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Imagine a universe where 1+1=3. This contradiction would trigger the effects of the principle of explosion, and thus, literally everything (possible and impossible things) could happen. If we lived in such world, we could prove anything (true) even if it were impossible...but we would be able to prove also the contrary.

I've talked with a scientist about these topics. I've asked him how we could get a universe where ONLY impossible things happen (i.e., we couldn't prove that possible things are true/exist in that world) by containing some of the effects of the principle of explosion. Or a universe where literally all impossible things and some/lots of/the majority of possible things happen/are true as well. But he told me that he thought that this was not possible, that the only way to do that was to have a universe where literally everything (literally all possible and impossible things) would happen. So we have the initial problem again.

So, given that he is not sure, is there any way to get such worlds?

PS: I'm not asking to get these worlds physically. I'm not saying that logical/illogical systems/things and the principle of explosion would make these worlds real. I'm just asking how could we get these worlds theoretically speaking.

closed as unclear what you're asking by Conifold, Eliran, Philip Klöcking Oct 10 '18 at 10:32

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    Possibility is something that cannot happen. What happens, can happen. The answer is "no way". The world you describe is not where impossible things happen. It is the world where impossible for our world things happen. So, I suspect you actually ask "Hoe could we get a world where only impossible for our world things happen?" The answer is not obvious. In fact, you need to get rid of logic. And it's very-very hard for a mind, maybe impossible. – rus9384 Oct 8 '18 at 12:36
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    Aren't we arguably in such a world already? There's no plausible or coherent explanation for anything at all. Once you drill down the cause of an event to the cause of the cause etc. you end up at "I don't know." Always. We live in an impossible universe. – user4894 Oct 8 '18 at 16:43
  • @rus9384. As I said in my question, I'm not asking how could that universe exist. I don't care if it is really impossible and cannot exist. If we assume 1+1=3 and we let the principle of explosion to be realized completely, it would "produce" (in a theoretical sense of course) a universe where literally everything (and truly impossible and possible things, not only relative to our universe) would exist. Thus, when you say "no way", in such universe since literally everything would be true (and false)... – Sue K Dccia Oct 8 '18 at 19:04
  • @rus9384 we could prove that you were wrong and actually it would happen (literally everything, and truly possible/impossible things would happen). We could also prove of course the contrary. So that is what I'm referring to with these universes. So my final question would be: How can we get (theoretically speaking) a universe where only impossible things (not possible and impossible things at the same time) would exist? How can we get a universe where impossible and some/lot of/the majority of possible things would exist (and vice versa)? – Sue K Dccia Oct 8 '18 at 19:04
  • @rus9384 Could we theoretically achieve it by containing certain of the effects of 1+1=3 contradiction (and thus, instead making all true and false at the same time, making certain things false/nonexistent; for example making false that literally all possible things would happen and making true that some/a lot of/the majority of them would be true/existing/happening)? – Sue K Dccia Oct 8 '18 at 19:05

10 Answers 10

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A very interesting question. I think that you are not committed to the existence of both the worlds, because you are looking at the impossible world (IW) from the perspective of the possible one (PW).

1+1=3 trigger the principle of explosion if our logic works, but it doesn't if logic is different as it should be in the IW. From the point of view of the inhabitants of the IW, our impossible is the possible, 1+1=3 is fine and it doesn't not require that 1+1=2.

An impossible world is first of all a different world, with by definition a very different logic at work. This logic has not to be symmetric to ours, it could be so alien that from here we can't simply judge it.

We could object that in the IW the principle of non-contradiction doesn't work, so we have true contradictions and we get the explosion. But this is just one IW: there could be, for example, an IW where the principle of non-contradiction doesn't work and also doesn't work the principle of explosion... sounds strange? Of course, it's impossible.

