#paraconsistency

[tension-squint]

There exists a world above. Look up. Through the pulsing palm fronds. Vultures guided by ancient thermals to take a circular path. See through their eyes. Sky. Yearning. Calm. The sureness of flesh. Sweet decay. It's all there, in your eyes. Look down with them. A crop circle arises from hostile patterns of concrete. Crop rustling out a spell, "sublimitas et humilitas," and there goes the wind, constituting itself as translucent architecture spiraling through corridors of field, pure field. You recognize this yearning-- is it a book or a face you recall? Or is it every book? Every face? The appearance of a nicotine craving, an undulating sphere, its subtle migration between stomach and heart. Look closer. It's the memory of a lover's touch. It's the twist of a stranger's lips felt as your own. Energy, pure energy. Flow of blood, presence of ghosts. Rendezvous: past, present, & what will we make of our future? Uncharted masses surrounded by sea. Return now to your body. Open arms and an always breaking heart. Agape. Leaking. How their voice shined when they said, "I love God." Capital G, you heard in the voice you wished in that very moment to kiss. How voice everywhere will soon come to shine. The rippling red of the buzzard's throat. Roadside corpse. Stench of transmigration. The slash pine hanging over it all, needles golden here so early in the day. How it all shines.

[sterwood]

We can do a purely logical exercise concerning the presidential pools, in France this year, in the States next year. Are these pools events? Is there in them a political truth? What is the logical context? Classical? Certainly not. Nobody thinks that with Segolene Royal or Hillary Clinton, the poor African or Mexican worker will become a new political hero. So the choice is between intuitionistic or paraconsistent. My own conviction is that pools, today, in our world, are always of the third type: false event, paraconsistency. Seeming like a contradiction, but being like an identity. But everybody knows that I am an ultraleftist. So after all, you can be courageous intuitionistic men and women.

Badiou, “The Three Negations”

I really need to understand Badiou’s relationship to paraconsistent and dialethic logics, because they would seem to be something that would be a major problem for Badiou, if they’re not addressed well enough.

This would also be a problem for Meillassoux, who holds on strongly to the principle of non-contradiction.

[studyinglogic]

A (non-technical) conversation with a twelve-year old on dialetheism

Her: What were you doing this week?

Me: I was reading up on dialetheism.

Her: What's that?

Me: It's a very uncommon view in philosophy: the idea that there that there are true contradictions.

Her: What's so surprising about that?

Me: Well...can you think of any statements like that?

[Two minutes pass]

Her: ...No.

Me: Well, take this statement - "This sentence is false". So if it's false, then it's false that it's false - so it's true. And if it's true, then it's true that it's false - so it's false. In other words, it's a true contradiction.

Her: Okay. So there are these true contradictions.

At this point, I was quite amazed. Most philosophers (like John Burgess) take dialetheism to be a completely implausible view, but here was a teenager calmly accepting it. And I thought to myself - this is great! Here's someone uncorrupted from learning classical logic, who just accepts it. Then -

Her: Are there any contradictions we can see, though?

Me: Hmmmm...here's an example from two philosophers named Beall and Colyvan. Imagine I'm standing in the threshold of a room. Am I in the room, or not in the room? Well, that might be a dialetheia, since I'm both in and and not in the room.

[Beat]

Her: But...! Obviously you're half in and half out!

So, in other words - she accepted dialetheias like the Liar sentence, but not empirical dialetheias. (Just like Graham Priest, so she's in good company.) But it was an interesting test of philosophical intuition, nonetheless - perhaps dialetheism only seems weird after a good dose of classical logic.

[idlnmclean]

Contradiction Tolerance

In contradiction intolerant methods, the existence of a contradiction in the conclusions of a proof indicates a false premise or sequent in the proof.

For example, (x - 3)(x + 5) &gt (x - 2)(x + 4). Assume the inequality has at least one end point. We find the end points algebraically by isolating 0 on one side of the inequality.

