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?

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    Welcome to Philosophy SE. This question is not completely clear in what you're asking and could benefit from some clarification. The simple answer is that true and false have specific meanings in certain contexts which would not admit other concepts; remember that mathematics and philosophy both add value to human thought primarily through what they disallow, not what they allow. Order is in effect the containment of thought to specific concepts deemed to be valid as a model within the universe that also seems to constrain events to a given valid set we interpret as order. – Tim B II Apr 9 '18 at 3:50
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    Maybe not; see Three-valued logic and Many-Valued Logic. – Mauro ALLEGRANZA Apr 9 '18 at 5:53
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    The ides that there are TRUE statement is very deeply grounded in our language, and also bivalence, i.e. the fact that what is not TRUE is FALSE. But, at the same time, the Vagueness phenomenon is widely present in our language an daily life, and this does not fit well with bivalence. – Mauro ALLEGRANZA Apr 9 '18 at 6:43
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    0 and 1 in logic are not numbers but Boolean values. – Mauro ALLEGRANZA Apr 9 '18 at 6:44
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    @Discretelizard I actually don't disagree with you about it being preferable in many cases to have bivalent logic; what I was saying above is that the title refers to there only being true and false, and then the body of the question not only shows that not to be the case, but then wants to know why the former is the case again, while asking if the latter can be a good idea. The question is actually a good one, but is written (IMHO) in a confusing manner. Your answer to it reflects a good understanding of the intent, but that doesn't mean the question's wording can't be improved. – Tim B II Apr 9 '18 at 11:32

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.

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  • What's the square of -i? – H Walters Apr 9 '18 at 4:46
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    (-i)^2=(-1*i)^2=(-1)^2*(i)^2=1*(i)^2=i^2=-1. The square of -i is also -1. – Jo Wehler Apr 9 '18 at 6:41

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)

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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.

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    I'm not sure if stochastics is that relevant here. Even in stochastics, each event either occurs or doesn't. There are still only 2 truth values. The only thing that is on an interval is our uncertainty of the event. Also, your axiom is wrong. Not only do the events need to be disjoint, they also need to independent. – Discrete lizard Apr 9 '18 at 9:49
  • Thanks for pointing out the wrong axiom. - I consider probability a generalization of the two discrete truth values. Probability does not only capture our uncertainty of the event. In quantum mechanics probability often captures a property inherent to the event, independent from our knowledge. – Jo Wehler Apr 9 '18 at 10:00
  • "Probability does not only capture our uncertainty of the event". Nevertheless, there are only 2 truth values in mainstream stochastics. I don't think anyone's opinion on what stochastics should be matters. Therefore, I think this answer is misleading at best. – Discrete lizard Apr 9 '18 at 10:09
  • What's wrong in your opinion with considering probability a generalization(!) of the two discrete truth values? – Jo Wehler Apr 9 '18 at 10:52
  • 1) This contradicts the interpretation(s) in mainstream stochastics. 2) The concept of a (probability) measure is a very different mathematical object than a logical variable or operations on such variables. 3) The mainstream generalisation you want is known as 'fuzzy logic', see here for the differences between this and probability. – Discrete lizard Apr 9 '18 at 11:12

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.

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    I agree about equivalence. But suggest a different mapping for i; as something intrinsically contradictory but set in an undetermined way to take on one of the other truth values. This is by analogy with en.m.wikipedia.org/wiki/Tachyonic_field As you say, just contrivances, and this suggestion is just for fun. – CriglCragl Apr 10 '18 at 11:48
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    @CriglCragl But something intrinsically contradictory should not have a modulus of 1. Because something with modulus 1 can never be zero. It needs to be somehow incomplete unless it is squared. You could follow down Re(i) == 0, and have orthogonal truths. But someone else already commented on multidimensional interpretations, and I didn't care for it. – user9166 Apr 10 '18 at 16:09
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    @CriglCragl You may be interested in the multidimensional logic I refer to in my answer, as that assigns logical value to paradoxical statement. A value is considered 'non-paradoxical' only if the norm (L1 norm, not Euclidean) is 1 and hence can be seen as a fuzzy logic value. – Discrete lizard Apr 10 '18 at 21:20

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.

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    This is a rather vague claim. I'm not sure if it is completely true. Even if it holds, the fact that we can 'simulate' other logics (whatever that means) doesn't mean other logics are useless. Sometimes, other logics are simply an easier way to describe or solve certain problems. This is one of the reasons why fuzzy logic has uses in practice. – Discrete lizard Apr 10 '18 at 7:17
  • Suppose you want to simulate 256-value logic. Then each "logical" operator could be represented by binary function on the the set of natural numbers less than 256. – Dan Christensen Apr 10 '18 at 12:55
  • Yes, but why would you? – Discrete lizard Apr 10 '18 at 17:35
  • @Discretelizard I can't think of any applications off hand, but I understand that 8-value logic can provide some processing efficiencies in computer chips. Of course, the same functionality could be made available using standard 2-value logic. – Dan Christensen Apr 10 '18 at 19:40
  • Perhaps I should be clearer. Why would you simulate some multi-valued logic with some numbers? Why not just use the multi-valued logic immediately. – Discrete lizard Apr 10 '18 at 21:16

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