or false.
I do agree that we can do nothing formal without identifying the
underlying logic system. Perhaps somebody can enlighten us?
> Thanks for the interesting discussion.Thank you.
The Geek wrote:
> I think the discussion all boils down to the
> definition of "consistent" and "contradiction". I was
> kind of hoping one of the authors would chime in at
> some point to save me, but... :-) On page 137 they
> define consistency in reference to CSP problems as "an
> assignment that does not violate any constraints".
> That was the reason for my A,B,C example - looking at
> it as if it were a CSP.
> Their argument assumes the sentance must either be
> "true" or "false", (no quantuum physics or "unknown"
> values allowed) and hinges on the
> belief/assumption/fact one cannot "consistently"
> assert that a sentance is true if it can be shown to
> be false. If you buy into that, then their statement
> follows naturally - if the sentance was false, then he
> could not consistently assert it to be true, which
> would therefore make the sentance true - a
> contradiction since we assigned a value of "false" to
> the sentance at the beginning. The only way you can
> assign a value to the sentance and have everything
> hold together is if you assume it's true, which then
> implies that everyone else can assert it without a
> problem, but he cannot.
> But, at this point I don't think we're going to make
> any headway. Since they didn't define the logic
> system they're using, we can't really do anything
> Thanks for the interesting discussion.
> Rob G.
> --- Paul Hsueh-Min Chang <avatar@...>
> > I will try to make some clarifications below:
> > > While they haven't spelled it out in detail, I
> > think
> > > they're using a simple logic system and definition
> > of
> > > "consistent". In other words, they're not trying
> > to
> > > allow for an "agent belief" representation where
> > we
> > > acknowledge that beliefs might be wrong. Instead,
> > if
> > > you assert a sentance it must be true or you've
> > > introduced a contradiction.
> > Maybe our sources are different. AFAIK a
> > contradiction is the negation
> > of a tautology (P. Tidman and H. Kahane, Logic and
> > Philosophy: A Modern
> > Introduction, 8th edition, p48). For example, p^~p
> > is a contradiction.
> > So when I assert a contingently false sentence whose
> > truth depends on
> > the world, I introduce no contradiction. But maybe
> > our definitions of
> > contradiction differ.
> > > If we take 3 sentances:
> > >
> > > A = B.
> > > B = C.
> > > A != C.
> > >
> > > If we try to assert all three sentances, we cannot
> > do
> > > it without creating a contradiction. Since this
> > is
> > > the definition of "consistency", they would say
> > that
> > > we cannot "consistently" assert all three
> > sentances,
> > > since doing so would introduce a contradiction.
> > It is true that we would introduce inconsistency,
> > but we would not
> > introduce a contradiction. Of course, any set of
> > sentences that contains
> > any contradictory sentence is inconsistent, but not
> > vice versa.
> > > In this case, as an agent **I** can consistently
> > > assert the sentance "Agent A cannot assert this
> > > sentance without being wrong" without introducing
> > a
> > > contradiction. However if Agent A tries to assert
> > the
> > > same sentance, he runs into a problem. If he
> > asserts
> > > it as true and is right, then the sentance is
> > false,
> > > so he's wrong.
> > I know what you are trying to say, but that I don't
> > think we are talking
> > about the same thing. The authors clearly intend to
> > show two things:
> > 1. The sentence is a tautology (i.e. necessarily
> > true).
> > 2. Yet, J. R. Lucas cannot assert it.
> > They offer two arguments to show (1). What I do not
> > understand is the
> > their second argument to show (1). You are talking
> > about (2), which I
> > find no problem.
> > > But the whole point of the illustration is simply
> > to
> > > show that **sometimes** one agent is unable to
> > > assert/know/do things that another agent can, but
> > that
> > > doesn't automatically imply that agent is
> > inferior,
> > > it's inability to assert/know/do may be related to
> > the
> > > specific situation.
> > I do understand the point; it is simply that
> > particular argument (the
> > one on p950 in parentheses) that confuses me. As a
> > philosophical
> > treatment, I think the chapter should reasonably be
> > taken literally, but
> > when I do so, I cannot make sense of that particular
> > argument.
> > Paul