0 &gt (x - 2)(x + 4) - (x - 3)(x + 5)

0 &gt (x^2 + 2x - 8) - (x^2 + 2x - 15)

0 &gt 7 or 0 &lt -7

Either way we try it, we get a statement which can not be true. Thus, our assumption in this example concludes in a contradiction; proof by contradiction intolerance allows us to erase our premises and conclude the negation; therefore, no end point exists. Testing the inequality for any value of x will produce a valid solution, so the domain of the inequality is from negative infinity to positive infinity.

In contradiction tolerant methods, the existence of a contradiction in the conclusions can reduce in consistent contexts to proof by contradiction intolerance, but in general for paraconsistent contexts, the existence of a a contradiction does not allow the erasure of the premises; a contradiction is an indicator that no end point exists, but we could not conclude that no end point can be defined; we can probably define a paraconsistent end point. Likewise, if we assumed no end points existed, and using the above methods demonstrated in the solution of the inequality, we found valid end points for a different inequality (e.g. |x| &gt= 3 end points are at 3 and -3) then we would prove a contradiction; in this particular case with a consistent algebraic context, I would think it make sense that NO paraconsistent points are definable as the system described has no place for paraconsistency to arise within it; it is closed, consistently well-defined, and decidable.

This is covered by the Creative Commons Sharealike 3.0 unported license November 6th 2013 by Ian D.L.N. Mclean.

[idlnmclean]

Free space unlike outer space has no properties by classical consistent reasoning. Outer space is a bound space which is approximately empty and devoid of properties. In quantum mechanics, outer space is modeled as the Dirac sea, and in general relativity, outer space is modeled by Einstein's field equations defining spacetime. Free space is not in either of these theories except as an unreachable and unphysical ideal. I was thinking about it today, and it sparked some thoughts about proof by contradiction tolerance.

[idlnmclean]

Twitter discussion with @ExiledSurfer about paraconsistency

exiledsurfer: I think you & I are in agreement. All methods describing our existence are belief-based attempts at describing the same phenomena.

Ian Mclean: The argument I want to make is that ANY argument but the most trivial arguments will lead to errors of reasoning. Apply Occam's Razor to logical method & take Vlatko's challenge to it: no presumptions.

exiledsurfer: You wanna call this combination 'parasimplicity' then?

Ian Mclean: The best I can come up with for that challenge is an axiom schema that asserts all axioms are partially members.

exiledsurfer: are then all members partial axioms as well?

Ian Mclean: That's one of the things I'm confused about. It seems that all theorems become partially axioms & likewise.

exiledsurfer: Then lets look for a definition of partiality which codifies it.

Ian Mclean: I've recently been thinking this fuzzy-complex form of the Empty axiom schema reflects as Russel's set. Or rather a paraconsistent form of Russel's set. Asking how empty a set is asking, "To what degree is it full?"

exiledsurfer: and again, at what holographic scale of resolution.

Ian Mclean: To my mind, our minds are reflections of the universe(s) we live in. Lower resolution holograms w/in a superhologram.

exiledsurfer: Agreed, and to mine, vice-versa. Chicken-egg is what we seek to unravel - aka prime motion, no?

exiledsurfer: The connection between program-size complexity & algorithmic probability: bit.ly/IlCv9Q Could you adapt this to reflect?

Ian Mclean: The first question is in what way?

exiledsurfer: without presumptions. possible? Is Occam's Razor not a presumption in and of itself?

Ian Mclean: the trick is to nest the presumption in a metasystem: "There are no presumptions in the object system". This is formally equivalent to using the axiom of the Empty set as an axiom schema.

exiledsurfer: From nothing springs everything. I like this - axioms as experimental variables rather than presumptions.

Ian Mclean: Normally, the set would either be Empty or not-Empty; however, fuzzy-complex membership means both to a degree.

exiledsurfer: back to holographic resolution - which becomes a super set of Empty Sets, no?