  • What is "a world where only impossible things happen"? Can we just take our world and reverse the truth values on all sentences? Everything true will be false and vice versa, we won't even have any contradictions. – Conifold Oct 10 '18 at 0:14
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    @Conifold In our world the sentence "this sentence is true" makes sense. Reversing it in another world we should admit that in that world the sentence "this sentence is false" is a meaningful (non-paradoxical) sentence. It is not that simple as you suggest. – rus9384 Oct 10 '18 at 5:46
  • we could even reverse the LNC itself, and we’ll have contradictions... but maybe these comments would be more useful attached to the question – Francesco D'Isa Oct 10 '18 at 5:50
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    In fact, this answer is beautiful as it shows that Explosion can't be applied to any world. This is what I argued: it is meaningless to call any statement (alone or within context) contradictive because laws of logic themselves might not work. – rus9384 Oct 10 '18 at 6:39
  • A statement is "contradictive" if it is admitted as both true and false, the laws of logic are irrelevant. In fact, if we reverse all the truth values the laws of classical logic will not hold (assuming they did originally), but there will be no contradictions (if there were none originally). – Conifold Oct 10 '18 at 17:46
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One serious possibility for understanding entropy and the arrow of time, given that all events in quantum field theory (QFT) seem to be reversible (equally workable forwards and backwards in time, with directly equivalent particles), is that time ordering is a property of minds, rather than the world. The nature of time is a major, maybe the major, source of tension between QFT & general relativity.

Causality, as considered in the -unresolved- Problem Of Induction, may not reside so much in the world, as in our making sense of it.

"Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." - Alice In Wonderland

We can imagine a world which does not make sense, but even in doing so we seek ordering principles, like Alice. Dodgson was satirising the 'new mathematics' of his time, with apparently impossible things like imaginary numbers.

Jung suggested that causality is a way of grouping phenomena, of finding patterns. And posited synchronicity as an alternative principle, linked to ideas like the astrological 'as above so below'.

We like to imagine meaning, causality and so on, as external to us. An alternative view, is that meaning is like the turbulence a fishes fins make in water, that we makd in the substance of our impressions of the world. In this view, we cannot but help make meaning when we navigate. To not do so, would be to stay stationary in our thoughts. But this picture draws attention to our freedom, our choices, in how we make sense of the world.

Carlos Castenada talks about the realm of the https://en.m.wikipedia.org/wiki/Nagual The non-conceptual realm, between the islands of meaning in which our interactions occur. It is a realm of 'impossibilities', but also one in which action and intention are still possible, but without many of the assumptions which in every-day mind obscure the world as it is in some ways.

Mathematics is a language. 1+1=3 is just redefining words, and doing so will be subject to a judgement of how good the new system of definitions is. Defining the complex number line is much more subversive, and in some ways was more unsettling to people when it was proposed. Whatever languages we have, whatever systems and organising principles, the world exists out there beyond them. And unbound by them. We can, we must, be able to explore it without the concepts neccessary to understand it. Or we could create no new concepts. Our experiences form fixed points on a larger landscape of the unexperienced, like the proofs of maths do, or the experimental results of science, and the unproven and unevudenced beyond. Outside of those, we can find new places to stand, which reinterpret and simplify those, explain them with deeper and more universal insights, which we call knowledge. The creative space of hypothesis generation can include far more subversive ideas than 1+1=3. To reach new high points on the landscape of ideas, we may need to cross valleys of chaos and impossibility. And we do.

  • thank you for your answer. Unfortunately I was unable to understand it completely. Could you rephrase or re-explain your point? – Sue K Dccia Oct 8 '18 at 19:41
  • @SueKDccia What we think is possible, involves making a model of the world which simplifies it, and makes it tractable to us. We can interact with the world in a different way, not projecting onto it, though with costs and requiring mental skills. Creativity always does this in some way, in going beyond the known, into what was 'impossible'. That isn't a place for tweaking definitions, but for shaking the foundations of everything. Old knowledge, like a limited idea of who we are, won't dissappear, it is part of history. But it can be radically altered, more so than by redefining 1+1 to = 3 – CriglCragl Oct 8 '18 at 20:26
  • I'm still having problems finding how does this relate to my question. Please don't get offended by that – Sue K Dccia Oct 8 '18 at 21:14
  • @SueKDccia Perhaps: The world is always beyond the dualism of possible and impossible, but our minds and the words we use, must diminish to what we call possible, for our own convenience, or sanity. – CriglCragl Oct 8 '18 at 21:17
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The question seems to be based on the elimination rule for contradiction or explosion.

