An itsy bitsy reflective model.

All propositional logic can be redefined as "assumption logic", where the outcome of every expression may be a result of about the assumed truth values of the proposition, or may not be. This fits Soros' statements of how some things ("its raining outside") are unaffected by our beliefs, and others ("this is a revolutionary moment") are.

We define two boolean unary functions, tautology, Ta(x) and contradiction C(x) as follows:
Ta(T) = T
Ta(F) = T
C(T) = F
C(F) = F
These can be built out of existing boolean operators: Ta(x) = (x OR (NOT x)) C(x) = (x AND (NOT x))
We define two more unary functions, Identity I(x) and Paradox P(x) as follows:
I(T) = T
I(F) = F
P(T) = F
P(F) = T
Each of these also defined in terms of existing unary operators I(x) = (x), P(x) = (NOT x)
Now, we assign to any proposition who's value is known, one of these 4 unary operators, which operates on x. X is our assumption about whether a given proposition is true, we can't be neutral in this model.
When the value of a proposition is Ta or C, it operates exactly as a classical boolean propositional logic - it ignores our assumptions. But when the value of a proposition is I or P, its dependant on our assumptions. So all of traditional boolean propositional logic is a subset of a reflexive propositional logic, through Ta and C. But the reflexive propositional logic adds new possibilities.

Russell's paradox (from Principia Mathematica) is an example of a proposition with values of I or P. The proposition is:

The set of all sets which do not contain themselves, contains itself. 

Assume it is true.  Then the set of all sets which do not contain themselves, contains itself, so the value of the proposition must be false.
Assume its false. Then the set of all sets which do not contain themselves, does not contain itself. So the value of the proposition must actually be true.

So the truth value of this proposition is a function of assumptions about the proposition. therefore it cannot be given a value of True or False, aka Ta, or C, but rather must be given the value of P. It negates its assumptions, just like the "Am I fat" question in the post below.  Now consider the opposite proposition:

the set of all sets which contain themselves contains itself. 

The former proposition stands out famously in history, because neither assumption we make leads to a logically consistent system. But this proposition is a much sneakier beast, because either assumption you make leads to a logically consistent system, so you'll think you've discovered the truth when all you see is just a reflection of your own assumptions: If you assume the set of all sets which contain themselves contains itself, it works. If you assume it doesn't, it works. So its real logical value is I. Mathematics is full of statements who's truth value is I, meaning whatever you assume leads to a logically consistent system. The late discovery of non-Euclidian geometry shows it clearly, a different consistent system emerges when you assume on of the axioms of Euclidian geometry is false.

So what does all this show?
1) A propositional logic of assumption (reflexivity) exists which contains all of classical logic as a subset of what's possible.
2) That logic is expressible in terms of the classic logic, by changing just a few things. None of the quantum logic gates or anything far out need be applied. So its common sense stuff.

This example is itsy bitsy though, meant to show how cute it can be. Its not modelling markets or doing anything like the more ambitious folks are doing. Really expressive and powerful things I think have to be probabilistic, and have to get a little quantumy. But more on this later.

Soros and Meta-Relexivity.

I am now reading a formalization of George Soros' theory of Reflexivity. I looked it up because on reading its general description, I felt I already was so familiar with the general ideas that I would only learn something by how someone characterized them with math, and that paper didn't let me down. But the reality is there's a million different ways to model reflexivity, and what you get out of it depends on how you model it. The basic assertions of reflexivity apply to the theory of reflexivity itself.

But what are the basic ideas? Well, in short, reflexivity is how God tortures people who believe in open and honest societies. The two principles can be summed up as 1) People have limited knowledge as to what's going on. 2) People decide something is true, and what they decide on effects the outcome of the situation. So for instance if we all got confused and decided the dollar was worthless, it really would be.

Its a torture for those who believe in open and honest societies, because it eventually leads you to consider situations where an optimal outcome is created by a lie. In the brief amount of reading I just did, Soros identifies situations where asserting a thing causes it to become true. This has a quality of prophecy when it happens, and doesn't totally feel like a lie... The thing is true, because we say its true. However, in a logic system based on reflexivity, its just as likely for the opposite to happen. This is the case where one's thin wife is considering subscribing to the lifetime supply of free pastries, hesitating before she does so and looking in the mirror: "But am I a fat person?" she asks.

If you say no, she'll order the pastries and become a fat person. If you say yes, (if you lie) she won't and will remain thin. That's an awkward example but it shows the simplicity of the idea of the Splendide Mendax concept which reflexivity requires, and how lies can create positive outcomes. Thus it tortures people who believe in open societies without a lot of secrecy, like Soros, Assange, myself and many many others. Which is not to group those folks together too much: Soros embraced reflexivity and became rich. Julian "Mendax" Assange rejected it and became locked in an embassy. I just sat on my couch and drank beer, going not one way or the other, and ended up marginalized. People all over the world have gone in their own ways, done their own things, but the core principles remain, and are incredibly interesting.

This is supposed to be my math/reason blog though, so I want to return to what I said in the first paragraph about meta-reflexivity. Why did I say there are many ways to model reflexivity? Because the axioms of reflexivity require it. Reflexivity models a relationship between flawed perception of the world, and the world. Accurate and complete modelling of that relationship would require accurate and complete knowledge of the world, which the axioms say we lack.

But at the same time, its powerful as hell. Knowledge is an asset. One of the most amazing things in the world to me is probability theory, the way it turns knowledge of lack knowledge into an asset we can work with. It quantifies unknowns, and according to some of the stuff I see coming from quantum computing, it thus connects us with some of the most profound truths about the universe we live in. Soros was shaped as a philosopher (which he is, I hope the world will remember him as such, not a hedge fund manager) by the works of Popper, and what I would speculate is that he saw from Poppers work the information theoretic aspects of the scientific enterprise, not as a tool to find "absolute truth" (which it doesn't) but as a way to manage and minimize unknowns in a universe where nothing can be absolutely known, a universe where the crown jewel of knowledge must be awareness of our own lack of knowledge, the closest we can get to absolute truth.

For regular people like me who like thinking about reason and math, reflexivity thus spells out F-U-N. Its axiomatically assumed that no formalization of it can be perfect or complete, yet the promise of real power and insight is there as well for any formalization we might come up with. The world of reflexivity models is thus a creative space, where we can experience math not as a submission to given absolutes, but rather in terms of Einstein's definition, as the "poetry of pure reason", a truly creative endeavor meant to bring enlightenment and insight.  And that's good stuff.

Hello New Blog

I am starting this blog to have a place to muse about reason, math, that kind of thing.