paradoxes and deductive theorem again
Fri, 10 Sep 1999 18:36:58 -0700 (PDT)

> > hmm... well a formal logic system that doesn't provide a 
> > fixed interpretation
> > is doomed from the start I think, so everything in the system has
> > a "constant value"; anyway, but this is a matter of symbols 
> > and denotation,
> > and shouldn't be involved in a definition of "proposition".
> Eh?  Well, it is.  If the process of reasoning itself changes the value of a
> statement, it's not a proposition.

yeah... what i've meant to say is that a proposition (refering to the
meaning of the sentence, not the textual or verbal sentence itself) is 
not even a physical thing and "change" is not even applicable...
one may loosely say that "the screen is blank" is true sometimes and
false othertimes, but if i say that, what i really mean is that
there are various interpretations of the sentence (even in the
English language) and that some are true and some are false.

> > You're trying to decide whether something is a proposition or
> > not by looking at the way it is represented.

(i did not intend this necessarily to be a criticism)

> True -- my explanation was unclear.  The problem here is that the two
> statements are not propositions; intuitively, this is clear because they
> don't have a clear true or false value.  Formally, you've given a proof.

Well. Suppose I said "I have orange hair". Would you consider it clear
that that statement is either true or false (even though you don't
know which); if so, would that then justify use of the deductive

> > But it seems like every
> > representation of a proposition is ultimately going to need to use
> > undefined terms, so you're not going to be able to show that anything
> > is a proposition it seems; can you give me an example of a 
> > situation in 
> > which you _can_ deduce that something has the excluded middle.
> What do you mean "has the excluded middle?"  The excluded middle is, by
> definition, excluded.  You can't have it.  I can't prove that anything has
> it.

By excluded middle I mean "is either true or false". As for the example,
if you answer that question about the orange hair, that should clear
things up...

> > > > [...]
> > > > So, Plato is not not telling the truth, and thus Socrates 
> > > > is not not
> > > > telling the truth (contradictory with what we previously proved,
> > > > that Socrates is not telling the truth). So, we proved a 
> > > > contradiction;
> > > In fact, you proved that both statements cannot be 
> > > propositions, because if
> > > they were, they would both be simultaneously true and false -- and a
> > > proposition may only have one value.
> > Actually, what I proved was: _Suppose_ they were; then they 
> > would both be
> > simultaneously true and false. This is different than saying 
> > "if they were
> > ...".
> What you did was _suppose_ that the statements weren't propositions, and
> reasoned from that that they must be either true or false.  You tried both
> cases, found that they could be neither, and from that you have to conclude
> that your assumption was false: they are not propositions.

ummm... i don't think that's what i did (:-)). here's what i did:
I took as premises that both Plato's and Socrates' statements were
propositions. Then, that led to a contradiction. 

I think what you were saying was that "well, then it must not be the
case that both Plato's and Socrate's statements were propositions".
this almost seems obvious. but then, the deductive theorem usually
seems obvious. What I'm saying is that just because X leads to a
contradiction doesn't mean that X _implies_ a contradiction (i.e., not X).
This would only be the case if we know that X (in this case, the
assumption that both Plato's and Socrates' statements were propositions)
was a proposition. Anyhow, that reasoning would be okay as long
as it was always a proposition to say that something is a proposition.
this does not seem obvious though.

before, it seemed obvious that there was a set of truths, such that
everything was either a member of the set or not. but assuming
this leads to problems (denying it for any specific instance also does).

> > After all, the precise statement of the deductive theorem is
> > that these two kinds of statements are always equivalent, and the
> > deductive theorem when accepted universally we know leads to problems.
> What two kinds of statements?  I missed something.

The deductive theorem says that "A leads to B by sound reasoning"
is equivalent to "if A, then B"... 

> > My question: is
> > it always a proposition to state that something is a proposition?
> Um...  Lets' see, I don't see any catches in that one.  I'm going to go out
> on a limb and say that it is.
> Now, I could be wrong.  It's possible that Godel's First Theorem proves the
> existance of some statements like that which are NOT propositions.  But I
> don't know enough about incompleteness.

neither do i...

> You were saying that you could build a compiler which checked for cycles in
> logic, and if any cycles were found it would reject the input as "not a
> proposition".  This would reject some actual propositions.

really, i'm not currently considering that... i don't know what i'm
considering anymore. i don't think self-reference is really at the
heart of the problem (but maybe it is)... aaack. i think i'm
going to settle for a system in which the deductive theorem
(probably the generalization rule as well) is postulated only
in special instances by humans, and in these cases only with much
prayer that no paradoxes will be derived. anyhow, a practical
automated logic system is going to have to do reasoning on uncertain
statements and should thus be used to getting contradictory results;
maybe it can just "ignore" the paradoxes, like people usually do...
of course, problems will still probably slip through the cracks
if this approach is taken.

> In that sense, logic is a falsehood.  It forks no lightning, as the poem
> says ("Do Not Go Quiet Into That Good Night").  Its only use is a tool for
> persuasion, not for truly proving anything.

oh, i've given up hope long ago of results of logic being deserving of my
absolute certainty... but it's hard to deny its practical uses.

> > - iepos
> -Billy


- iepos