Consider the following cute little set of carefully numbered (but self-referential) statements. We will skip the axioms and definitions required to specify ``English'' as the language of discourse and presume that we all recognize a ``statement'' to be a well-formed English sentence and so on. Here is a bit of fun we can have with the Laws of Thought and Gödel:
This is a set of statements that frames an argument that is obviously understandable both in human language and in logic. They are reasonable, whatever that means. Every term in the sentences used has a well understood meaning, the sentences are well-formed grammatical constructs, and the logic used is impeccable. We have used the basic rules of formal logic, in the context of a well-defined system of symbolic logic, to contradict themselves with a clear demonstration of the contradiction!
However, it is extremely important to make it very, very clear just what we have discovered and illuminated with this example, because even if classical logic based on the Laws of Thought is inconsistent in any nontrivial language (one capable of formulating statements such as statement number one) - which includes, as Gödel noted, all formal systems of reason capable of formulating arithmetic, which incidentally includes axiomatic set theory via a mapping between the power set construction and the natural numbers - those laws are often empirically useful, seem intuitively true, and we'd like to understand why and how they end up being useless and untrue some of the time if only to be able to put a well-defined fence around the hole yawning in the fabric of reason, threatening to flush all of mathematics, all of logic, all of reason itself! Has reason proven unreasonable, has the weight of symbolic argument created a black hole into which all reasoned discourse must slide?
The answer, perhaps unsurprisingly is ``yes and no''! This argument shows that symbolic reasoning does have such a black hole at its center, waiting to trap the unwary. The name we have carefully given that hole is Mu. We see that in order to build a consistent system of symbolic reason, we must extend the Laws of Thought to acknowledge , that it is not enough to have a simple boolean dichotomy of true exclusive or false for all objects and constructs in any nontrivial system. We have also shown that the rule of material implication cannot be permitted to work the way we have blithely used it without further restrictions because once one demonstrates a statement where A implies not A and not A implies A (so that the statement is , undecidable), one can follow the usual method used to show that admitting a contradiction into a theory allows one to prove the truth of any statement to show that admitting an undecidable statement into a theory similarly makes all statements undecidable. This is a point that seems to have been missed - uncertainty is just as contagious as contradiction in any logical symbolic theory that contains modus ponens and modus tallens as valid algebraic rules for determining contingent ``truth''.
Take any statement, no matter how outrageous, that we wish to prove: The moon is made of green cheese. Invert it: The moon is not made of green cheese. Now we add statement one above: ``This statement is false''. Now it is obviously true that if ``This statement is false'' is true, then it is also true that if the moon is not made of green cheese then ``This statement is false'' is true. The fact that it would still be true of the moon were made of green cheese is irrelevant to the process of formal logic. However, we can transform this logical statement into if ``This statement is false'' is false, then the moon is made of green cheese.
In ordinary arguments this is no problem. As I type this, today is Thursday. It is certainly true that if I were the supreme ruler of the Universe, it would be Thursday because it is, most definitely, Thursday. We don't care in the slightest that this means that if it were, say, really Monday that I logically could not be the supreme ruler of the Universe because it isn't Monday, it is Thursday. We only get into trouble and arrive at a (note well) true conclusion - that I am not supreme ruler of the Universe - if I lied and the premises of my original argument were false, or if they were contradictory - today is Thursday and it is not Thursday, it is Monday, in which case I am supreme ruler of the Universe too and equally well its contradiction, a mere flyspeck on a backwater of a planet in a tiny corner of the cosmos. We hate it when the latter happens so we require logical systems to admit only statements that are never both true and false they have to be one or the other.
In our perfectly sensible argument, however, we are in a different kind of trouble. Which is it? Is the moon made of green cheese or isn't it? If we look at statement one, we find that if we assume that it is true, we conclude that it is false. We therefore conclude that the moon is made of green cheese. If we assume that it is false, we conclude that it is true. The moon is not made of green cheese. If we assume that it is false, prove that it must then be true, and conclude that it must therefore be false (iterating one more level in the process of decision) then the moon is made of green cheese again. Unless we try the opposite, or iterate one more level, in which case we conclude that the moon is not made of green cheese! Ad nauseam, if not ad infinitum.
Neither. Both. The existence of one undecideable statement in any system of formal logic makes all statements in that system undecideable. We can use the very power of formal logic like a poison, working backwards using nothing but permitted algebraic steps to contaminate every single axiom upon which any theorem is based, making them as undecideable as that undecideable statement. If, when confronted with any question in a system of logic we ever answer ``I don't know'' or ``Mu'', or whack the asker on the head with a banana and run away, giggling - all the rest of the answers to all of the other connectable questions becomes suspect. The basis of formal reason is formally unreasonable.
Let's see if we can rescue it, if only because it does seem to work so well, most of the time. To fix things up, we are going to have to clear our head of cobwebs, shake off the seductive allure of logic, and fall back on a higher level view of what we're trying to accomplish. The easiest way to see where we are going is to look carefully at the difference between the argument that used ``Today is Thursday'' and the argument that used ``This statement is false''. The former refers to something that is objectively true in an external set Universe. For that matter, so is the statement ``The moon is (not) made of green cheese'', in English! Who the hell cares about logical games like the ones above - they are bullshit. There is only one correct answer to the question ``What is the moon made of?'' That answer is I do not know!
At least not ``know'' as in ``can prove to be undubitable truth using nothing but pure reason in any system of logic''. We can (as we will see) find ways of providing highly plausible answers to what the moon is made of, and those ways will in no way depend on whether today is Thursday, or whether ``This statement is false'' is formally undecideable. The answer to the question of what the moon is made of is formally undecidable, and so are all other well-formed questions any human has ever formed, because formal logic, even if used extremely carefully and avoiding the truth-sucking pit of undecidability, leads to contingent, not absolute truth as the argument above formally demonstrates.
