(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!

(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.