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.