paradoxes and deductive theorem again
iepos@tunes.org
iepos@tunes.org
Fri, 10 Sep 1999 15:06:13 -0700 (PDT)
> 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".
so, your main criterion is that it be true or false. this seems sort
of reasonable; of course, intuitionnists will argue that there are
some things that clearly have the deductive theorem even though
we can't show that they have excluded middle (are true or false).
i'm not sure about this though; in fact, i'm not convinced that
this restriction on the deductive theorem is strong enough; it isn't
really clear that the deductive theorem follows from excluded middle,
and I wonder if there are some things that clearly have excluded
middle, but that lead to problems when deductive theorem is assumed.
> > 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. 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.
> > [...]
> > 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
...". 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.
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?
>
> > 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 ...
> 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.
> 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...
>
> > Oh, and by the way, that computing system thing I was working on
> > earlier is still coming, at least in my head. I started coding in
> > C and almost went insane and have started working in Java (for lack
> > of anything better) since I don't particularly care about speed.
>
> Python is far better than Java for slapdash coding. Once you get something
> solid, you can either move parts into Java using JPython or into C using
> CPython.
Hmm... i've never learned python. Maybe i should then...
> You may (i.e. WILL) want to look at the Evolving Algebras/Abstract State
> Machines home page, at http://www.eecs.umich.edu/gasm/. Insanely good
> stuff.
Hmm... i will take a look sometime then...
>
> 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";
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.
> There are interpreters for it, and even GUI design tools.
Well... i will take a look. it sounds interesting.
> > - iepos
> -Billy
thanks for your response...
- iepos