# Two positions at once is logically impossible

Assume that the position of an object is x1 and x2 at once, where x1 does not equal x2.

Let's define the statements A and B as:

• A: The position of the object is x1.

• B: The position of the object is x2.

Then, from law of non-contradiction, this is impossible because A is A and A is not A at the same time. So one can't be at two places at once because it is logical necessary. Is this true?

I don't want to involve quantum mechanics. I just want an answer from a philosophical (Logic) point of view.

• If you are referring to the famous wave & particle distinction then you may have to surf many answers and points of view. There are several views or contexts because there is not enough information besides what you give. The immediate issue with the object is ARE WE SURE this object did not transform into something else such it was a particle & it became a wave. In such a case we have TWO OBJECTS not one. So object 1 is different if it transforms into object 2 & THEREFORE no issue arises being 2 objects are at 2 different locations. It is not the case the SAME OBJECT is at two locations. – Logikal Feb 26 '20 at 18:14
• The law of non contradiction expresses something contextually different outside philosophy. In philosophy the concept means nothing can be A & NOT A in the SAME RESPECT, SAME CONTEXT, & THE SAME LOCATION all at the same time. That is if you are claiming object Z is at two or more locations simultaneously then you must prove Z IS IDENTICAL in both places.That is if I say Dog Fido is in two distinct locations simultaneously I need to show the dog in both place is the same Fido & not another Fido dog. No switcharoos please.Chances are there is a difference in quality, content, time, place, etc. – Logikal Feb 26 '20 at 18:23
• No, it is not logically necessary, and not because of quantum mechanics. You need a premise of the sort: if Pos(x) ≠ Pos(y) then x ≠ y. This is a non-logical axiom describing a property of position-of-object, so whatever necessity is involved it will be weaker than logical. Indeed, we can easily conceive of objects that are at multiple places at once, such as abstractions. The shape of a circle is at any place where there is a circular thing, yet it is a single shape at all those places at once. So the above axiom probably only applies to physical objects. – Conifold Feb 27 '20 at 11:38
• If some of the answers below satisfies you, please accept it. – Mauro ALLEGRANZA Sep 7 '20 at 14:16

You are already presupposing that an object can have only one position at once by using the phrase "the position of". You could have said instead:

The positions of the object are x1 and x2.

There would be no contradiction then.

A mere logical analysis will not tell you whether an object can have more than one position. Especially not if you presuppose it from the start. It all comes down to what you mean by 'position', and I don't think it has a clear answer, at least not given our everyday use of this concept.

Suppose I am standing with my feet apart. What is my position? Is it where my right foot is? Where my left foot is? Is it the area of both combined? Is it the center of my body? Maybe it is the entire area I am occupying in space? In that case, if a position is a point in space, then I have many -- infinitely many -- positions. For other purposes a position may be defined differently, and then I might have just one position. Like many problems in philosophy there could be a lot of confusion if you don't first clarify the concept you are talking about.

• The idea of object x being in two distinct places simultaneously is what the original proposition expressed. You happen to use the "I am standing on the borderline or multiple places, so where am I" counter. If there are clear distinct boundaries that are so specific that there are no gray areas the proposition can be answered objectively. That objective answer can't have multiple solutions. Only one objective answer. Remove the gray borderline examples & you have an undoubted answer. Make the points as far away as possible & be at two locations simultaneously. It is hard to pull off. – Logikal Feb 28 '20 at 13:29
• @Logikal Of course if you remove the cases where there is more than one answer or the answer is unclear, then there will only be one clear answer. But how can you justify ignoring those cases, when they are so numerous? – user253751 Feb 28 '20 at 18:30
• @user253751, my point was that with more specific details the ambiguity disappears and then I can ignore the numerous unclear answers. If we don't have enough specific details the answer we arrive at could be false. We may reason I correctly and not see the error. Therefore part of my point is also that if I do arrive at the wrong conclusion that I can trace where I went wrong and how do o correct the error. Details can make the difference. – Logikal Feb 28 '20 at 19:43
• @Logikal "What position is the 3rd atom in the knuckle of my left big toe?" would be more specific, sure. But that isn't what was asked. (anyway how do you tell which atom is number 3?) – user253751 Feb 28 '20 at 22:46
• @user263751, communication is the key. If you have detailed information then that information should be given. If you know which atom is number 3 then you should give out that information & remove the guess work from other people.You still are with holding information if you give me a lil bit then ask me how would I know or find out something else which you may already know. The question asked does not specify what body parts are in a location. The question seems to lean towards the ENTIRE OBJECT being in more than one distinct location. No feet spread wide & claiming you are in 2 rooms. – Logikal Feb 28 '20 at 23:16

An appropriate formalization (with natural language terminology instead of logic symbols) of your sentences in a language of predicate logic with term equality `=`, a function symbol `pos`, a variable `y` for the object and two variables `x1`, `x2` for the two positions is

``````S1: (pos(y) = x1) and (pos(y) = x2)
S2: not(x1 = x2)
``````

Your usage of "the position" justifies the translation as a function expression that has exactly one value. One could question now whether or not "position" is indeed something that behaves as a function -- this may be more of a question for physics than for philosophy, and if the answer is no, then already the initial wording of the situation would have been inadequate and the conclusion will not be provable with the methods used here. But for the sake of demonstrating from a logical point of view how a semi-formal proof can be carried out with the premises as you worded them, as per your request, let's assume that this formalization is indeed adequate.