Ian Mclean: I'm uncertain of how to proceed on that front. I'm not sure if what I end up w/ is simply a CFV universal set or multiset. I've spent time thinking of it as a generalized Empty set w/ members of various non-zero degrees. I've deep concerns about the definition of vacuous truth and circular reasoning. And that would be recursive which is seemingly different from circular reasoning.

exiledsurfer: wait, i mean holographic resolution is a superset of partially empty & partially full sets, the "projectors" measure their states.

Ian Mclean: I think of holographic resolution as being the cardinality of the set. How much information is available to project?

exiledsurfer: granted, agree; you state it more formally than my intuitive grasp of where we are headed.

Ian Mclean: The biggest problem I've come across is the problem of infinity. Something has to change. What is an infinitely Empty set? Comparatively, what is an absolutely unempty set? Universal set? The thing I can not escape in any of this is it all leads to taking paradoxes as theroems and axioms to some degree.

exiledsurfer: para-paradoxicality and para-axiomacy can be created and defined as well. why. the. fuck. not. This has bothered me since i was first introduced to sets as a child, and got a ruler across the back of my hand for asking.

Ian Mclean: I loved the idea of sets when I was introduced to them. I especially love recursive sets. Infinitely recursive sets. What I didn't like is what happened next. Along comes transcendental numbers & non-constructable sets.

exiledsurfer: when we first started talking, i introduced you to Hofstaedter's I am a strange loop, i forget if you bit into it or not...

Ian Mclean: I got a copy of it that I've skimmed, but I've been focused on Popper Selections and the Undecidable.

exiledsurfer: Because we are both looking for a understanding of a model of infinity from a finite point of view, no?

Ian Mclean: If the critics of inconsistency and dialethia were wrong about the necessity of explosion in logic, what else? What if we accept paradox face on with some additional caveats? What if we drop the assumption of contradiction?

exiledsurfer: Doing so would certainly prove the simultaneous existence / non existence of god,

Ian Mclean: Prove and refute in a sense. Degrees of divinity.

exiledsurfer: holographic scales of divine resolution.

Ian Mclean:: From sound of 1-hand clapping or of a tree falling in an indeterminate forest to the bite of an apple or a drink of water. Would it be bad form to call this method "grokking"?

exiledsurfer: absolutely NOT. maybe even the best form.

Ian Mclean: In the long run, I think I'm going to have fun with the syntax and naming. Zen mechanics and grokking in a multiverse. The proof of non-countable numbers might become a proof of paraconsistent recursive numbers. Numbers of paradoxical recursion.

exiledsurfer: This is good. i like this. i like it a lot.

Ian Mclean: I wonder what a bootstrap set might look like in terms of mathematical properties. It has no discernible origin.

exiledsurfer: We can also postulate, and accept as a given that there is no need for a point of origin, or of discernability.

Ian Mclean: I wonder what the proof might look like. You start building this system & find that you can't find the beginning. I don't think it would be as startling to find a proof that you could find multiple origins for an event or object. The thing to realize, I think, is the resulting object, U, generally would include the case where ∃U∀axiom(axiom∉U). That case is when axioms are asserted with 0 degrees. Zero point. Suppose axiom = U then Russel's set. So we assume Occam's razor at the metasystem level. What I think of as the bootstrap logic.

exiledsurfer: Assuming this about quantum events is probably a good thing, a correct thing, intuitively.

Ian Mclean: My current view of time is not as a line but as lighting clouds. Phase spaces of possible states with decisions through them.

exiledsurfer: I feel like a midwife at the birth of an idea. This explanation of time is sublimely brilliant, creative, descriptive. wonderful.

Ian Mclean: I've been thinking about negligible error and infinity. Suppose we have an error rate approaching 0. Suppose error ⊕ 0. If we begin speculating about infinite processes, our error rate should begin approaching ¬0 somewhere near ∞. If our logic is explosive, wouldn't that mean inconsistency is unavoidable near or at that point? And what reason do we have to believe that (¬0)⊕(∞)? If the erroneous process is an infinite number of iterations then, in what sense is the error less than infinite?

exiledsurfer: AND, we are both perceptually aware that finiteness and infiniteness are one and the same - but how do we demonstrate it?

