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NancyLebovitz comments on Reflection in Probabilistic Logic - Less Wrong
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Another dilettante question: Is there a mathematics of statements with truth values that change? I'm imagining "This statement is false" getting filed under "2-part cycle".
Yes. There is an overview of revision theories of truth (and many other approaches to defining truth) at the SEP.
I've never heard of it but my intuition says it doesn't sound very promising - any correspondence definition of "truth" that changes over time should reduce to a timeless time-indexed truth predicate and I don't see how that would help much, unless the truth predicate couldn't talk about the index.
The truth predicate could talk about the past (but not the present), but I believe that reduces down to other things that have been explored quite thoroughly (iirc, an idea of a hierarchy of math with higher levels being able to talk about lower levels but not vice versa was around for Bertrand Russel to endorse). And at that point "time" is just a label for what level of explanatory power you are talking about.
(This reminds me of when I short-circuited a NOT gate, curious to see what would happen and suspecting it might oscillate between the two values. Actually, it stabilized to a voltage somewhere between “0” and “1” and stayed there.)
You could look up George Spencer Brown's "Laws of Form". It's...odd. (Poking this in through a phone otherwise would go into more detail.)
I read it long ago, but didn't understand the parts that looked like circuits. I should probably give it another try.
If you posit that: "This statement is false" (i.e. the fixpoint μ x. ¬x ) oscillates or "switches" between TRUE and FALSE, then this is pretty much how digital logic works in electronics, because the NOT operation is implemented as a logic gate with a finite delay.