George Boole’s “Laws of Thought” have been extremely useful in many disciplines, but I contend that they are nevertheless incomplete. In their dominance over most of Logic and Philosophy, they have caused a conceptual ‘blind spot’ in the many fields of research which use or emanate from such reasoning – including the Foundations of Mathematics. It’s time we set the records straight. I propose one law, which for now I call “The Law of Existence”, and show how the Laws of Thought emerge naturally from its consequences.
Perish the Thought!
A “contradiction” in Logic is a situation where one line of reasoning affirms a statement, but another line of reasoning denies that same statement (or affirms its opposite). This is called an “inconsistency” and for centuries it been seen by ‘conventional thinking’ (i.e. Western analytical logic and philosophy, but even the mundane and day-to-day thinking of most Westerners) as being the absolute pitfall of thought. If your theorem or reasoning contradicts itself, then it is unsound and worthy of rejection outright.
This somewhat extreme reaction is actually an inevitable consequence of the way Logic works. Statements follow one another, in an intimate chain of dependence. To put it very simplistically: the truthfulness (what logicians call the ‘truth value’) of one statement depends on the truthfulness of the previous one. And so if there is one statement that is both True and False, then that failure propagates backwards, through the whole chain of arguments, like the burning fuse of a stick of dynamite right to the final explosion of the theorem as a whole.
In fact, this problem of inconsistency is even called the Principle of Explosion – where if your system of logic allows even so much as a single statement to be proven both True and False, then ‘anything goes’ and thus your whole system of reasoning ‘explodes’ into every statement being provable to be True! This state, where all statements can be proved to be True is called Triviality (from the Latin “tri” + “via” – three roads – like at a crossroads where you can go in any direction) leads to utter absurdity or ‘nonsense’. So such extreme rejection of inconsistency, I think you’ll sympathize, is quite understandable. It is, however, wrong.
Reality is stranger than fiction
Indeed, were inconsistency not a Fact of Reality, we would have a very static universe – devoid of anything that could even slightly oscillate, vacillate, or change in any way. “Good-bye Music!”
“Good-bye crashing waves and surfing!” “Good-bye Light!” Oh, and “Good-bye Matter!”.
OK, so maybe I’m being a tad melodramatic, but you get my point: Reality is not ‘ideal’ nor perfectly clear-cut. It is messy, and changing all the time! Even our thoughts are changing and shifting. It would be a stroke of hypocrisy and hubris to call these ideas “Laws of Thought” and not allow for contradiction or inconsistency. But I don’t think George Boole even knew what he was missing (that ‘blind spot’ I mentioned earlier) – again, unto themselves, they do seem very down-to-earth, very ‘reasonable’ laws – so why not call them such? I don’t blame him. I do, however, blame us as a species for not having the courage to face the music – not having the grit to just deal with it. Inconsistency is here to stay – there’s no going around it and there’s no avoiding it.
How can I be so sure? Kurt Gödel’s Incompleteness Theorems aside, it will become quite obvious in just a moment. Let us look at the first Law of Thought:
The Law of Identity
“That which is, is.”
Duh! Right? Quite. But there is an entire ‘other half’ of that law which was forgotten or deliberately ignored:
“That which is, is not that which it is not.”
Another Duh! Right?
To draw directly (with a minor correction) from Wikipedia’s entry for the Law of Identity: “Each thing is the same as itself and different from another.” The problem with the present day usage of the Law of Identity is that everything after the “and” in that sentence above has been forgotten. There is something incredibly important that we’ve just dropped: the implicit and necessary mutual existence of another thing from which to be different!
But even more flagrant is the inherent contradiction that a Thing which is, simultaneously is not! That is to say a Thing cannot “be” without also “not being” something else. So all extant Things both are and are not. It is a necessary condition of existence!
Think of it this way: I am a human. But also, simultaneously, I am not a fish (nor a cat, nor a door, etc.). These are two statements which are both True about me. By being what is human, I am also necessarily not (negative) what isn’t human.
So my point: the Law of Identity, as it is used today, is expressed solely from a cataphatic perspective; the apophatic perspective having been completely ignored or forgotten. You will immediately see that the Law of the Excluded Middle (A Thing only ever is or is not) and the Law of Contradiction (Nothing is and is not) unfortunately do not hold, given the first Law of Identity – once it is stated correctly.
One Law to Rule them All
Apophatic Thinking (identifying a Thing by what it is not) is sufficient because one can attain Cataphatic Thinking (identifying a Thing by what it is) by ‘Apophatically apophasizing’ (identifying a Thing by what it is not not). That is the nature of the negation – it is an involutory function (a function which is its own inverse, such that the second application cancels both).
Let’s try “All Things are not.”
That leaves an enormous domain of possibility open: A thing is not… another Thing, all Things, the Thing itself? What is the Thing itself? How can it be not itself?
What about “Of All Things, at least one is not.” (For all x, there exists at least one which is not)
Now that’s an interesting way to put a spin on Things. We’ve turned it inside-out and excluded at least one – thus defined the existence of at least one Thing.
Yes, you deserve a break. More coming soon!
Thanks and heartfelt regards,