No process of logic - and pay careful attention here, because this is very important - no process of pure logic will answer this or any other question. Not even if I go to a ``higher order'' logic and fix up the silly problems with self-referential undecidable statements, restrict my domains suitably, introduce the -pit and dump all Gödellian knots into it so they can't sully the landscape and purity of our reasoning process. This was argued convincingly by David Hume two hundred and fifty years ago and philosophers, logicians, and mathematicians have been in a state of acute denial ever since. The question of what the moon is made cannot be definitely, or certainly answered in our imaginations, and all symbolic reasoning occurs in the imagination.
The moon is what it is! Which may be nothing at all like what we imagine it to be, nothing like what it appears to be. The moon may not have the objective reality we presume for it at all, let alone have specific properties such as ``being made of green cheese''. Or not. It may be the illusion of a moon, a moon that exists only in our sensory perceptions. Or it may have perfect existential external reality and be made of what it is made of, quite independent of what we ever think it is made of.
What matters, then, is not so much the validity of the system of reasoning we use to ``prove'' things about the moon, as the correspondance between the results of that reasoning and our experience of the moon. If we take the system of reason used too seriously, it contains a black hole and will swallow itself. If we admit that the black hole is there and simply keep away from it, we'll find that it is quite useful and results in a remarkably good and consistent correspondance, a compelling correspondance, but not a logically necessary correspondance, between our imaginations and our sensory experience of what may well be an objectively real external Universe.
The deepest foundation of any system of symbolic reasoning, then, must be basically ``black hole repellent''. is there, waiting, and we have to wrap our minds quite literally around it and accept the contingent and uncertain nature of all knowledge, including knowledge based on formal reason and mathematics but especially inductive or deductive knowledge of a presumed existential reality. We must begin with assumptions, with axioms, and if we are not careful to avoid overstressing the lack of certainty of the axioms, those axioms will cause our system of knowledge to ``self destruct'' as we come up with predicates that do not close within the system, that must map into (carrying everything else along with it).
What we should conclude from all of this is that formal logic is rather insubstantial, existing in our imaginations, and yet there is something that is non-null, an existential set Universe that at least contains our sensory experiencing including our imaginations. The latter can be likened to a big (really really big) messy (really really messy) room, in which our ``selves'' wander around, trying to pack it all away in neat little boxes we fold out of the trash paper we find on the floor, finding unsurprisingly that some of our boxes have holes through which the contents fall, and that no matter where we wander, the room itself seems to just go on and on so that we have a very hard time building a box large enough to contain the room itself: no matter how large a box we make up, we can easily find an object that won't fit into it. Remember, our imaginations themselves are at least a subset of the contents of the room, we can in one breath imagine a box large enough to hold everything, and then imagine an object just a bit too big to fit that box, or write a sentence that claims that it is false that is true, therefore false, therefore true, therefore neither, therefore both.
So let's be sensible, and focus on the process of making boxes, of organizing whatever we can see, without worrying so much about whether the boxes we build are perfect boxes, without holes, always large enough to hold all possible contents. Indeed, let's fall back on existential set theory - the set theory appropriate to a real objective existential Universal set (whether that set is open, infinite, finite, closed or most likely of all - unprovably any of the above) and stick with only the Law of Identity and the power set construction applied to the Universal set. The latter specifies the set of all possible sets, all possible sets of sets, etc. of this actual set Universe. These ``sets'' are, whether or not we imagine them - the partitionings we might try to predicate or otherwise construct are imaginary, but the set objects are real. The only processes that are necessary for building the various orders of power sets are iteration and permutation of set objects, and we should never confuse our fanciful attempts to pack subsets in the power set away into neat imaginary categories, boxes constructed out of spiderwebs and fairy dust with the underlying reality where the boxes themselves are just high level manifestations of structure that are at best a part of the reality.
Or, as a General Semantician might say, the map is not the territory. But even this is misleading in an attempt to metaphorically represent knowledge, cognition, semantics, epistemology versus the world. The map may or may not be the territory it represents (reality can be thought of as a perfect map of itself), but maps of disjoint territory are, in fact territory in their own right within the global territory that contains the maps and the territory they are mapping. The essential existential condition of a Universal set is that it is the only map of itself, it has no legend, and that all lesser maps must begin with a legend to establish symbolic correspondances between information-compressing constructs in the territory of the map with a (usually much larger and information rich) territory that is outside of and disjoint from the map. Any such map (if it is honest) will always contain Terra Incognita - unknown territory - along with an intrinsic inability to be a perfect map of the global territory including itself, no matter how carefully the legend is constructed.
What is this ``legend'', the code that specifies what the lines on the map are supposed to mean, and how to we establish and test that meaning, decide if the map we have built is a good map or a bad one? We need just two things: Axioms to specify an imperfect and mutable system of reason and enough common sense to come in out of the rain, or more relevantly, to not take our imperfect system too seriously, to accept its limitations, and to keep away from the black holes that inevitably appear when we try to make a self-portrait, a map that is at least partly a map of itself. Reason works pretty well when it refers to something else, but the minute you apply reason to reason or expect it to produce something besides contingent truth (on a good day, with a tail wind), you discover that it is unreasonable and that its conclusions can always be doubted.
Common sense is, we must hope, common. If you don't have it, I'm unlikely to be able to help you discover it (although later I'll certainly try to quantify it). Axioms, however, are not common. Or are they? Axioms could be as common as dirt (and the basis of common sense itself) and most people would never know. Hmmm, at long last it appears to be time to look into this ``axiom'' thing. Time, in fact, to ask...
What's an Axiom?