Logic

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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:

- Statement number one is false. (Self-Referential Statement)
- No statement can be true
*and*false. (Law of Contradiction) - All statements must be true exclusive-
*or*false. (Law of Excluded Middle) - Statement number one is neither true nor false. (If statement
number one is true, it is false, but if it is false, it is true.
It is not a member of the set of ``true statements in English''
*or its complement*, the set of ``false statements in English''.) - The Law of the Excluded Middle is therefore
*false*in the domain of ``statements in English''. It is*definitely not true*that all statements must be true*exclusive-or*false, as we have constructed one that is not. - Statement number one is arguably both true
*and*false by*implication*-*if*it is true*then*it is false,*if*it is false*then*it is true, suggesting that it is somehow*both*in a symmetric way - and therefore*the Law of Contradiction is false as well!*Or at least it*would*be false if it weren't for the fact that we just threw out the Law of Excluded Middle - it is no longer sufficient to show that it is not*true*in order to be able to conclude that it is*false*.

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 connect*able* 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?*

What's an Axiom?

Logic

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