Imagine a universe where 1+1=3. This contradiction would trigger the effects of the principle of explosion, and thus, literally everything (possible and impossible things) could happen.

Suppose we start with a contradiction, then we can derive as a valid proof any conclusion we want as the following shows:

enter image description here

Here my premise is a contradiction (⊥) on line 1. R could be anything. We could let R be the sentence "1+1=3".

From that contradiction I was able to derive "R" on line 2 using explosion (X), but what does that mean? It does not mean that "R" is true. It only means that the conditional sentence "⊥ → R" is true, not that "R" is true.

As the authors of forall x put it: (pages 119-20)

Still, isn’t it odd that from a contradiction anything whatsoever should follow? Not according to our notion of entailment and validity. For A entails B if there is no valuation of the atomic sentences which makes A true and B false at the same time. Now ⊥ is a contradiction—it is never true, whatever the valuation of the atomic sentences. Since there is no valuation which makes ⊥ true, there of course is also no valuation that makes ⊥ true and B false!

When using explosion it does not mean that B becomes suddenly true. It retains whatever truth value it originally had. What is true is only the conditional.

Consider the question in the title: How could we get a world where only impossible things happen?

We do not get an impossible world using explosion. From explosion we only get that we can validly conclude anything from a contradiction. Some of those things may be false, but the explosion argument does not make them true nor manifest them in some way.


Reference

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

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A few points that didn't get enough attention:

The deductive reasoning invoked by OP has been proven (by Kurt Gödel) to be insufficient to describe our world. Specifically he proved that any Formal system powerful enough (to describe a significant portion of the world) will be either incomplete or contain contradiction. As we have not witnessed a physical manifestation of a contradiction we usually accept that the Formal systems we have devised are incomplete. Hence a postulated contradiction "1+1=3"/(1+1≠2) doesn't tell us much about that world, since we don't, and cannot, have a complete picture from deductive logic for this world. Let alone applying a formal system to a proposed "impossible world".

@FrankHubeny touched on another issue: When a contradiction is proven we can then deduce anything from there, validly. The "principal of explosion" is about the Form of an argument, it says nothing about the Truth of any statments. https://en.wikipedia.org/wiki/Validity_(logic)

So what is "Truth" in the possible world? Is it what everyone agrees it is? Is it what corresponds to what is real? I could go full skeptic and deny the truth of everything I see, then this would be the impossible world. But if I change my theory of truth to something more consensual, I could make sense of the world again. There are many interpretations of Truth and that determines our conceptions of possibility, a theory of truth need to be decided before OP question can be addressed.

I like @CriglCragl answer for this question which I think boils down to: There is interaction / interdependence between 'Mind' and 'Matter', i.e. we need a framework of possibilities /actualities in order to sustain cognition. Thus our cognition may well be unable to grasp the very nature of possible/impossible.

In short there are too many gaps in our understanding (of everything) to form a cogent theory or conception of a rich and complex impossible world. However you can simply specify a "small world", a limited domain of interest where you can adjust the rules as needed.

One more helpful resource in framing this question is the ontology of Alexius Meinong: https://en.wikipedia.org/wiki/Alexius_Meinong

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Unless impossible means the opposite thing in such universe, we already exist in a universe where impossible things happen, every instant, under our eyes. But perception does not allow us to experience impossible things.

