(Re)Search

I’ve been thinking about the relation, or possible identity, between paradox, contradiction, and duality. As such, I’ve been doing some research into a few concepts from logic. ^1.

I came across this take on the rule that anything follows from a contradiction. While the student’s exposition doesn’t explain it formally like on the wikipedia page, it’s nice to see someone who gets a sense of what follows from this principle: literaly anything is possible. And that’s what has always intrigued me since the first time this rule was shown to me, back in Logic I, some nine years ago. Perhaps others in my class were puzzled and incredulous like the author mentions in her piece, but me, hell, I was smiling because this explains everything.

Well, I bet I didn’t quite think exactly that at the time—I do know that in my head it entirely justified why magick works, and was directly linked to a cornerstone principle of occult thought: nothing is true, everything is possible—it took me a little longer to recognize the importance of this principle.

Another facet of logic I’ve been reacquainting with is proof by reductio ad absurdum. It’s apparent that there is a relationship between the structure of the reductio and the Principle of Explosion mentioned above: they both rely on contradiction, the structure of which can be expressed as A & ~A.

Of course, this brings in my old friend and sparring partner, the law of the excluded middle, which states that for anything, x, x either has the property P or it does not. In predicate logic, this is written Px v ~Px, which in the more basic symbolic logic is P v ~P, and we can simply substitute A for P and get A v ~A.

So we’ve got A & ~A, and A v ~A: like complements of One & Other, like a duality.

But then here’s a bit of self-referencing of sorts because I feel the structure of an understanding of duality goes something like ((A & ~A) & (A v ~A) ) & ((A & ~A) v (A v ~A) ). Or perhaps an even longer sentence, but adding more conjuncts and disjuncts simply seems to expand the point that, somehow, this ties together to create an infinitely rich tapestry.

Anyway, a paradox is basically the same thing as the case when the conjunction of A with its negation is true, so we could say that anything follows from a paradox. On the other hand, a reductio derives a truth so long as it discovers a contradiction in some set of premises: it proves the truth of the negation of some assumption which was used to derive the absurdity. So in both cases, we see how contradiction gives rise to some thing.

I guess where I’m trying to go with this, in part, is the idea that paradox and contradiction have an identical logical structure, and it is from this that everything else is created (derived). If anything follows from paradox, this includes self-consistent systems, i.e., an internally consistent set of sentences—a ‘true’ thing, say—can come from contradiction.



1. For potential readers, the symbols used in this entry are parsed as follows: & is ‘and’, v is ‘or’, and ~ is ‘not’. Hrmm, I ought to make something in the side bar about these things!

Duality Rough Sketch

I posted this on a different blog, but knew I'd eventually re-post it here.

Any given duality is fourfold:

1. It is two things in mutual exclusion: left or right, night or day, male or female, on or off, for some examples.

2. It is a single thing which forms a space for manifestation between empty polarities: left/right, night/day, male/female, on/off, for some examples.

3. It is three things when counting the polarities plus their (essential) relationship to One and Other: {left, right, and}, {night, day, and}, {male, female, and}, {on, off, and}, for some examples.

4. It is nothing without relation to some other duality.

Dualities are squares and crosses.

And of course these more specific claims need to be fleshed out. The above model of duality is merely the skeleton.

Notes

New category of entries called "Notes," intended to capture rough sketches of scattered thoughts for possible later reference. Difference from (Re)Search posts will be the obvious lack of links to outside sources: simply pouring out some contents from my mind.

So I find myself circling around relationships in terms of the number one and the number eleven. 1 & 11. One is, obviously, the thing of itself, and eleven being two thing in themselves standing together in a relationship. Could be a human to a tree, or a fish to the sea, or a collection of bikes to a set of roads. Basically anything collected as 1 unit in relation to another collection becomes 11.

* 11 as the tarot card Justice.
* 1 as the tarot card The Magician.

Thinking within the tarot, I also bring in 2, which is what 1 + 1 come together to form, and get reference to The High Priestess.

Here I arrive at a model of duality: two things together become one thing, this one thing defined by the set of relationships between the two things—the ‘&’ in One and Other. Note that One as singularity, Other as singularity, with reference to the idea that a singularity is empty, yields that the whole of the manifestation is in the ‘&’, and that One all by itself is equal to nothing, or 1 = 0.

So we can bring in The Fool to this little tarot oriented structure, an interpretation of duality.

At the singularity that is formed at the meeting point of the empty polarities, there is a ‘balanced’ or ‘just’ relation: it is when the singularities are non-empty—already being influenced by the tugs of Other polarities—that the ‘just’ relationship between One and Other can become ‘unjust’ or ‘unbalanced’.

* Just & Unjust form a duality, as do Balanced & Unbalanced.

In the Rider-Waite deck of the tarot:
* The High Priestess holds The Law half concealed: 1 / 2; 1 whole, 2 parts; 1 in relation to 2; 2 = 1 + 1; 1 / 1 + 1; 1 and 1 and 1.
* There are the 2 pillars, One white and the Other black.
* A fourfold relationship between the pillars, the Priestess, and the Law.
* Connection between Law and Justice.

Note that ‘numerology’, as displayed throughout this post, seems as a three-fold mental tool, kind of like acrobatics, sleight of hand, and a filing system for the mind. The meaningful relations it creates are both illusory and real (much like any other meaning created by any other means), and, like all interpretation, are ultimately in the ‘I’ of the beholder.

* I as singularity is empty, I as Jungian Self is empty.
* I = 1 = 0.
* I = Interpretation.
* Interpretation dependent upon One and Other.
* O & O = I.

I need to reformulate some of my expressions. It seems better to express Martin’s “M = D + P” as M <=> D & P, and the same would hold for my “E = I + O.” This would make more sense out of Energy = Input + Output, mentioned in a previous post, which I now realize doesn’t quite hold with the notion of the Law of Conservation of Energy, as written about in said post—will have to amend that.

* I & O can also be thought about in terms of One & Other.
* Other is not-self (not One); i.e., O & ~O is equivalent to One & Other.
* O & ~O <=> M, where M = E.

For potential readers:
* <=> is parsed as 'if and only if'.
* ~ is parsed as 'not'.

(Re)Search

I think I'm gonna' start a new category of posts titled "(Re)Search," which will be mostly a collection of links and maybe some commentary on the content or inspiration for posting said links. These will likely be related--perhaps sometimes very abstractly--to ideas explored in this blog.

Tonight I found myself thinking about figure/ground ordering in perception, as well as its connection to Gestalt psychology.

I also discovered that Gestalt therapy is somewhat different from, and not entirely connected to, Gestalt psychology.

'Gestalt' being defined as: a physical, biological, psychological, or symbolic configuration or pattern of elements so unified as a whole that its properties cannot be derived from a simple summation of its parts.

There are dualities involved in the creation of the whole from its parts (aside from "whole & parts," see also : mereology): figure/ground being one aspect, in this case, of perception.

Also noted in some of the above links is the duality of Self and Other. I particularly liked the following from Gestalt therapy:

"...self...is a comparison with 'other'. Without other there is no self, and how I experience other is inseparable from how I experience self,"

which also makes think of Jung. Indeed, part of what led me on this search was the duality that can be perceived between the consciousness and the collective unconsciousness--as a form of the figure/ground relationship--with respect to interpretation of experiences.

As an aside, this applet of a stereoscopic animated hypercube is a fascinating interplay of figure/ground in 3d! But the viewer requires those blue/red 3d glasses for full effect.