Suppose theres a statement LP such that:
1) LP = ~True(LP)
2) True(LP) = True(~True(LP))
3) True(LP) = ~True(LP)
There is no such statement LP. (QED)
The assumption that statements are either true or false assumes simple binary logic applies in all domains.
It seems that an infinite set of non-binary logics are possible.
Two simple trinaries are:
There is also the possibility of probabilistic logics, in which all things are expressed as probabilities, because nothing may be known with absolute certainty.
It seems that the game space of all possible logics is infinite.
Within any possible logic the space of all possible theorems seems infinite.
If one takes a mathematical expression of all possible postulates (as Wolfram has done) and starts exploring all possible theorems, then if one had started doing that when the universe formed, and one had all the matter of the universe arranged as computronium, then given our current understanding of the limits of computation implied by quantum mechanics, the current integer being explored would take two and a bit lines of an A4 page to print in an 11 point font.
Any infinity is so much larger than that.
An infinity of infinities (all possible theorems expressible in all possible logics) is so far beyond that, that whatever understanding we might as individuals posses, is a close approximation to total ignorance, and must always be so.
And none of that removes our personal responsibility to make the best approximations we can to whatever it is we seem to be and be within.
And the mathematics and logics we have available to us are certainly the best modelling tools we have for that job.
It seems that it simply doesn’t pay to get too over confident about the relationship of any particular logic or theorem within any particular logical or mathematical realm to whatever reality actually is.
Our brains seem to be heuristic machines selected by the process of differential survival in this reality we find ourselves in.
That seems to be the most accurate statement we can make about them at this time.
Not a useful question.
Some useful questions are:
What sort of computation would we expect to see from a brain that is working within very tight constraints of time and energy available to produce survivable outputs consistently?
What sort of heuristics and oracles would we expect to find in such a machine?
[followed by Yes it is possible to extend logic with truth values and it has been done! But as as far as I know there is nothing that can be said in multivalued logic that can not be said in binary logic.
So they do not ADD anything vital!]
One can model a trinary logic in a binary logic by creating a model in which a bit is set to represent the state of undecided.
But is having a bit being true in a model really an accurate representation of a reality of a state of being that is neither true nor false but is undecided?
It is the best model that can be made, within the binary world.
Within that binary world it is equivalent.
In the world of the trinary, it is nothing at all like the reality.
[followed by How can we tell
whether our world is a binary world or not?]
Is that a useful question?
What if our world is in some aspects fundamentally unknowable?
Isn’t a better question something like:
Which of the available logics gives the greatest probability of generating a survivable outcome given the time and computational constraints of our current context?
There will certainly be situations when time constraints demand simplicity.
We seem to be in a context where we have just enough time to look more deeply at the depths of complexity we seem to be facing, and to make survivable choices.
And it is difficult.
Most will not be able to break the shackles of the binary – it will be too uncomfortable at a subconscious level that directs our limited ability for attention.
[followed by different subthread]
“IF it is a state of the world THEN it is decided.”
That is a statement that can be true in a binary world.
If the world is not binary, then that statement need not be true.
You seem to have constructed a model of the world based on binary logic, and seem unable at present to imagine anything else (“Nor do I understand what you mean by BINARY WORLD? “).
I have explored many classes of non-binary and probabilistic logics, and have created world models in some of those that seem to be a better fit than any available in binary.
And I get that, within a binary logic world, such a statement has no meaning, because everything is, by definition, binary.
I do get that I am asking a lot of anyone to step outside of that box.
It is a profoundly uncomfortable experience, to lose all certainty.
It is amazing how addicted to certainty our neural networks become.