In this universe, possible means that tends to exist, and therefore remains along time, and impossible means that dissipate as soon as it exists. If you think on it, this principle is essentially Darwin's natural selection. What is possible persists; what is impossible, dissipates.

First, observe things at a macroscopic scale. What is impossible must occur (at least for an instant), but as said, unless impossible would be equivalent to "that exists forever", impossible things follow the principle of explosion, the lack of coherence, and perish fast. In consequence, it is not possible for our mind to know the impossible, because we only can know what is possible.

If natural selection has ever produced a giraffelephant, it was condemned to perish fast, because it was not possible for it to persist. In the macroscopic scale, it would be difficult to observe that 1+1=3, but that's a problem of our perception, not of physics. "1" represents a thing; but things don't exist as such out there, out of our mind; everything is just atoms interacting. What "1" really means is a bounded abstraction of a spacetime region. If boundaries move without us perceiving them, actually that phenomenon happens here. Remember the vanishing leprechaun? That's not as dramatic as 1+1=3, but that proves that 1+1+1 + 1+1+1 + 1+1+1 = 8 (there are 9 ones) is actually real!. [1]

At the microscopic scale, such phenomenon also occurs, and perhaps 1+1=3 happens more frequently than we think. But, to be coherent, if impossible objects occur, they must quickly dissipate.

Or maybe 1+1=3 happens under our eyes and it is us who don't know how to count. At a microscopic scale, the idea of things is naive (one, "1" thing, is not valid; quantum physics has proven that particles are not objecs). Imagine two particles having attraction have a relativistic energy of 1 Joule each. Now, provide them of 1 additional Joule of kinetic energy within both. Now, permit them to be attracted and form a unit. The system will have 3 Joules of energy. We're just counting incorrectly. Of course, that has not been proven at such scale. The only notion we have of such behavior is the idea of synergy, what id usually related precisely with the idea of 1+1=3.

So, impossible things happen, as macrostates, or microstates. But impossible things can't persist, and possible things persist. Impossible is part of this universe, although it is not able to be known from our rational experience.

[1] https://services.math.duke.edu/~blake/leprechauns/vanishing_leprechaun1.htm

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As has been said by @user4894 there would be a sense in which we already live in the world you describe. Time, space, motion, change, existence and most other ideas do not make sense in metaphysics and on analysis appear to be logically impossible. These things are not actually impossible, of course, but our usual idea of them is such that they are demonstrably impossible. This is what we learn from metaphysics.

Thus if we examine the world-view of most people the world they think they live in is impossible. This is well-known to metaphysicians and it is the main logical argument for the 'advaita' or 'mystical' world-view.

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According to the example stated: the universe where 1+1=3 is true. Let's consider this universe as a space X where [1+1=3]===true is X's reality, And another universe Y where [1+1=3] is false. These two different universes have two different realities. That means what is possible in one world may be impossible in another one or what is true may be false and vice-versa. So let's consider Z as a traveller from Y to X. Z will cross one reality to another and the second one will not be his reality but X's one. All what Z knows about Y is acquired from Y reality and therefore this will not be demonstrable in X since X has its reality. And once Z back on Y, every thing he can tell about X will be impossible. And according to Y's people, Z will have lived the impossible.

Since he could not leave in X and Y on the same time, he lived the impossible and the possible but not the same time. That means the possible and the impossible cannot exist on the same context at same time. The impossible is my limit something that I can outstrip and live it as possible in another context other time.

Thank you, that's my point of view.

  • I'm not asking that. You are assuming the existence of a universe where only impossible thongs would happen. I'm exactly asking how could we get it (theoretically). Besides, even if it cannot exist, theoretically speaking, in a universe where literally everything (possible and impossible things) can happen (produced by a contradiction like 1+1=3 and letting the principle of explosion to be realized completely with all its consequences), what you describe to be impossible, that literally all possible and impossible things would exist simultaneously, would certainly happen. @gratienasimbahwe – Sue K Dccia Oct 8 '18 at 19:08
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If your mind is made of immaterial non-stuff, you might have the ability to create a world of impossible things with your imagination. On second thought, if you bring this world into existence then it will become, and becoming is possible or we wouldn’t exist.

