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.
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 paradox4.6.
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 more4.8.
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 know4.9...
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...