Revision of SRH (2024) - fuller definition

Now:

∀x ∣ P(x) ∨ ¬P(x)

is true because whatever P(x) decides ¬P(x) complements it.

Meanwhile:

∀x ∣ P(x) ∧ ¬P(x)

is false because if P(x) is true then by definition ¬P(x) is not true, and vice-versa.

And we would say that

∀x ∣ x ∈ U

Which is to say that every x belongs to a special set called the Universe of Discourse (U) which covers the boundaries, and context of what we are talking about.

If someone says : "what do you think about Harry?", you would need to get from them the U. "What are we talking about?" is Harry a dog, a celebrity, a member of the British Royal family, a train, a comet etc. Without establishing U nothing makes sense.

A bit more precise we could also point out that there is nothing outside U.

∀x ¬∃y ∣ x ∈ U ∧ y ∉ U 

Which is to say that while everything x is in U there is making sure that there are no x outside U.

But this is the subtle issue. In the conversation above once we define U as "royal family members" then we can say that there is no one in the current conversation who is not a Royal. Often what happens in a real conversation is a quick change of U when someone talks about someone else like Meghan Markle who was not Royal, became royal by marriage, but has become unpopular with Royals; or more complex when someone points out that David Beckham is no longer a friend of Prince Harry. Now U has expanded to include non Royals, but is still about Prince Harry. The moving of U is essential to meaning, and is the basis of a huge proportion of humour:

"What is brown a sticky?" The listener is scanning for U and given the usual meaning of "sticky" has limited the conversation to "sticky" things. How many brown sticky things are there? Not many: chocolate? poo? ready to find it funny. The answer (if anyone says anything else) is "a stick" completely changing the listeners U because in English adding a "y" to a noun makes an adjective. This change of "U" to something unexpected is a pleasant surprise, technically in information theory has "surprisal", and the delight in that is a huge chunk of humour especially when it is referring to things that are naughty and we are not supposed to talk about. We are all well familiar with U.

Indeed much of literature would be the same, using words from one U in another U. When John Steinbeck in "Of Mice and Men" describes Curly's wife as having hair "like sausages" our normative U of hair which includes things like "locks, curls, ringlets" gets unexpectedly swapped for greasy meaty food. At a stretch we can see a parallel and our prior U (to borrow from Baysian Inference) has a small overlap where we could see a bunch of hanging sausages like this:

But the beauty of language is how effortlessly we can overlap and switch contexts (U's).

So the point of this is to show how central the unseen U is to discourse. It is always hidden, requires intelligence, and mental activity to create and maintain a working U during a conversation and is the great marvel of modern AI that it can curate the U so successfully.

So there is a really important issue here looking at the trivial logic statements at the start. What are we quantifying over? Or to put another way how big is our U?

Now SRH is about the size of the Universe of Discourse, and specifically (following Godel) when self-reference is possible, there is no "complete" U. And since all systems of any power have self-reference you cannot have a fundamental Universe of Discourse, a definitive U. To put another way: we cannot know or conceive of Everything, or have a single conversation with Total scope. In more sociological terms there is no Ultimate Narrative. That is the death knell of naïve Right-Wing thinking that likes to posit a single Truth. It is the end of Judaism which is regularly criticised in this Blog for trying to maintain a single truth, race/ethnicity, relevance and God (with the caveat that the Jews at least recognise at some level that G-d is non doctrinal). It is the end of the European Enlightenment, in particular, which set out in the 17th century to finish the work of the Renaissance and establish a single model and understanding of the Universe, but after continual assaults during the 20th lies in tatters due to increasing moves to uncertainty (Boltzmann, Bohr, Godel especially) and the ultimate impact on Western Civilisation of Post-Modernism which has evolved into a present malaise and nihilism (not least because it suits current powers to "End History" and maintain themselves in power as the holders of Liberal Democratic Capitalism).

Now SRH proposes that because U cannot be defined by its own elements then the existence and definition of U can only be constructed within a greater U (call it U1) and so on. This is exactly what Cantor found when he tried to define the size of infinite sets and it is the essence of Diagonalisation. This very same feature is then the basis of all the paradoxes and Proofs by Contradiction from Self-Referential from Russel's Paradox to Godel's Paradox, to Tarski's Paradox, to Turing Halting Paradox. I'm adding the word "Paradox" where it is not usually applied because I want:

"Paradox" here to be identical with any argument which shows that the application of the rule to itself is a contradiction. And specifically because the Domain of Discourse (U) is insufficient.

Specifically:
(1) Russel: "Set of all sets that do not include themselves". This set does not belong within any U.
(2) Godel: "This statement is unprovable". Actual statement in U that it does not belong in any U.
(3) Tarski: "This statement is false". Truth/False cannot exist within any U.
(4) Turing: "the program P will loop if it halts, and halt if it loops." Halting cannot be decided within any U.

There are others. But the thing to note is they are all the same.

