1

What I mean by this, is something which is exactly opposite to say both positive and negative, just as much as the two are in opposition to each other.

This idea arose and evolved when I was thinking of axes. Now, as many of you may know, axes aren't in opposite directions to each other, but perpendicular (90 degrees to each other instead of 180), but somehow my brain thought they were for a split second, and since there's three axes... Well, my mind gave birth to the idea of 3 opposing forces.

Then I got to thinking of how opposites always come in pairs, obviously something can't be its own opposite, and we know of things like pos vs neg, right vs left, good vs evil etc, but could you have 3 things/idea where all three are in exact opposition to the other two?

3

In Boolean Logic, the answer is no. For any statement (A) in a set of possible statements (or propositions), there is exactly one unique opposite statement (¬A).

There can be multiple distinct propositions B,C,D,....such that A ∧ B = FALSE, A ∧ C = FALSE, A ∧ D = FALSE. And there can be multiple distinct propositions I,J,K,...such that A ∨ I = TRUE, A ∨ J = TRUE, A ∨ K = TRUE,...

But there can be only one unique proposition that satisfies the conditions:

  • A ∧ ¬A = FALSE
  • A ∨ ¬A = TRUE.
  • ¬¬A = A

This is because the rules of Boolean Logic lead to a mathematical structure called an orthocomplemented lattice.

Consider the set of days of the week:

The proposition "Today is Monday" is not the opposite of "Today is Tuesday" because even though the statement ["Today is Monday" and "Today is Tuesday"] is always False, the statement ["Today is Monday" or "Today is Tuesday"] is not always True.

The proposition "Today is Tuesday" has one unique opposite proposition which is "Today is either Monday, or Wednesday, or Thursday, or Friday or Saturday, or Sunday".

However, in other forms of logic, such as some - but not all - Fuzzy Logics, or Quantum Logics, it is possible for a single proposition to have more than one opposite. This is because such logics lead to different structures which are described by orthomodular lattices instead of orthocompelmented lattices.

In particular, in Quantum Logic, the reason why one proposition can have more than one opposite or complement is because just as in your initial example, statements are represented as vectors in multiple dimensional Hilbert Spaces, instead of being represented as subsets of a Boolean Algebra.

0

Let us say that two things are in opposition if presence of one excludes the presence of the other.

This is easy to generalize to several things: The presence of one excludes the presence of all others.

If we want to make this more quantitative, we might say that two things (numerically measured in some vector space such as the real line) are in opposition if the sum of their measurements is constant, say zero. This is also easy to generalize: several things are directly opposed if the sum of their measures is constant, say zero.

Coordinate axes are not in direct opposition, since you can have a point such as (1,1) (or i+j, if you prefer that notation).

Three things that oppose each other are often drawn as a triangle in which you can select a point and then see how close or far away it is from the edges or the corners. This corresponds to ability to pick any two of the three, but is conceptually similar to a trichotomy.

An example where you can select two out of three: http://en.wikipedia.org/wiki/Project_management_triangle#.22Pick_any_two.22

0

I don't know if it really answers your question but in physic there is something that might illustrate this.

When you look at electromagnetic charge, it can be either positive or negative, so you have some kind of direct opposition. You have two possibilities, and when you add an electric charge to an opposite charge of the same intensity you are left with an electrically neutral object (like an neutral atom made of electrons and protons of charge -e and +e).

But when you look at strong interaction, the quarks can carry three different charges, and they are called color charges (it's not like real life color, but it's a good analogy). In a proton for example, you have three quarks, and they have three different colors : blue, red and green, and the result is a "white" proton, which means that the proton is neutral in regards of the strong interaction. So we could say that the opposite of one color charge is the sum of the other two. The cool part is that this is how every proton and neutron are made, so this example is everywhere around us. Actually, you have also anticolor. For example, pions are light particles made of only one quark and one antiquark, and they can carry for examples red-antired or blue-antiblue, so that, again, the result is neutral particle.

The assumption here is that the opposite is defined by : the summation over opposite charges is zero. And I feel sorry that all the answers are either about math or physics.

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