A proposition is basically a hypothesis. For example (to review a
classic syllogism in logic) a logician puts forward the proposition that
``Socrates is mortal''. This is a hypothesis (put forth as a
proposition). To prove (or disprove) this hypothetical statement to be
true (or false) we require an axiom: ``All men are mortal'' (unprovable
assertion), a bunch of unstated definitions (for mortality, men, and
being - basically to lay out a set-theoretic framework of categories)
and a premise: ``Socrates is a man'' (instead of a woman or a
razor-bearing space alien controlling the mind of the President). From
which we can conclude (using rules of inference for predicate logic)
that yes, ``Socrates is mortal'' is *true*.

By which we *really* mean that *if* in fact all men *are*
mortal - a thing we can certainly not prove or even imagine proving -
*and* if Socrates (whose genes are now lost beyond recall but who
could have at least conceivably *not* been a man in a variety of
interesting genetically accidental ways) *then* Socrates was, indeed
mortal. Even without all the conditionals put back in place, we all
feel somehow that this argument is really a bit silly. A scientist (as
opposed to logician) would say instead ``Are we talking about the same
guy? He's *dead*, for God's sake. Can't get much more mortal than
that.''

Now consider the following labelled set of statements.

- All statements must be true or false.
- Statement 3 is true.
- (Therefore) statement 2 is false(?)

There is clearly no problem with either statement 2 or statement 3 -
statement 2 is an assertion. It is the premise of the argument, not
really an axiom. If we try to build a truth table, we attempt to map
the truth of statement 2 into the truth or falsity of statement 3 to
arrive at our premise-conditional solution. Statement 3 is the
proposition, and hence is framed with a metaphorical question mark. It
*must* be either true or false (according to statement 1) and its
truth-value must be determined from statement 1 as a rule and statement
2 as a conditional premise.

Suppose statement 2 is *true*. Then statement 3 is true. It says
that statement 2 is *false*, which is a contradiction. Suppose
statement 2 is *false* then. Then statement 3 is false - *technically* it is merely not true, but *statement 1 then comes into
play and tells us it is false*. So statement 2 must be *true* which
is *again* a contradiction. Gentlefolk, we have a
paradox^{4.6}.

So what *is* the solution? There are two possibilities. One is to
say that the Law of Contradiction is *wrong* as it is framed and
reframe it (perhaps) so that any statement (or statement *set*,
since the ``all'' is a category, remember and can only be interpreted in
terms of sets and subsets) can be true or false or *null*.
Obviously the correct truth-value of statement 3 is then null. This
fixes the problem, but makes the application of our ``self-evident''
rules of inference difficult, which is a shame as they work so perfectly
*well* for arguments where statement 3 is ``It is a waste of time to
argue with two physicians''^{4.7}.

The other is to place restrictions or the permissible class of
statements; to modify set theory itself. This requires additional
axioms, and those axioms ultimately are very complex, as one can make a
*chain* of statements of arbitrary length with arbitrary back and
forward reference and at the *very end* of the chain insert the key
statement that closes the entire loop to make it logically reduce to one
of the many, many forms of closure that result in null. Note that I'm
making no effort at all to present all of axiomatic set theory here or
this *would* be a math text - there are a pretty wide range of
things people try to do to avoid the kinds of problems we're discussing
and I don't ``like'' most of them as they seem to be motivated by a
desire to keep set theory closed at all costs, which to me seems pretty
difficult to accomplish anyway when dealing with things like ``being''
and ``nonbeing''. Basically one has to add a bunch of axioms that
ultimately add up to ``all argument chains must be well-formed'' where
well formed means ``have a consistent truth table'' and where you have
axioms to deal with recursive generation of sets, infinite sets, and
more^{4.8}.

I personally prefer a *naive* set theory to which one can *add
axioms* to construct more complex set theories and have already
presented some (I hope good) reasons to throw out any insistence on the
Laws of Contradiction and Excluded Middle *inside* a closed set
theory as long as the logical language one wishes to establish on top of
it is going to reference concepts like objects that are not in the set
theory where in a closed theory sets of ``objects'' and ``not in the
theory'' are self-contradictory concepts. There are other good reasons
to lighten up even the Law of Contradiction in logical systems, or at
least generalize it further. I can easily imagine an *trinary*
computer based on a trit (-1,0,1) instead of a bit (0,1) where *all
logic* is done on the basis of 1 = true = not false, -1 = false = not
true, and 0 = true *and* false (or not true and not false, or
``maybe'', or ``unknown'').

Whoa, you say. Can you do that? I don't know^{4.9}...

Sure, we do it all the time in human rhetoric and judgement and quantum mechanics and even in computer science in Artificial intelligence systems. Dualism is one of our nasty Western Philosophy inheritances that just plain gets one into a lot of trouble. The real number line isn't dualistic or even state countable. One gets in real trouble with measure theory if one tries to consider (for example) the probability of hitting any given real number point with a random toss of a real number coin (a uniform deviate generator with infinite range and infinite precision, whatever that means) in a naive way.

Even in a *trinary* sort of extended logic, though, there will be
undefined/null loops. The problem of self-referentiality and null loops
and undefined operations is much deeper than mere duality - it arises
from trying to *conceive* of the duality of existence and
non-existence, where the latter *does not exist* so that statements
predicated upon ``something nonexistent'' are already self-contradictory
and bound to get you into trouble. It also arises (in computation and
trying to work out these paradoxes in your head) from trying to resolve
an internally contradictory execution series as if they are in some
sense temporally or operationally *ordered*.

What we should conclude from all of this is that formal logic really
should be simplified to be a kind of set theory with the null set used
instead of the empty set in the Laws of Thought, and the treatment of
the rules of inference as *axioms*. Logic reduced to definitions
and axioms, that is, where inference rules themselves are never *a
priori* assumed to be invariable and universally applicable in all
systems of logic (at least if one wants to avoid having part of your
paycheck withheld two months in a row to pay off some bill, yet another
pair o' docks, or if you are averse to taking two wiener-dogs out for a
walk)^{4.10}. Of
course the problem with *this* is that if you asked any given
mathematician or logician if they were axioms already, they might well
say yes (and *still mean that they were self-evident truth*).

Hmmm, at long last it appears to be time to look into this axiom thing. Time, in fact, to ask...