paradoxes and deductive theorem again

[btanksley@hifn.com]

> From: iepos@tunes.org [mailto:iepos@tunes.org]
> Subject: Re: paradoxes and deductive theorem again

> > > 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...

Sure it is.  It's quite possible to have something true one day and not true
the next.  All you have to do is have a mathematics capable of describing
the context change.

> 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.
> whatever...

No...  You can formally define the statement and still have it change
values.  Furthermore, you can create an evolving algebra for it and wind up
being able to use it as a proposition, in spite of the changing value.

There's no vagueness involved, nor nihilism.

> > > 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)

But it was one anyhow -- and an accurate one.

> > 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
> theorem?

If we can come to agreement on what 'orange' is, yes, it would make a good
proposition.

> > > 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".

...not both.

OH!  WAIT!  I think I figured it out.  When you say "has the excluded
middle," you _really_ mean /follows the law/ of the excluded middle.  Boy,
that one threw me for a LOOP.

> As for 
> the example,
> if you answer that question about the orange hair, that should clear
> things up...

If it's a proposition, it follows the law of the excluded middle.  Some
things which are not propositions might very well follow that law, though.

> > > > > [...]
> > > > > 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. 

That's what I just said you did.  If an assumption leads through logical
steps to a contradiction, then the assumption must be incorrect.

> 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.

Wait!  You're asking too much.  You don't have to prove that EVERY statement
of that form is a proposition; you only have to prove that that statement
is.

In fact, though, the only statements of that form which aren't propositions
are (hypothetical) statements involving infinite sets.

> 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).

I don't think you have a problem there -- again, you have to be careful how
you define "everything".  For example, the set of all sets isn't a member of
the set of truths, nor is it not a member -- it's a contradiction.

As with the rules of logic, you have to check types :).

> > > 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"... 

Okay, then: what problems?

> > > 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...

Sounds like a question for sci.logic.

> > 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.

No, it's not.  It's simply the most obvious way to make a paradox.

> 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.

If you're going to do this, perhaps you'd best ignore all the other rules of
logic as well.  Use fuzzy logic instead.

It's actually a contradiction to ignore one of the rules of logic but accept
the others.

> - iepos

-Billy