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The content of this blog is my personal opinion only. Although I am an employee - currently of Imagination Technologies's MIPS group, in the past of other companies such as Intellectual Ventures, Intel, AMD, Motorola, and Gould - I reveal this only so that the reader may account for any possible bias I may have towards my employer's products. The statements I make here in no way represent my employer's position, nor am I authorized to speak on behalf of my employer. In fact, this posting may not even represent my personal opinion, since occasionally I play devil's advocate.

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Saturday, July 09, 2011

Paradox and Consistency

I am fascinated by paradox. I am interested in the possibility of logical systems which are not consistent but where the inconsistency is somehow limited, so that it does not propagate and pollute the whole system. I am unhappy with systems where, given any inconsistency, any theorem can be proven.

I suspect that there may be much existing work in this area. Certainly I am aware of Russel's "set of all sets that do not contain themselves". And with Godel (although I probably do not understand Godel as well as I would like.). I should probably research the literature.

But since this is a hobby, I thought I might begin by writing down some simple observations. Not new - not to me, and probably well known.

But fun.

Consider simple systems of statements:

A single statement system:

S0: "This statement is false." The classic Liar's Paradox. The statement is neither false nor true: some say that it reflects an extra logic value, "paradox". Three valued logic.

A two statement system:

S1: "S2 is false"
S2: "S1 is false"

This system is self referential, but not necessarily paradoxical. It might be that S1 is true, in which case S2 is false, confirming that S1 is true. Or, vice versa. Thus, it is not contradictory. Either situation is consistent. But, from the outside, we don't know which.

What should we call this? Bistable? Metastable, if more than 2 stable states? Unknowable?

A three statement system:

S1: "S2 is false"
S2: "S3 is false"
S3: "S4 is false"

This is paradoxical, in the same way that the single statement Liar's Paradox is paradoxical.

Conjecture: I suspect that be a true paradox, then the feedback loop must have an odd number of stages. Whereas to be bistable or metastable, it must have an even number of stages.

Compare to inverter rings or storage bitcells with cross coupled inverters in computer electronics.

Q: can you construct a system that has interescting self referential rings of odd and even length?

2 comments:

Jake Vortex said...

I have been thinking about the sentence "This sentence is True". It could be True or False. It doesn't seem like anything else in the system could influence which truth value is more appropriate. Making it a sort of logical free variable.

Andy "Krazy" Glew said...

As I suspected, my thoughts are amaeurish, and betray more my lack of education:

http://plato.stanford.edu/entries/paradoxes-contemporary-logic

Nevertheless I am not discouraged. "Amateur" means "lover of" - and I do enjoy thinking of this. So what if I am more than 100 years behind the state of the art?