Why do we use only true and false? It is possible to have many states in-between in fuzzy logic and other many-valued logics.
If we assign numbers to true and false, such as 1 and 0 respectively, what would be the logical interpreation of -1, i, j or k (with i,j,k as defined for quaternions)? Is there any reason for this dichotomy? What type of statement would have these truth values if such a statement existed?
The question whether we could have a logical system that can be represented with complex numbers raise an interesting point: Are the logical systems where multiple dimensions are useful?
The answer turns out to be yes. Consider the multidimensional logic of Carlos Gershenson.
Here, each logical variable is a pair from the 'square' [0,1] x [0,1]. The reason that a 2-dimensional representation is chosen is such that we can assign a truth value to even paradoxical statements such as "This phrase is false." The basic idea is that if for the pair (x,y) we have x+y=1, then this is considered a non-paradoxical value within fuzzy logic1. Otherwise, the truth value is paradoxical, but can still be represented and computed with. (for more information, see the link provided)
But let me answer your actual question. One of the main reasons that most of mathematics uses a two-valued logical system is that most of mathematics is concerned with proving something either true or false. Nothing else. Hence, as mathematicians only wish to speak about two logical values for their statements, a two-valued logical system is the simplest system that allows them to do that.
1: Here we see a parallel with the 'imaginary numbers', they were introduced in Cardano's formula as an 'algebraic trick' to have some 'nonsense' in the middle of a derivation, but a correct result at the end)
Actually in some logics, specifically in continuous model theory, we can consider the interval [0,1] instead of the usual proposition set {0,1} and take 0 as indicating the truth value and 1 as the false value because sup[0,1]=1 and inf[0,1]=0. Also, while there are many complex number values there is only one number whose square is -1, and that is i.
The reason for this dichotomy is because logic is normally looked at algebraically instead of geometrically, thus forcing us to consider how to construct truth-value systems.
As for what type of statement would have truth value of i and negative one, well it would definitely have to be mathematical formalizations of some type of dialectical logic that relies heavily upon idempotents(mathematical objects whose iterations equals itself) to build its truth-value system. I do not know about the specifics since such a system is not yet known/proved to exist yet.
The domain of stochastics relies on generalization the two discrete truth-values 0 and 1 to the continuous interval [0,1] of probabilities, i.e. all real numbers between 0 and 1 are possibile probabilities. Choosing "0" and "1" as two distinguished truth values is a suitable convention - remember the dual system in computing. Probabilities have to satisfy certain axioms, e.g. for disjoint sets A and B of events
p(A union B)= p(A) + p(B)
Therefore one cannot choose quite arbitrary numbers for probabilities and truth values.
Clearly, there isn't.
There are certainly theories that would admit a -1, or the range of integers, or an interval of reals, or an infinite vector space (the space of state matrices in quantum physics) as proper representations of some logical state.
But logic seeks a basis for thought. It is looking for what can be seen as most basic. And for most humans, that is a binary comparison.
Within that Boolean context, what behavior could i have? First, you would have to decide how the math maps. In Boole, addition means 'or' and multiplication means 'and'. So in that world -1 = 1. A truth value of i or -i then would have to be 'alternate units' in the algebraic sense, two things that are neither true nor false separately, but when both apply, they establish a true statement.
Then instead of there being exactly three such things, there would really be an infinity of them, and they may not be very useful. But they might be fun to contrive.
It seems that all other forms of logic (e.g. with 2+ values), should you ever really need them, can be simulated with ordinary mathematics based on good old-fashioned true-or-false logic.
Perhaps the simplest way to answer the question here is to consider the nature of mathematical language and what it is meant to represent. Mathematical language is largely characterized by the absence of the typical phenomena that usually motivate the introduction of many-valued systems. It is meant to be non-vague, to contain only referring terms, to describe eternal/non-temporal truths, etc. In response to some previous proposals/comments, I suppose one should generally maintain a clear distinction between the value of a variable meant to represent some physical or non-physical quantifiable property and the "truth-value" of the corresponding statement ascribing to some object(s) the relevant quantifiable property. Not everyone agrees. The vagueness literature, for example, contains lines of argument that precisely fail to respect that distinction categorically. Some authors' thinking moves from considerations about the degrees to which objects can possess a given property to conclusions about the degrees of truth of the corresponding statements. That is one way to motivate degree-theoretic treatments of vague predicates. I've never been convinced it makes sense to do that.
