To recap it hypothesises that "true" self reference is impossible. In other words a statement cannot gain meaning from refering to itself.
The progress is the realisation that self-referential statements are true a priori. Because if they were false you get a NULL statement. The classic example of this is the liar paradox.
"This statement is true." Is apparently a solid statement and we gain apparent security from its claim of truth which only goes to support our security in the statement. However
"This statement is false." Leaves us with a problem. We assume it is true when we read it but find that it is telling us that it is false. So we reread it with this in mind and negate the truth value and find that infact it is true. It is neither true nor false but is undecided. It is meaningless. It is NULL.
The most recent argument regarding the SRH is that if the second statement is NULL then the first statement can't be true!
The reason? True is used in a dialectic with false. When I say that potassium burns with a crimson flame this is only false because I should have said it burns with a lilac flame. Likewise if I say that potassium burns with a lilac flame this is ONLY true because it illiminates all the other things I could have said like crimson etc.
However "This sentence is true." can only be said one way. Say it the other way and it is nonsense. It seems that rather than true and false at work here it is more Existence and NULL. Self-referential sentences are either existing (self supporting) or NULL (self-defeating). This is much like real systems then which are not true or false but working or not-working. Real things where there is feed back require that feedback to not blow the system apart otherwise the feedback ceases to be and such systems no longer exist.
Taking my old favourite the T = {T} the argument runs as follows.
With non-self-referential sets consider.
E = {a,b,c}
We can define a set A = {a,b}
Now A is meaningful because we could have defined A = {b,c} or anything else. Most importantly we could have assigned it to the complement of the set A = {a,b}' which means A = {c}. By choosing from amongst the possibilities in the context we are adding structure and meaning.
Now let us define a new set T = {T}. I'll jump to the chase we can't define T = {T}' - it is meaningless. Infact T is now bound intrinsically to the definition - it depends upon itself. If T fails then the whole definition fails. It is necessary for T must be valid for the definition to be valid. The definition is thus on very artificial legs - it can't fail. My argument is that it can't then suceed in a meaningful way either!
Lets turn to Russel and Whitehead quickly. The set of all sets that contain themselves. We would naturally want to add the set to itself. All appears fine.
But led by that appearance they end up considering the "set of all sets that do not contain themselves". It's the classic problem. If we do not contain the set itself then it belongs to itself, and if we do then it doesn't belong to itself. So it becomes meaningless. Where did the problem begin?
Well my suggestion is that the problem began with the construction of the concept of a "set that contained itself". Being dependent upon itself it can't fail. It becomes a priori, lacking content at best and worse - meaningless. Self-reference leads to errors - avoid.
Yet we apply self-reference all the time. "I think that ...". "I want that...". Now these could be simple statements about the state of our bodies and minds. But this is not what we mean is it? We mean that in some unjustifiable way these thoughts and wants come from nowhere and are justified simply in coming from us. That "I" want something is justification in itself of its value. The fact that we may have been persuaded by an advert or a friend is always over looked. That "I" want it, becomes a reason in itself. I begets I. But the problem as argued above is that such a situation is relying upon meaninglessness. For who really says that "I want what you want"? Only those who have escaped self-reference can say this.
Now I'm not saying that human existence isn't entirely free and every permutation is possible and escaping self-reference is a common event - I'm just refering to that particular state that some humans get stuck in which begins the process of evil. In Hindu mythology the devils all issued from Brahma when he mistook his role as the vessel of existence as the grander (an impossible) creator of all existence. How could something both be existing and create existence? Likewise when humans come to think that they are both judge, jury and executioner they fall into self-justification, self-reference and meaningless - this way lies evil.
On more theoretical territory the same problem plagues Western thought and academia- and via these the concept of economy used in the West and which is trying to be deployed globally. In self-referential terms their exists a point where human knowledge will reign supreme with theories that completely explain reality, where economic systems are tamed and everyone is rich and technology has risen to give humans every freedom they want. But such a dream rests upon something being Absolute - be it the imagined True Universal Reality or Ultimate Economic Goal or even something abstract like Perfect Human Happiness (TUR, UEG, PHH three nice new TLAs)- and that Absolute will either remain Mysteriously-God-Given (MGG) or will require its own explanation. This will either lead us into further investigations (ad infinitum) or will eventually end up completing the circle and explaining itself (my old Ouroboros). It is the Ouroboros that I'm defeating here: if it does end up explaining itself then we have explained nothing and our theory stands next to meaningless as its closest ally.
What humans don't like about the MGG aspect of the universe is that this is where we began our enquiries all those millenia ago - the Universe is an extraordinary place appealing to our understanding but also always suprising and holding hidden depths. More troubling it means that we won't be able to explain ourselves and gain freedom from our place in the world. The world birthed us and where the world goes so we go. A modern eco message in there. However not all is lost because realising this 100% dependence on our circumstances and our inability to escape this is actually the door to freedom that the sages have taught us. These teachings have often been handed down in organised groups called religions which suffer all the problems of status and membership discussed in the blog. But the teachings themselves are free.