Now Godel notes that where Self-Reference is present then Incompleteness (or inconsistency) follows. But this is a new statement and formulation of Truth (and one that is probably self-referential too)! SO SRH was to generalise this and say that:

For any U where self-reference can be constructed on its elements, then there is a P(x) which contradicts the definition of U.

To put that another way if we ever try to define a U over elements, in such a way that enables self-reference then there will be elements that do not belong in U.

That is why it was called the crudely call the "Self-Reference Hypothesis" because it flagged an issue with self-reference. A way around thus would be to ban self-reference. This was the original plan: there is no self.

NOTE: first noted SRH in the context of knowledge systems knowing themselves. AIME was a pattern recognition system that was designed to work on its own output. What emerges is that such "self-referential" systems have fixed points around which Bifurcation and Chaos often exists. This is directly analogous to contradiction in logical systems. But because a dynamic system exists in "time" the "contradiction" of a static system looking for a fixed result, transforms into oscillation and non-determinism. It is all the same issue caused by "self-reference" either statically or dynamically. 

But there is a glaring issue here. SRH (as many others have noted) declares that self-reference is a potentially breaking issue, and becomes more dangerous as a system becomes more complex.

How do you define "Self-Reference?" Is there a function R(x) which takes definitions and decides if they allow self-reference?

Well if R(x) exists does it itself allow self-reference? This is simple.

R(R) is a tautology because it can apply to itself so is definitely self-referential. And the SRH flags this as an issue and so R() is incomplete.

R(R) = false then it is a contradiction already.

And even if we mitigate and define U to not allow the expression R(R) we are admitting already that R is incomplete.

so R() is a problem. But if there is no complete definition of R() then SRH fails as it depends upon a definition of Self-Reference.

However we thought of one way out of this (see next blog for summary of the below):

Let us define:
F(x) decides if x is a rule.
T(x, y) decides if rule x is true for y.

So we can say:

∀x | S(x) → ∃y ¬T(x,y)

Which says that if x is a statement then there exists a y such that x(y) is false.

Or "for every rule there is an exception". That is you can always find something that contradicts a rule.

We can apply SRH to this because as a rule itself it is clearly self-referential.

But if we say it is false which gives us:

∃x | S(x) ∧ ∀y T(x,y)

Which states that:

There exists a rule x for which everything is true. That is: there is a tautology.

HOWEVER before we apply SRH consider the statement:

∃y ¬T(x,y)

There must exist a y such that T(x,y) is false.

So the statement is actually saying that it itself must contradict something.

well if it contradicts itself then  there must be a tautology.

So while it explicitly says that all rules are incomplete, this implies that there must be at least one tautology!

SRH says that all statements with self-reference are incomplete.

And the statement that makes SRH incomplete is a tautology.

This is the only way to construct a statement that generates true U free from paradox.

This is the plan. But it is informal at this stage, and also there is no language capable of expressing the U of all U's call the Q.

Cantor decided it was open ended and without an expression. There is no limit to the expressions of set size. That is not to say that sets have no size limit, the very ability to define a set has no limit. You can always define the power set of the previous set and that be by Diagonalisation a new order of size.

Even the English language that Structuralism would argue was the limit of expression--in that what cannot be said does not exist--is incomplete most famously and simply through "This is false." The more sophisticated and larger a language's expressiveness the easier it is to break and show it is incomplete, not the harder.

Like Cantor do we need to admit that there is indeed "no limit". And I don't mean a simple "no limit" which would be to seemingly make an expression. "No limit" as in inexpressible.

And this is why SRH also got the names Hamlet Hypothesis, Plus-One Hypothesis or God Hypothesis. What we are trying to formally say is that whatever you think of is incomplete and can be shown to be so, with the exception of this rule itself which becomes its own contradiction.

Buddhism would say grasp nothing (not even that statement). Or perhaps better grasp only within a given U, but never try and grasp at a U as this simply requires a new U. Everything has its day, and when that day is over, so is it. Whatever we grasp is only ever a tenancy. Time will come when we must move on. Things are all impermanent. And it is tempting to grasp at that and say well that is "one thing we got" (to quote "Deep Blue Something") but SRH is all about not grasping. If you wish to grasp anything grasp that there is nothing we can grasp in all circumstances and to map to reality forever. Buddha calls this Anatta - non-self/no lasting essence. Indeed the Three Marks of Existence are all linked to SRH:

1. Anicca (impermanence),
2. Dukkha (commonly translated as "suffering" or "cause of suffering", "unsatisfactory", "unease"),[note 1]
3. Anattā (without a lasting essence).
   

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Another way to see this is to ask whether a statement has meaning without defining a Universe of Discourse (U).

Suppose we just made a statement that requires the listener to just include everything with no context.

But of course it has a context because I can just refer to it. Since college friends and me have always just called this Q.

Does Q include itself? And now we are into Russel Paradox immediately and trivially.