Anyway, there can be no life, movement, stillness, color (including black and white), light, darkness, shapes, or sounds. If some “thing” that’s not really a thing does “move”, you can just say that it’s not movement in any sense of the how we would use the word, because it’s impossible.

I’ve tried to do this before, and I was unsuccessful.

  • but in the impossible worlds I'm talking about, where truly impossible things would exist/happen/be true, literally everything (possible and impossible things) would be right. SO even if that you say it is impossible (like bringing this world into existence, keeping it impossible and allowing us to exist or calling "movement" when a "thing" that is not really a thing "moves"), it would be possible (and impossible at the same time of course) @anonymouswho – Sue K Dccia Oct 9 '18 at 21:34
  • @SueKDccia Okay, you may want to edit the question though because the heading and your discussion with a scientist says “ONLY impossible things happen”. So in this world, does 1+1=3, or does every equation equal all numbers? – anonymouswho Oct 10 '18 at 2:54
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No matter where are you, although the symbols may change, the addition and multiplication tables would be the same. Even if we left everything the same, except for interchanging "2" and "3" as in your example, we would have essentially the same tables. No contradictions, no explosions. Sorry to burst your balloon. :^(

  • Well, we can even change tables. Then that would be just another operation called "multiplication". – rus9384 Oct 9 '18 at 19:19
  • If you change the tables substantially, they would be useless.It would be quite an advantage to have addition and multiplication tables that actually worked. – Dan Christensen Oct 9 '18 at 19:33
  • But the OP does not ask about usage. And they might not work in a parallel universe. Funny note: you say it is connected with usefulness, which implies they are made in order to predict (while many mathematicians tell maths are not tied to the real world). – rus9384 Oct 9 '18 at 21:14
  • @rus9384 I'm fairly certain that culturally advanced, sentient beings in any physical universe will have essentially the same addition and multiplication tables as us. – Dan Christensen Oct 10 '18 at 3:43
  • Isn't this faulty generalization? I am sure, it is. There are things which cannot even be imagined (by us), let alone analyzed (by us). – rus9384 Oct 10 '18 at 5:37
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A world where for example 1+1=3 would be correct, would not be an "impossible world". for example lets say that we would be in the world where instead of 1+1=2, 1+1=3 is true, it would no longer be "impossible" what we perceive as "impossible" in our own world would no longer "impossible".

If we take a bit of a closer look on what "Impossible" means in essence, "impossible" is usually referred as one of 3 different definitions: 1.

not able to occur, exist, or be done.

2.

very difficult to deal with.

3.

(of a person) very unreasonable.

the first one being the most common one.

Now lets presume we are talking the first definition of impossible here "not able to occur, exist, or be done.

If it can happen in the parallel world we are talking about here, the "impossible things" that happen could no longer be considered as "impossible" for they have happened which contradicts the meaning of impossible.

So basically a world where "Impossible things happen" is not a very possible world for it contradicts itself and is not "valid" of a world even as a parallel world.

  • and why a world where 1+1=3 is true would not be an impossible world, if that can never happen, in any universe? @Veraen – Sue K Dccia Oct 10 '18 at 8:31
  • Because if it happens somewhere it makes it possible @SueKDccia – Veraen Oct 10 '18 at 8:38
  • then all impossible can be converted into possible? @Veraen – Sue K Dccia Oct 10 '18 at 8:41
  • technically it is possible yes, but here we take into consideration that it would be a parallel world, where "impossible things happen" but if they happen there they would no longer be impossible things but possible things, but we don't live in a world that has "impossible things" happening so technically without taking parallel worlds into consideration there can be impossible things but not in a world where impossible things happen. @SueKDccia – Veraen Oct 10 '18 at 8:48

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