The set of premises is not immediately contradictory: To be able apply the law of non-contradiction, which states that `not(P and not P)`, one of the two `and`-ed statements in S1 would have to be the direct negation of the other, so that `S1` has the form `P and not P`; but `pos(y) = x1` and `pos(y) = x2` are just two independent formulas that are not immediately visible as contradictory.

So if we want to prove the argument rigorously, we need a few additional inference steps. The idea will be that if we have equality statements `=`, then we are free to combine any of the terms between the equality symbols, which eventually leads to the observation that if the first assumption is true, we must have `x1 = x2`, which contradicts the second assumption that `x1` and `x2` are different positions.

More precisely, we are exploiting the following two properties of equality:

• symmetry: If `A = B` , then `B = A`.
• transitivity: If `A = B` and `B = C`, then `A = C`.

In addition, we need

• conjunction elimination left: If `A and B`, then `A`.
• conjunction elimination right: If `A and B`, then `B`.
• contradiction introduction: If `A` and `not A`, then `⊥` (= contradiction).
• negation introduction: If `A` leads to `⊥`, then `not A`.

With these rules, we can prove the validity of your argument formally:

``````1. pos(y) = x1 and pos(y) = x2        (assumption S1)
2. pos(y) = x1                        (conjunction elimination left on line 1)
3. x1 = pos(y)                        (symmetry on line 2)
4. pos(y) = x2                        (conjunction elimination right on line 1)
5. x1 = x2                            (transitivity on lines 3 and 4)
6. not (x1 = x2)                      (assumption S2)
7. ⊥                                  (contradiction introduction on lines 5 and 6)
8. not (pos(y) = x1 and pos(y) = x2)  (negation introduction on lines 1 to 7)
``````

Note that the contradiction arises not between the two position assertions, but between `not (x1 = x2)` and `(x1 = x2)`, the latter of which is derived from the assumption `S1`.

If we are convinced that the initial formalization of your situation is adequate, and accept the individual inference rules, then the above reasoning indeed proves that the object can not be at positions x1 and x2 at the same time. The key of the proof lies in the implicit assumption that "position" is indeed a function, in the sense that every object can only have one position, so that we can use `=` and argue about how equality works.

We can now actually go a step further. Up until now, we only proved that the set of premises given, with the concrete positions `x1`, `x2` and object `y` is inconsistent. But we may want to prove something more general:

For any object y and positions x1, x2, if x1 and x2 are different, then y can not be at x1 and x2 at the same time.

To do this, we need two more inference rules:

• conditional introduction: If `A` leads to `B`, then `if A then B`.
• universal introduction: If `A` holds for arbitrary `x`, then `for all x: A`.

In the proof above, the assumption `6. not(x1 = x2)` led to the conclusion `8. not(pox(y) = x1 and pos(y) = x2`. So we can do a conditional introduction and state that `if not(x1 = x2), then not(pox(y) = x1 and pos(y) = x2)`, thereby resolving the dependency on the assumption `6.`:

``````...
9. if not(x1 = x2) then not(pox(y) = x1 and pos(y) = x2)      (conditional intoduction on lines 6 to 8)
``````

At this point, conclusion `9.` does no longer depend on any assumptions about `y`, `x1` and `x2`: The second premise `S2` is discharged because we now said more generally that if `not(x1 = x2)`, then `not (...)`, which no longer depends on whether or not `not(x1 = x2)` is actually true. And the first premise `S1` was discharged in line `8.` where we concluded that this assumption must have been wrong since it led to a contradiction. So `y` and `x1`, `x2` are indeed arbitrary objects and positions, the conclusion does not depend on what exactly these variables look like, so the proof should work for any object and positions. This is why we are allowed to a final universal introduction:

``````...
10. for all y, x1, x2: if not(x1 = x2) then not(pox(y) = x1 and pos(y) = x2)      (3x universal introduction on line 9)
``````

We now formally proved that for all objects `y` and all positions `x1`, `x2`, if `x1` and `x2` are different, then the position of `y` can not be `x1` and `x2` at the same time.

