This Xmas was gezellig and raccolto.
I spent Christmas mostly at home. It was a quality time best described as “gezellig”. That would mean Gemütlichkeit I guess.
I was physically relaxed but my mind kept working. So it was a fruitful time too.
Since my 1st thesis on “indicative conditionals” had already reached its perfection and maturity, what remained troubling me in the last two weeks was my perfectionistic wish to simplify the formalism in my 2nd thesis on “causal conditionals”. Maybe thanks to all the candles they lit in my living room, or maybe for my Mass time prayers, Santa Claus brought me good luck in breaking through:
Last night, 25th of December, just before I went to a party in which I did not stay long, I recalled to myself that my previous idea of formulating the “production ranges” essentially came from my subconscious attempt to make the lower-bounds and the upper-bounds for a narrowing function’s images stay monotone with the preimages so that a certain “ordering” can be well captured in the model, then why not directly use the general order-preserving functions to denote that ! (It is funny how I “rediscover” what I have arrived at before. My thinking had always been through “pictures” and that was why I didn’t translate them well into algebraic language, until now.) Then I again noticed that my previous definition of “causally closed systems” etc was essentially a fixed-point condition, and this recall hinted me to reapply my “saturation” definition to images of these narrowing functions and it worked! Following these two new clues I discovered that the causal structures can be equivalently defined with a pair of conditions - a universal condition that every possible narrowing function has a saturated image and is lower-monotone, and an existential condition that there is one narrowing function that is upper-monotone.
I came back home by midnight and wrote them down, I worked till this morning, 26th of December. The idea was confirmed correct but the next thing was wrestling with the relationship between the “semilattice” condition pair and the order-preserving condition pair, and my question came twofolded at once:
- Are they correspondingly compatible?
- Is the former included in or reducible to the latter?
At first I thought yes and yes, when I woke up at noon and took my shower.
Then there were great movies and more cozy times and Thai food.
Then I resumed working and came to my correct answers: no and no.
(Thank God, 3 months after my formulation of “determination and contribution”, I gave myself this time to put all pictures together, so that I avoided publishing the mistake I had made. This correction is crucial to the coherence of the entire theory I have for causation, especially with the “Asymmetric Potentiality” condition I had formulated as a definitional source for the Contribution Theorem.)
Now here is the final assertion: the narrowing functions in a causal structure are not necessarily intersection-semilattices at all! So all I need to do is omitting that condition in my general definition. And this pair of semilattice conditions (therefore a lattice condition) naturally coincide with (and are therefore reduced to) the saturated order-preserving condition in those “informational” narrowing functions as special case anyway!
I typed in the main formulae, and leave my remaining writing to Sunday, 27th. Time to wrap it up!
I expect to put both theses online for everyone in early January, maybe as Technical Reports if the office approves.
Buon Natale e Felice Anno Nuovo!
