Brief Update on Paradox and Quantum Information Logics
This is the part where I disclaim about how almost everyone will not actually read this, and almost no one will respond to it though someone will probably "like" it. It is my fault really because I am choosing to post this in the wrong context where there isn't really an audience for it. But frack the contexts and communities in which there is an audience for it; they aren't worth my time or effort as a general rule of thumb, and the peers that can appreciate what I am doing are too busy with their own work to actually try to read and comprehend what the fuck I am doing. So you get this borderline passive aggressive disclaimer. Thanks for reading this far.
Due to limitations on my ability to transcribe notations in this context, you get a plain text and informal description of my latest developments rather than the formal sequent syntax that no one here can probably even read anyway. That's the trade off I am almost always making with these posts. I can write what I am thinking in very precise formal language that almost no one on Earth can read--and I mean that given my experience with my peers at the 5th World Congress on Paraconsistency. Or I can write somewhat accurately in informal language by relying on you, the reader, to infer and interpret filling in the lexical gaps. I spent several years reading Zizzi's thesis on the formal language, and didn't understand jack fucking shit for the first few years of reading, so it is perfectly reasonable to expect that you're going to read my work for a few years without understanding it but fervently trying, right?
Got to imagine Sysiphus is happy.
Anyway. Enough with my depression.
Zizzi's logic, Lq, introduces a complex sequent and many valued, complex consequence relation. Lq's sequents come in three distinct kinds. Atomic assertions which are asserted with a complex truth value. Single context, single proposition composite assertions which are asserted with real truth values which are in general between 0 and 1, so they are called partial truth values and partially true assertions. And finally, contexts on left and right, asserted with truth value one xor zero; these are simply a special case of classical material conditional assertions which are truth preserving for assertion value 1 and false preserving for assertion value 0. This summarizes Zizzi's Lq quantum metalanguage of entailment; Lq then is an odd mix of fuzzy logic (partially true assertions) and classical logic (true assertions) with a special complex valued logic.
At the 5th World Congress on Paraconsistency, I attended a thirty minute presentation in which the presenters cited a research result showing that fuzzy paraconsistent logics which use classical, truth-preserving consequence relationships are logically explosive. They presented a sequent logic of their own design which allows tactical and strategic control of the explosive qualities of fuzzy paraconsistent logics by using degree-preserving consequence relationships instead. As Lq is a fuzzy paraconsistent logic, it follows that Zizzi's logic has to either violate or obey the cited theorems. It turns out that Lq is fuzzy, complex, and degree-preserving up to but not including reduction to classical sequents; at the point where logical explosion occurs for fuzzy paraconsistent logics, Lq collapses into classical logic. Zizzi briefly explores in her thesis the concept of generalizing the metadata constraint which imposes the reduction to classical sequents, but she gives no conclusions about it and only a guide to further research and development. By combination of her work and the work of Carles and Rodriguez, the logical extension of Lq and the generalization of the metadata constraint would replace the classical sequents of Zizzi's Lq with degree-preserving consequence relationships like those defined in Carles and Rodriguez's work. This functions for all real values including for transfinite real truth values. The monkey wrench is when you try to apply Carles and Rodriguezs' work to Zizzi's complex valued degrees which due to their complex numerical properties can only be partially ordered; Carles and Rodriguezs' work depends on the total order properties of the real numbers.
I realized recently that the complex numbers have a single element at the intersection of the real and imaginary numbers. The significance of this is that degree-preserving ordering relationships can be established from the imaginary numbers to themselves and from the real numbers to themselves, and they can jointly be ordered against their common element, 0. This allows an extension of Carles and Rodriguezs' work to imaginary degree-preserving consequence relationships. So we get numerous new metalogical assertion and denial sequents by generalizing Zizzi's Lq sequents and the metadata constraint.
When a sequent is asserted with a real valued degree, the sequent is a real assertion but an imaginary denial. When a sequent is asserted with a purely imaginary valued degree, the sequent is an imaginary assertion but a real denial. When a sequent is asserted with a complex degree with no parts equal to zero then the sequent is a complex assertion. When a sequent is asserted with a negative imaginary part, the sequent is imaginably untrue. When a sequent is asserted with a negative real part, the sequent is realizably untrue. When a sequent is asserted with either a negative imaginary part or a negative real part, the sequent is complexly untrue. Positive imaginary parts or real parts result in a sequent which is complexly true, and it needs to be noted that a given sequent can be both complexly untrue and complexly true; it should also be noted that a sequent which is complexly true can also be either an imaginary or real denial. Finally, a sequent which is asserted with degree 0 is a denial; the turnstile is reflected from facing right to facing left, and the degree of the denial is in general a non-zero complex value which mirrors the patterns of assertion, so there are for example imaginably untrue denials. A denial with a zero degree inverts likewise to become an assertion asserted with a non-zero complex degree.
The notions of true and untrue are distinctly different from notions of true and false. Degree value 0 on an assertion is roughly identified with falsehood. If you tell someone something is false and your assertion is sound and valid in classical reasoning then you've added absolutely nothing to their pool of knowledge. You've also not subtracted anything from their pool of knowledge either. Untruths are assertions which actually leave you with less information than you started with. You can think of this like malicious lies which are meant to deceive or misled; assertions meant to manipulate people's naivity or gullibility for example. I think of untruths like negative soup from the Phantom Tollboth; the more you eat, the hungrier you get. So untrue assertions are still assertions rather than identified exclusively with falsehood, negations, or denials.
If you've read to here and you have spent time reading up on the cited literature on sequent calculi, you might actually know something that almost no one else in the world knows assuming that this whole post isn't an elaborate untrue assertion of some kind. You're welcome.