• +1 very nice analysis. I believe in the two places where you write "pos(b) = p2" it should be "pos(a) = p2" before your proof. – Adam Sharpe Feb 26 '20 at 20:54
• @lemontree Thanks for the answer. So can we say that every quantity that can be measured it will be only single valued? – ado sar Feb 26 '20 at 21:07
• @ado sar I'm not sure what you mean by "measurable quantity". The key point of the proof relies on the fact that "position" is a function, in the sense that every object has exactly one position value, if that's what you mean. The same will hold for properties such as weight, length, temperature, ... Of course, not every property is a function, other attributes may allow for more than one value (for example, an object may have more than one owner), in which case the proof will not work -- that depends on the nature of the property measured. – lemontree Feb 26 '20 at 21:48
• @Adam Sharpe Thanks; fixed it. – lemontree Feb 26 '20 at 21:52
• The immediate observation is you have used the same function name twice with two distinct values: namely S1 (pos(y)=x1 and (pos(y) = x2). So if we analyze the CONTENT in reality you are saying pos(y) is x1 and not x1 simultaneously. Whatever pos(y) is in reality then the truth value must be constant or you must admit something has changed at some point. What happens if nothing changed? Well then I will take the right to assume there is MORE THAN ONE pos(y) existing in the real world. Something has to give here. Functions are supposed to be unique I thought. – Logikal Feb 27 '20 at 0:01

Assume that the position of an object is x1 and x2 at once, where x1 does not equal x2.

Yes, this is a contradiction. No physical theory, including quantum mechanics, matters; it's a contradiction regardless.

I just wanted to comment on a common source of confusion..

A person can stand in four different US states at once by going to Four Corners:

The states are Colorado, Utah, Arizona, and New Mexico.

So say Alice is at Four Corners. Here're some example arguments:

1. An incorrect argument comparing individual states Alice is in:

1. state(Alice) = Colorado

2. state(Alice) = Utah

3. Colorado does not equal Utah.

4. Therefore, position(Alice) != position(Alice).

5. The law of identity is contradicted.

2. A correct argument comparing ensembles of states Alice is in:

1. state(Alice) = { Colorado, Utah, Arizona, New Mexico }

2. state(Alice) = { Colorado, Utah, Arizona, New Mexico }

3. { Colorado, Utah, Arizona, New Mexico } equals { Colorado, Utah, Arizona, New Mexico }.

4. Therefore, position(Alice) == position(Alice).

5. The law of identity isn't contradicted.

3. A correct argument comparing individual states Alice is in:

1. state(Alice) = { Colorado, Utah, Arizona, New Mexico }

2. randomElement(1, state(Alice)) = Colorado

• Note: "1" stands for the random seed, to book-keep the context of this random call.
3. state(Alice) = { Colorado, Utah, Arizona, New Mexico }

4. randomElement(2, state(Alice)) = Utah

• Note: "2" stands for the random seed, to book-keep the context of this random call.
5. Colorado does not equal Utah.

6. Therefore, randomElement(1, state(Alice)) != randomElement(2, state(Alice)).

7. The law of identity isn't contradicted.

The fallacy I wanted to point out is that some folks would think that state(Alice) could have different returns; but, it can't if it's supposed to return the states Alice is in. If Alice is in multiple states, then the return shouldn't be one of those states, but all of them.

### Regarding false descriptions of multiple positions in quantum mechanics

Okay, regarding quantum mechanics: one of my pet-peeve fallacies is that folks often think that Quantum Mechanics is somehow so outside of reality that it's even an exception to logical reasoning – which is just plain silly.

I stress this because I want to be clear that, if someone says,

But particles can exist at multiple locations in quantum mechanics!

, this could very easily be mistaken (by the speaker or/and listeners) to mean something incorrect. It's probably best to just disregard such statements as being loose with words.

The very first sentence of the relevant Wikipedia article fixes this misconception:

In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not admit an interpretation in terms of a local realistic theory.

"Quantum nonlocality", Wikipedia

This is, a nonlocal entity isn't a local entity that's duplicated in multiple locations or something else like that; rather, it "do[es] not admit an interpretation in terms of a local realistic theory".

For example, say that in the examples above, the function `state(Alice)` couldn't return an ensemble of states; it had to return only 1 state. Then, when Alice is standing at Four Corners, `state(Alice)` shouldn't return at all; Alice's position doesn't "admit an interpretation in terms of a being-in-a-single-state theory".

Anyway, I rant about this because the question statement started with

Assume that the position of an object is x1 and x2 at once, where x1 does not equal x2.

, then went on to specify that quantum-mechanics isn't on-topic. So I just wanted to be clear that neither quantum-mechanics nor any other physical theory is relevant; this is a logical contradiction, which is beyond the scope of physical theory.

The question is first (implicitly) asked by Zeno, in his attempt to understand the nature of what constitutes change in the phenomenonal realm. Aristotle offered an answer in terms of potentiality and actuality in both his Physics and Meta-Physics, which you might want to look at.