# paradoxes and deductive theorem again

iepos@tunes.org iepos@tunes.org
Fri, 10 Sep 1999 21:38:02 -0700 (PDT)

```> > 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.

hmm... i'm not going to argue over this anymore. to me, change is
something that applies only to physical things; i only imagine
propositions and don't see any reason to believe they have physical
existence... anyhow, it might well be useful to interpret it the
other way though.

anyhow, when i say, for instance, that my favorite food has changed,
I don't mean to imply the physical existence of some "my favorite food"
which has had a change brought upon it; rather, i mean that
the phrase "my favorite food" now denotes a different thing for me
than it used to. to me, this is a funny feature of the English language that
can cause logic problems (but may still be worthy of study, of course).
A good formal language should fix interpretations to its symbols
(independent of the context the are uttered in).

similarly, when i say that it is no longer true that "i am hungry",
although it was awhile ago, i don't mean to imply the physical existence
of some "i am hungry" proposition which has switched truth-values;
rather i mean that the symbol "i am hungry" now denotes a different
proposition because, essentially, I'm speaking a different language
(the word "am" takes on a different meaning, to reflect the change
in time).

>
> > 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.

well... it certainly seems reasonable to me also that these sorts of
things would be propositions. but if i understand your reasoning,
you seem to accept sentences to represent propositions as long as
you have a clear understanding of what all the components of the
sentence refer to. this reasoning is dangerous i think. for,
if you have combinators (particularly "Y", the paradoxical combinator),
then you can construct paradoxical sentences out of clearly understandable
parts. (if you don't know what i mean by "combinator", see my
introduction at http://www.tunes.org/~iepos/abstractions.html...).
combinators then allow sentences to refer to themselves without using
"this" or assigning the sentence an identifier.

> > > 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.

heh heh. oh it was kinda unclear, wasn't it? ;-)

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

well, maybe incorrect... but if you go so far as to say "not true",
then you're toast. for consider the liar paradox, "this statement is
that it is not true also leads to a contradiction. the reasoning
you're trying to make is an application of the deductive theorem,
and needs to be restricted to the case when the assumption
is a proposition i guess.

> > 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.

that's true. i didn't mean to say you did have to; but if you did, then
the specific reasoning would be okay. but anyhow it doesn't seem obvious
in this specific case either.

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

Hmm... i don't understand why you say that...
are you still talking about whether it is a proposition to say something
is a proposition?

> > 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

Well saying that everything is either true or not true (member of some
"set of truths" or not) is saying that the excluded middle is
universal (everything has it)... but i thought you'd concluded
that only propositions obeyed the excluded middle.

> 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.

(???). why is it not "not a member"? this doesn't seem paradoxical...
of course, there is the "set of all sets which are not member of
themselves"... which doesn't seem to be a member of itself but
also doesn't seem to not be a member of itself, but if you take a stand
either way (that it is or isn't a member of itself), you're doomed.
anyhow, this was just the point of restricting the excluded middle.

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

> Okay, then: what problems?

Well, there's the Liar paradox... or this simpler one, which doesn't
involve negation: Consider the statement "If this statement is true,
then Z is true" (where Z is some arbitrary absurd statement). Suppose that
it is true; the condition would be met, and thus Z would follow.
So, by unrestricted use of the deductive theorem, we conclude that
if that funny statement is true, then Z is true. But then,
we realize that that is precisely what that funny statement stated,
and thus we have admitted the funny statement, and thus Z (anything).

The self-referential statment can be formalized using the "Y" combinator.
Specifically, it would be "Y (\x.x -> Z)", or "Y (C P Z)", where "C"
is the "converse" combinator, "P" is implication, and "Z" is that
absurd statement. So, that's the problem with the deductive theorem...

> > 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.

hmm... well some sort of uncertainty logic at least. "fuzzy logic" i guess
usually means a boring logic in which everything is categorized
from 0.0 to 1.0 in probable truthfullness; well maybe there is some
usefullness in it (our brains seem to work using some sort of it,
don't they?).

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

oh no, (gasp!), a contradiction. well, it won't be the first one i've
seen... actually, i don't see how that would be a contradiction.
anyway, if i did implement such a system (which
i probably won't), i would probably accept all the favorite logic
rules, except i would toss out specific instances because they