paradoxes and deductive theorem again

[btanksley@hifn.com]

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

> > No -- the standard definition of a proposition is that it 
> > may be either true
> > or false (but not both), and its value is constant.

> > For example, the statement "x is mortal" is not a 
> > proposition, because
> > although its value is true or false, it's not constant.

> 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.

> > > For consider this dialogue:
> > > Socrates: What Plato is about to say is not true.
> > > Plato: Socrates has spoken truely.

> > Both statements cleverly fail to be propositions by leaving 
> > terms undefined.
> > Imagine that you're a compiler -- you have to determine 
> > whether everything
> > you reach is constant and either true or false.  At the 
> > time you reach
> > Socrates' statement, you can't know what Plato is about to 
> > say, so you've
> > got a forward reference.

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

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.

> 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.

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

> 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.

> However, if we know that it is a proposition to state that 
> something is a proposition, then the deductive theorem is licensed
> (at least, that seems to be your point of view).

> 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.

> > > Another question: if the deductive theorem is restricted 
> > > to some sort
> > > of proposition-hood, is it necessary that the assumption _and_ the
> > > hypothetically proven thing both be propositions, or is it 
> > > only necessary 
> > > for the assumption. I think this may depend on the way 
> > > "proposition-hood"
> > > is formulated, but I'm guessing that only the assumption will need
> > > the restriction.

> > No, both need to be propositions, according to formal logic.

> Hmm... not sure what you mean by "according to formal logic". 
> anyway ...

The rules of formal logic?

a -> b or c is equivalent to (!a or b) or c which is equivalent to !a or (b
or c).  If b wasn't a proposition this transformation would be invalid.

> > You're considering ruling out forward reference.  As you 
> > note, it's possible
> > to make valid forward references merely by re-arranging 
> > text.  A check for
> > single-valued constantcy is sufficient instead.

> Hmm... I'm not quite sure what you mean by that. But then again, I'm
> not really sure anymore what at all i was considering ruling out.

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.

> > Anything which is NOT defined is not possible to use in a 
> > proposition,
> > because its logical value is also not defined (and thus not 
> > true or false).

> Hmm.. are you saying you refuse to reason about propositions unless
> you already know specifically that they are true or that they are
> false? This restriction seems pretty harsh, and would rule out the
> deductive theorem entirely (except when you didn't need it, when
> you already knew the truthfulness of the involved propositions).
> Maybe i've misunderstood though...

No, because I'm only refusing to reason about things which are not
propositions -- or more accurately, I'm refusing to use the laws of logic to
manipulate things which are not propositions.

No, it's not important that I know whether they're true or false.  It's only
important that I know that they are constant and one or the other.  One good
way to check that is to test that they could be consistently assumed to be
one or the other.

It's possible to have a statement which is a proposition if it's true, but
fails to be a proposition if it's false.

> > Remember how propositions can only have one value at a 
> > time?  EAs formalize
> > a way to allow propositions to change over time, the way 
> > they do in real
> > life, so that their value can change.  Furthermore, they do 
> > so in a very
> > simple language which looks like computer programming.

> Hmm.. well i don't really want my "propositions to change over time";

Yes you do!!  consider the proposition: "The screen is blank."  You
certainly want that to change from "true" to "false" during bootup.  This is
directly modellable by ASMs.

> on the other hand, in some situations (when the language is not needed
> for use in formal logic) i might not mind a symbol representing a
> proposition to represent different propositions over time, which is
> what their system does i suppose. it is good when a system 
> can formalize
> sentences like "I am hungry" the truthfullness of which changes over
> time; my view on the matter is that the sentence (meaning the symbol,
> the string of characters) does not represent a specific claim 
> (proposition)
> even under the English language; the time and the speaker must also be
> known before conventions supply an interpretation. That is, i don't
> think it is particularly meaningful to try to supply a partial
> interpretation to the sentence "I am hungry" when the speaker and
> time are not known; you might as well do so without evening knowing
> what language it was written in.

That's correct; the statement is not a proposition until it has context.
This is true for every potential proposition, of course.  Formal logic has
no solution for that -- formality doesn't make any statement true or false,
it only forces all parties to examine their contexts.

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.

> thanks for your response...

My pleasure!  I'm glad those classes were any good for someone... :)

> - iepos

-Billy