Ian Mclean: Why do they exist? Because inconsistencies must not exist lest total contradiction entail. The existence of paraconsistent logics contradict this argument.

sdv_duras: There is no reason to consider it not deterministically. after all dont the laws of thermodynamics apply ...?

Ian Mclean: The determinacy of thermodynamics is an open question.

sdv_duras: do you really believe that the laws of thermodynamics are an open question?

Ian Mclean: I believe whether they are deterministic or not is an open question. Quantum Mechanics & particle physics pose strong arguments for indeterminancy.

David Blackberry:so how do we model indeterminacy in such a way that it recreates the present univ?

Ian Mclean: We might also see it as consistent vs paraconsistent indeterminancy. For perspective, consistently applies locally, hypothetically. Hypothetically, paraconsistency applies to questions beyond event horizons. I'd say we model a universe in which it is possible to perceive (p&¬p). Such a universe would accept degrees of contradiction in language. Keeping in mind that we're talking about abstract language. Language of physical signs of which verbal language is particular. Physical language is a reflection of an abstract language. There are non-physical signs which are meaningful in the language. At least meaningful to the language. At the moment, I'm structuring it in terms of the major hypotheses and trying to represent dependencies by nesting hypotheses. Discovering and noting the dependencies of the hypotheses wherever possible. Noting current "known" theoretical properties. To my mind, this goes back to the arguments I was making about paraconsistence and problems with consistency. The argument strikes at the root of formal scientific and deductive methods, so the argument needs to be sound.

exiledsurfer: which means we need to "formalize" descriptions of intuition inside of expected (accepted - consensual) existing paradigms.

Ian Mclean: If we are to attack the origins the problem, I would prefer to attack it with the problematic methodology. Fold the argument in on itself and show that it leads to what it says it must exclude and include that which it must not.

exiledsurfer: what a gödelesque approach. We must also say, "there is no 'problem', only a state (or multiple states) to be described.

exiledsurfer: the inconsistency being the quantum-relativity gap. Which projector or set of projectors can describe the state paraconsistently?

Ian Mclean: The inconsistency being the point at which the continuous becomes discontinuous in the logic of extremes. A singularity. My guess is that a singularity is an extremely distorted reflection of a paraconsistent computer in itself.

exiledsurfer: yet that inconsistency becomes necessary to describe the intersection of continuity/discontinuity, and therefore consistent.

Ian Mclean: Paraconsistency ensures consistency in some limit.

exiledsurfer: as it ensures inconsistency - but is there a limit to the inconsistency it insures, or is the inconsistency unlimited?

Ian Mclean: That is an exceedingly good question. Preliminarily, I expect that total inconsistency is still excluded. Which amounts to saying that inconsistency is limited to partial or unique inconsistency.

exiledsurfer: Then to what is there 'no limitation' - is paraconsistency 'unlimited'? Why accept any limitations to states?

Ian Mclean: Paraconsistency though may lead to an unbound multiverse. So far, I've only seen Zizzi's renormalized paraconsistent logic. She finitely bounds her system and forbids the object system from capturing the metasystem to maintain distinctness. The next part in the argument about TOE that I want to make is that in principle, no system is more powerful than the TOE.

exiledsurfer: Can you be sure that the TOE is not a subset of Everything and nothing, or just an expression of one potentiality?

Ian Mclean: If TOE describes everything & nothing and only everything & nothing then I would expect it could have no other non-trivial metasystem. So at this point in time, I can only speak from the consistent view point. Paraconsistence changes the argument. Once again though, the point is to make the argument from the edge of consistency. Show that if ∃TOE ⊢Meta(TOE) and Meta(TOE)⊢TOE.

exiledsurfer: Again, this makes distortion a natural and required state - if it exists, it MUST exist, and cannot not exist.

Ian Mclean: And more importantly, it needs to be as complete as possible.

[idlnmclean]

Dialethia 3: Methodological Problems

[idlnmclean]

Dialethia 2: the Liar's Paradox